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Dynamical Texture Synthesis to Probe Visual Perception

Dynamical Texture Synthesis to Probe Visual Perception

Talk at Neuromathematics seminar, EITN, Paris

Gabriel Peyré

June 15, 2015
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  1. Dynamical Texture
    Synthesis to Probe
    Visual Perception
    Gabriel Peyré
    www.numerical-tours.com
    Joint work with:
    Jonathan Vacher, Laurent Perrinet, Andrew Meso

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  2. Statistical Image Models for Stimulation
    aac Meso, L. Perrinet, G. Peyr´
    e CEREMADE–UNIC–INT
    obing Visual Perception 22/05/2015 5 / 20
    Psychophysics experiments

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  3. Statistical Image Models for Stimulation
    Context: Electrophysiology and Optical Imaging
    Author: J. Vacher, A. Isaac Meso, L. Perrinet, G. Peyr´
    e CEREMA
    Dynamic Texture for Probing Visual Perception 22/05/2015
    aac Meso, L. Perrinet, G. Peyr´
    e CEREMADE–UNIC–INT
    obing Visual Perception 22/05/2015 5 / 20
    Voltage Sensitive Dye Optical Imaging
    Psychophysics experiments

    View full-size slide

  4. Statistical Image Models for Stimulation
    Context: Electrophysiology and Optical Imaging
    Author: J. Vacher, A. Isaac Meso, L. Perrinet, G. Peyr´
    e CEREMA
    Dynamic Texture for Probing Visual Perception 22/05/2015
    aac Meso, L. Perrinet, G. Peyr´
    e CEREMADE–UNIC–INT
    obing Visual Perception 22/05/2015 5 / 20
    Voltage Sensitive Dye Optical Imaging
    Psychophysics experiments
    Drifting gratings:
    too simple.

    View full-size slide

  5. Statistical Image Models for Stimulation
    Context: Electrophysiology and Optical Imaging
    Author: J. Vacher, A. Isaac Meso, L. Perrinet, G. Peyr´
    e CEREMA
    Dynamic Texture for Probing Visual Perception 22/05/2015
    aac Meso, L. Perrinet, G. Peyr´
    e CEREMADE–UNIC–INT
    obing Visual Perception 22/05/2015 5 / 20
    Voltage Sensitive Dye Optical Imaging
    Psychophysics experiments
    2
    Drifting gratings:
    too simple.
    Natural images:
    too complex.

    View full-size slide

  6. Statistical Image Models for Stimulation
    Context: Electrophysiology and Optical Imaging
    Author: J. Vacher, A. Isaac Meso, L. Perrinet, G. Peyr´
    e CEREMA
    Dynamic Texture for Probing Visual Perception 22/05/2015
    aac Meso, L. Perrinet, G. Peyr´
    e CEREMADE–UNIC–INT
    obing Visual Perception 22/05/2015 5 / 20
    Voltage Sensitive Dye Optical Imaging
    Psychophysics experiments
    2
    Drifting gratings:
    too simple.
    Natural images:
    too complex.
    ! need
    random
    stimuli
    with
    parameterized
    complexity.

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  7. Natural Textures, Stationarity and Scales
    IEEE TRANSACTIONS ON IMAGE PROCESSING
    g. 1. Some examples of micro-textures taken from a single image (water with sand, clouds, sand, waves with water ground, pebbles). The emplacements
    the original textures are displayed with red rectangles. Each micro-texture is displayed together with an outcome of the RPN algorithm to its right. These
    icro-textures are reasonably well emulated by RPN. Homogeneous regions that have lost their geometric details due to distance are often well simulated by

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  8. Overview
    • Gaussian Texture Synthesis by Example
    • Spot Noise Models and Motion Clouds
    • Stochastic PDE Models
    • Bayesian Brain and Prior Estimation

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  9. Exemplar f0
    Texture Synthesis
    Problem: given f0
    , generate f
    “random”
    perceptually “similar”

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  10. analysis
    Probability
    distribution
    µ = µ(p)
    Exemplar f0
    Texture Synthesis
    Problem: given f0
    , generate f
    “random”
    perceptually “similar”

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  11. analysis synthesis
    Probability
    distribution
    µ = µ(p)
    Exemplar f0
    Outputs f µ(p)
    Texture Synthesis
    Problem: given f0
    , generate f
    “random”
    perceptually “similar”

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  12. analysis synthesis
    Probability
    distribution
    µ = µ(p)
    Exemplar f0
    Outputs f µ(p)
    Gaussian models: µ = N(m, ), parameters p = (m, ).
    Texture Synthesis
    Problem: given f0
    , generate f
    “random”
    perceptually “similar”

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  13. Input exemplar:
    d = 1 (grayscale), d = 3 (color)
    N1
    N2
    N3
    Images Videos
    f0
    RN d
    Gaussian Texture Model
    N1
    N2

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  14. Input exemplar:
    d = 1 (grayscale), d = 3 (color)
    N1
    N2
    N3
    Images Videos
    Gaussian model:
    m RN d, RNd Nd
    X µ = N(m, )
    f0
    RN d
    Gaussian Texture Model
    N1
    N2

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  15. Input exemplar:
    d = 1 (grayscale), d = 3 (color)
    N1
    N2
    N3
    Images Videos
    Gaussian model:
    m RN d, RNd Nd
    X µ = N(m, )
    highly under-determined problem.
    Texture analysis: from f0
    RN d, learn (m, ).
    f0
    RN d
    Gaussian Texture Model
    N1
    N2

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  16. Input exemplar:
    d = 1 (grayscale), d = 3 (color)
    N1
    N2
    N3
    Images Videos
    Gaussian model:
    m RN d, RNd Nd
    X µ = N(m, )
    Texture synthesis:
    given (m, ), draw a realization f = X( ).
    highly under-determined problem.
    Factorize = AA (e.g. Cholesky).
    Compute f = m + Aw where w drawn from N(0, Id).
    Texture analysis: from f0
    RN d, learn (m, ).
    f0
    RN d
    Gaussian Texture Model
    N1
    N2

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  17. Stationarity hypothesis: X(· + ) X
    (periodic BC)
    Spot Noise Model [Galerne et al.]

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  18. Stationarity hypothesis: X(· + ) X
    (periodic BC)
    Block-diagonal Fourier covariance:
    ˆ
    y( ) = ˆ( ) ˆ
    f( )
    y = f computed as
    where ˆ
    f( ) =
    x
    f(x)e
    2ix1⇥1
    N1
    + 2ix2⇥2
    N2
    Spot Noise Model [Galerne et al.]

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  19. Stationarity hypothesis: X(· + ) X
    (periodic BC)
    Block-diagonal Fourier covariance:
    ˆ
    y( ) = ˆ( ) ˆ
    f( )
    y = f computed as
    where ˆ
    f( ) =
    x
    f(x)e
    2ix1⇥1
    N1
    + 2ix2⇥2
    N2
    Maximum likelihood estimate (MLE) of m from f0
    :
    i, mi
    =
    1
    N
    x
    f0
    (x) Rd
    Spot Noise Model [Galerne et al.]

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  20. Stationarity hypothesis: X(· + ) X
    (periodic BC)
    Block-diagonal Fourier covariance:
    i,j
    =
    1
    N
    x
    f0
    (i + x) f0
    (j + x) Rd d
    ˆ
    y( ) = ˆ( ) ˆ
    f( )
    y = f computed as
    where ˆ
    f( ) =
    x
    f(x)e
    2ix1⇥1
    N1
    + 2ix2⇥2
    N2
    MLE of :
    Maximum likelihood estimate (MLE) of m from f0
    :
    i, mi
    =
    1
    N
    x
    f0
    (x) Rd
    Spot Noise Model [Galerne et al.]

    View full-size slide

  21. Stationarity hypothesis: X(· + ) X
    (periodic BC)
    Block-diagonal Fourier covariance:
    i,j
    =
    1
    N
    x
    f0
    (i + x) f0
    (j + x) Rd d
    ˆ
    y( ) = ˆ( ) ˆ
    f( )
    y = f computed as
    where ˆ
    f( ) =
    x
    f(x)e
    2ix1⇥1
    N1
    + 2ix2⇥2
    N2
    = 0, ˆ( ) = ˆ
    f0
    ( ) ˆ
    f0
    ( ) Cd d
    is a spot noise = 0, ˆ( ) is rank-1.
    MLE of :
    Maximum likelihood estimate (MLE) of m from f0
    :
    i, mi
    =
    1
    N
    x
    f0
    (x) Rd
    Spot Noise Model [Galerne et al.]

    View full-size slide

  22. Cd C
    Input f0
    RN 3 Realizations f
    Example of Synthesis
    Synthesizing f = X( ), X N(m, ):
    = 0, ˆ
    f( ) = ˆ
    f0
    ( ) ˆ
    w( )
    Convolve each channel with the same white noise.
    w N(N 1, N 1/2Id
    N
    )

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  23. Input distributions (µ0, µ1
    ) with µi
    = N(mi, i
    ).
    E0 E1
    Ellipses: Ei
    = x Rd \ (mi x) 1
    i
    (mi x) c
    Gaussian Optimal Transport

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  24. Input distributions (µ0, µ1
    ) with µi
    = N(mi, i
    ).
    E0 E1
    W2
    (µ0, µ1
    )2 = tr (
    0
    +
    1
    2
    0,1
    ) + ||m0 m1
    ||2,
    T
    0,1
    = ( 1/2
    1 0
    1/2
    1
    )1/2
    S = 1/2
    1
    +
    0,1
    1/2
    1
    T(x) = Sx + m1 m0
    where
    Ellipses: Ei
    = x Rd \ (mi x) 1
    i
    (mi x) c
    Theorem:
    If

    ker(⌃0) \ ker(⌃1)? = {0},
    ker(⌃1) \ ker(⌃0)? = {0},
    Gaussian Optimal Transport

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  25. The set of Gaussians is geodesically convex:
    µt
    = ((1 t)Id + tT) µ0
    = N(mt, t
    )
    Gaussian Geodesics
    µ1
    mt
    = (1 t)m0
    + tm1
    t
    = [(1 t)Id + tT]
    0
    [(1 t)Id + tT]
    µ0
    0,1
    = ( 1/2
    1 0
    1/2
    1
    )1/2
    T(x) = Sx + m1 m0
    S = 1/2
    1
    +
    0,1
    1/2
    1
    Input distributions (µ0, µ1
    ) with µi
    = N(mi, i
    ).

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  26. Geodesic of Spot Noises
    Theorem:
    i.e. ˆ
    i
    ( ) = ˆ
    f[i]( ) ˆ
    f[i]( ) .
    f[t] = (1 t)f[0] + tg[1]
    ˆ
    g[1]( ) = ˆ
    f[1]( )
    ˆ
    f[1]( ) ˆ
    f[0]( )
    | ˆ
    f[1]( ) ˆ
    f[0]( )|
    Then t [0, 1], µt
    = µ(f[t])
    Let for i = 0, 1, µi
    = µ(f[i]) be spot noises,

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  27. t
    f[0]
    f[1]
    Geodesic of Spot Noises
    0 1
    Theorem:
    i.e. ˆ
    i
    ( ) = ˆ
    f[i]( ) ˆ
    f[i]( ) .
    f[t] = (1 t)f[0] + tg[1]
    ˆ
    g[1]( ) = ˆ
    f[1]( )
    ˆ
    f[1]( ) ˆ
    f[0]( )
    | ˆ
    f[1]( ) ˆ
    f[0]( )|
    Then t [0, 1], µt
    = µ(f[t])
    Let for i = 0, 1, µi
    = µ(f[i]) be spot noises,

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  28. Input
    Spot Noise Barycenters

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  29. Input
    Spot Noise Barycenters

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  30. Dynamic Textures Mixing

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  31. Dynamic Textures Mixing

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  32. Dynamic Textures Mixing

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  33. Dynamic Textures Mixing

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  34. Overview
    • Gaussian Texture Synthesis by Example
    • Spot Noise Models and Motion Clouds
    • Stochastic PDE Models
    • Bayesian Brain and Prior Estimation

    View full-size slide

  35. Static Spot Noise
    “Texton” pattern
    g
    .
    (
    Xp)p2N
    2-D Poisson point process of intensity ,
    E
    (#
    {Xp
    2 U}
    ) =
    |U|
    Spot noise: I
    (
    x
    ) def.
    =
    1 X
    p
    g
    (
    x Xp)

    View full-size slide

  36. Static Spot Noise
    “Texton” pattern
    g
    .
    (
    Xp)p2N
    2-D Poisson point process of intensity ,
    E
    (#
    {Xp
    2 U}
    ) =
    |U|
    Spot noise: I
    (
    x
    ) def.
    =
    1 X
    p
    g
    (
    x Xp)
    (
    x
    ) =
    Z
    g
    (
    y
    )
    g
    (
    x y
    )d
    y auto-correlation of
    g
    .
    Proposition: I is stationary of covariance ⌃(x, x
    0
    ) = (x x
    0
    )

    View full-size slide

  37. Static Spot Noise
    “Texton” pattern
    g
    .
    (
    Xp)p2N
    2-D Poisson point process of intensity ,
    E
    (#
    {Xp
    2 U}
    ) =
    |U|
    I !+1
    ! I1
    a stationary Gaussian field of variance ⌃.
    Spot noise:
    EORY AND SYNTHESIS 3
    rithms
    SN for
    I
    (
    x
    ) def.
    =
    1 X
    p
    g
    (
    x Xp)
    (
    x
    ) =
    Z
    g
    (
    y
    )
    g
    (
    x y
    )d
    y auto-correlation of
    g
    .
    Proposition: I is stationary of covariance ⌃(x, x
    0
    ) = (x x
    0
    )

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  38. "Gabor" Noise
    To appear in the ACM SIGGRAPH conference proceedings
    Granite vase. Textile cushion. Straw hat. Leather boot. Rusty car. Wooden chair. Snake skin. Tree bark.
    [Lagae et al. 2009]

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  39. Dynamical Spot Noise
    static micro-textures [5] and dynamic natural phenomena [17]. The simplicity of this
    allows for a fine tuning of frequency-based (Fourier) parameterization, which is desira
    interpretation of psychophysical experiments.
    We define a random field as
    I (x, t)
    def.
    =
    1
    p
    X
    p
    2N
    g('Ap (x Xp Vpt))
    where
    'a :
    R2 ! R2 is a planar warping parameterized by a finite dimensional vector
    a
    .
    this model corresponds to a dense mixing of stereotyped, static textons as in [5]. The or
    two-fold. First, the components of this mixing are derived from the texton by visual trans
    'Ap
    which may correspond to arbitrary transformations such as zooms or rotations, ill
    Figure 1. Second, we explicitly model the motion (position
    Xp
    and speed
    Vp
    ) of each
    texton. The parameters
    (Xp, Vp, Ap)p
    2N are independent random vectors. They acco
    variability in the position of objects or observers and their speed, thus mimicking natural
    an ambient scene. The set of translations
    (Xp)p
    2N is a 2-D Poisson point process of inten
    The following section instantiates this idea and proposes canonical choices for these v
    The warping parameters
    (Ap)p
    are distributed according to a distribution P
    A
    . The speed
    (Vp)p
    are distributed according to a distribution P
    V
    on R2. The following result show
    model (2) converges to a stationary Gaussian field and gives the parameterization of the c
    Its proof follows from a specialization of [4, Theorem 3.1] to our setting.
    3
    “Texton” pattern
    g
    .
    (
    Xp)p2N
    2-D Poisson point process of intensity ,
    E
    (#
    {Xp
    2 U}
    ) =
    |U|
    (
    Xp, Vp, Ap)p2N
    independent random vectors.
    Warpings 'a : R2 ! R2.
    Spot noise:

    View full-size slide

  40. Dynamical Spot Noise
    static micro-textures [5] and dynamic natural phenomena [17]. The simplicity of this
    allows for a fine tuning of frequency-based (Fourier) parameterization, which is desira
    interpretation of psychophysical experiments.
    We define a random field as
    I (x, t)
    def.
    =
    1
    p
    X
    p
    2N
    g('Ap (x Xp Vpt))
    where
    'a :
    R2 ! R2 is a planar warping parameterized by a finite dimensional vector
    a
    .
    this model corresponds to a dense mixing of stereotyped, static textons as in [5]. The or
    two-fold. First, the components of this mixing are derived from the texton by visual trans
    'Ap
    which may correspond to arbitrary transformations such as zooms or rotations, ill
    Figure 1. Second, we explicitly model the motion (position
    Xp
    and speed
    Vp
    ) of each
    texton. The parameters
    (Xp, Vp, Ap)p
    2N are independent random vectors. They acco
    variability in the position of objects or observers and their speed, thus mimicking natural
    an ambient scene. The set of translations
    (Xp)p
    2N is a 2-D Poisson point process of inten
    The following section instantiates this idea and proposes canonical choices for these v
    The warping parameters
    (Ap)p
    are distributed according to a distribution P
    A
    . The speed
    (Vp)p
    are distributed according to a distribution P
    V
    on R2. The following result show
    model (2) converges to a stationary Gaussian field and gives the parameterization of the c
    Its proof follows from a specialization of [4, Theorem 3.1] to our setting.
    3
    on 1. I
    is stationary with bounded second order moments. Its covariance is
    t
    0
    ) = (x x
    0
    , t t
    0
    )
    where satisfies
    8
    (x, t)
    2 R3
    , (x, t) =
    Z Z
    R
    2
    cg('a(x ⌫t))
    P
    V (⌫)
    P
    A(a)d⌫da
    (3)
    = g ? ¯
    g
    is the auto-correlation of
    g
    . When !
    +
    1, it converges (in the sense of finite
    al distributions) toward a stationary Gaussian field
    I
    of zero mean and covariance

    .
    nition of “Motion Clouds”
    here this model where the warpings are rotations and scalings (see Figure 1). This allows
    t for the characteristic orientations and sizes (or spatial scales) in a scene with respect to
    er
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +, 'a(x)
    def.
    = zR ✓(x),
    “Texton” pattern
    g
    .
    (
    Xp)p2N
    2-D Poisson point process of intensity ,
    E
    (#
    {Xp
    2 U}
    ) =
    |U|
    Proposition: I is stationary of covariance ⌃(x, t, x
    0
    , t
    0
    ) = (x x
    0
    , t t
    0
    )
    where cg = g ? ¯
    g.
    I !+1
    ! I1
    a stationary Gaussian field of variance ⌃.
    (
    Xp, Vp, Ap)p2N
    independent random vectors.
    Warpings 'a : R2 ! R2.
    Spot noise:
    EORY AND SYNTHESIS 3
    rithms
    SN for

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  41. Motion Clouds
    72
    73
    74
    75
    76
    77
    78
    79
    80
    81
    82
    83
    84
    85
    86
    87
    88
    89
    90
    91
    92
    93
    94
    95
    96
    We detail here this model where the warpings are rotations and scalings (see Figure 1). This al
    to account for the characteristic orientations and sizes (or spatial scales) in a scene with respe
    the observer
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +, 'a(x)
    def.
    = zR ✓(x),
    where
    R✓
    is the planar rotation of angle

    . We now give some physical and biological motiva
    underlying our particular choice for the distributions of the parameters. We assume that the dist
    tions P
    Z
    and P

    of spatial scales
    z
    and orientations

    , respectively (see Figure 1), are indepen
    and have densities, thus considering
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +,
    P
    A(a) =
    P
    Z(z)
    P
    ⇥(✓).
    The speed vector

    are assumed to be randomly fluctuating around a central speed
    v0
    , so that
    8

    2 R2
    ,
    P
    V (⌫) =
    P
    ||
    V v0
    ||(
    ||
    ⌫ v0
    ||
    ).
    In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen
    define the texton
    g
    to be equal to a standard receptive field of V1 , i.e. an oriented Gabor-like a
    having a scale and a central frequency
    ⇠0
    . Since the rotation and scale of the texton is handle
    the
    (✓, z)
    parameters, we can impose without loss of generality the normalization
    ⇠0 = (1, 0
    the special case where !
    0
    ,
    g
    is a grating of frequency
    ⇠0
    , and the image
    I
    is a dense mix
    of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositio
    Its proof can be found in the supplementary materials. We call this Gaussian field a Motion C
    (MC), and it is parameterized by the envelopes
    (
    P
    Z,
    P
    ⇥,
    P
    V )
    and has central frequency and s
    (⇠0, v0)
    . Note that it is possible to consider any arbitrary textons
    g
    , which would give rise to m
    complicated parameterizations for the power spectrum
    ˆ
    g
    , but we decided here to stick to the sim
    case of gratings.
    Proposition 2.
    When
    g(x) = ei
    h
    x, ⇠0
    i, the image
    I
    defined in Proposition 1 is a stationary Gaus
    Rotations + zooms:

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  42. Motion Clouds
    72
    73
    74
    75
    76
    77
    78
    79
    80
    81
    82
    83
    84
    85
    86
    87
    88
    89
    90
    91
    92
    93
    94
    95
    96
    We detail here this model where the warpings are rotations and scalings (see Figure 1). This al
    to account for the characteristic orientations and sizes (or spatial scales) in a scene with respe
    the observer
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +, 'a(x)
    def.
    = zR ✓(x),
    where
    R✓
    is the planar rotation of angle

    . We now give some physical and biological motiva
    underlying our particular choice for the distributions of the parameters. We assume that the dist
    tions P
    Z
    and P

    of spatial scales
    z
    and orientations

    , respectively (see Figure 1), are indepen
    and have densities, thus considering
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +,
    P
    A(a) =
    P
    Z(z)
    P
    ⇥(✓).
    The speed vector

    are assumed to be randomly fluctuating around a central speed
    v0
    , so that
    8

    2 R2
    ,
    P
    V (⌫) =
    P
    ||
    V v0
    ||(
    ||
    ⌫ v0
    ||
    ).
    In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen
    define the texton
    g
    to be equal to a standard receptive field of V1 , i.e. an oriented Gabor-like a
    having a scale and a central frequency
    ⇠0
    . Since the rotation and scale of the texton is handle
    the
    (✓, z)
    parameters, we can impose without loss of generality the normalization
    ⇠0 = (1, 0
    the special case where !
    0
    ,
    g
    is a grating of frequency
    ⇠0
    , and the image
    I
    is a dense mix
    of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositio
    Its proof can be found in the supplementary materials. We call this Gaussian field a Motion C
    (MC), and it is parameterized by the envelopes
    (
    P
    Z,
    P
    ⇥,
    P
    V )
    and has central frequency and s
    (⇠0, v0)
    . Note that it is possible to consider any arbitrary textons
    g
    , which would give rise to m
    complicated parameterizations for the power spectrum
    ˆ
    g
    , but we decided here to stick to the sim
    case of gratings.
    Proposition 2.
    When
    g(x) = ei
    h
    x, ⇠0
    i, the image
    I
    defined in Proposition 1 is a stationary Gaus
    + a ✓
    where
    R✓
    is the planar rotation of angle

    . We now give some physical and biological motivat
    underlying our particular choice for the distributions of the parameters. We assume that the distri
    tions P
    Z
    and P

    of spatial scales
    z
    and orientations

    , respectively (see Figure 1), are independ
    and have densities, thus considering
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +,
    P
    A(a) =
    P
    Z(z)
    P
    ⇥(✓).
    The speed vector

    are assumed to be randomly fluctuating around a central speed
    v0
    , so that
    8

    2 R2
    ,
    P
    V (⌫) =
    P
    ||
    V v0
    ||(
    ||
    ⌫ v0
    ||
    ).
    In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sense
    define the texton
    g
    to be equal to a standard receptive field of V1 , i.e. an oriented Gabor-like ato
    having a scale and a central frequency
    ⇠0
    . Since the rotation and scale of the texton is handled
    the
    (✓, z)
    parameters, we can impose without loss of generality the normalization
    ⇠0 = (1, 0)
    .
    the special case where !
    0
    ,
    g
    is a grating of frequency
    ⇠0
    , and the image
    I
    is a dense mixt
    of drifting gratings, whose power-spectrum has a closed form expression detailed in Proposition
    Its proof can be found in the supplementary materials. We call this Gaussian field a Motion Clo
    (MC), and it is parameterized by the envelopes
    (
    P
    Z,
    P
    ⇥,
    P
    V )
    and has central frequency and sp
    (⇠0, v0)
    . Note that it is possible to consider any arbitrary textons
    g
    , which would give rise to m
    complicated parameterizations for the power spectrum
    ˆ
    g
    , but we decided here to stick to the sim
    case of gratings.
    Proposition 2.
    When
    g(x) = ei
    h
    x, ⇠0
    i, the image
    I
    defined in Proposition 1 is a stationary Gauss
    field of covariance having the power-spectrum
    8
    (⇠, ⌧)
    2 R2 ⇥ R
    , ˆ(⇠, ⌧) =
    P
    Z (
    ||

    ||
    )
    ||

    ||2
    P
    ⇥ (
    \
    ⇠)
    L
    (
    P
    ||
    V v0
    ||)


    ||

    ||
    ||
    v0
    ||
    cos(
    \
    v0
    \
    ⇠)

    Rotations + zooms:
    Independency:

    View full-size slide

  43. Motion Clouds
    72
    73
    74
    75
    76
    77
    78
    79
    80
    81
    82
    83
    84
    85
    86
    87
    88
    89
    90
    91
    92
    93
    94
    95
    96
    We detail here this model where the warpings are rotations and scalings (see Figure 1). This al
    to account for the characteristic orientations and sizes (or spatial scales) in a scene with respe
    the observer
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +, 'a(x)
    def.
    = zR ✓(x),
    where
    R✓
    is the planar rotation of angle

    . We now give some physical and biological motiva
    underlying our particular choice for the distributions of the parameters. We assume that the dist
    tions P
    Z
    and P

    of spatial scales
    z
    and orientations

    , respectively (see Figure 1), are indepen
    and have densities, thus considering
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +,
    P
    A(a) =
    P
    Z(z)
    P
    ⇥(✓).
    The speed vector

    are assumed to be randomly fluctuating around a central speed
    v0
    , so that
    8

    2 R2
    ,
    P
    V (⌫) =
    P
    ||
    V v0
    ||(
    ||
    ⌫ v0
    ||
    ).
    In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen
    define the texton
    g
    to be equal to a standard receptive field of V1 , i.e. an oriented Gabor-like a
    having a scale and a central frequency
    ⇠0
    . Since the rotation and scale of the texton is handle
    the
    (✓, z)
    parameters, we can impose without loss of generality the normalization
    ⇠0 = (1, 0
    the special case where !
    0
    ,
    g
    is a grating of frequency
    ⇠0
    , and the image
    I
    is a dense mix
    of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositio
    Its proof can be found in the supplementary materials. We call this Gaussian field a Motion C
    (MC), and it is parameterized by the envelopes
    (
    P
    Z,
    P
    ⇥,
    P
    V )
    and has central frequency and s
    (⇠0, v0)
    . Note that it is possible to consider any arbitrary textons
    g
    , which would give rise to m
    complicated parameterizations for the power spectrum
    ˆ
    g
    , but we decided here to stick to the sim
    case of gratings.
    Proposition 2.
    When
    g(x) = ei
    h
    x, ⇠0
    i, the image
    I
    defined in Proposition 1 is a stationary Gaus
    + a ✓
    where
    R✓
    is the planar rotation of angle

    . We now give some physical and biological motivat
    underlying our particular choice for the distributions of the parameters. We assume that the distri
    tions P
    Z
    and P

    of spatial scales
    z
    and orientations

    , respectively (see Figure 1), are independ
    and have densities, thus considering
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +,
    P
    A(a) =
    P
    Z(z)
    P
    ⇥(✓).
    The speed vector

    are assumed to be randomly fluctuating around a central speed
    v0
    , so that
    8

    2 R2
    ,
    P
    V (⌫) =
    P
    ||
    V v0
    ||(
    ||
    ⌫ v0
    ||
    ).
    In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sense
    define the texton
    g
    to be equal to a standard receptive field of V1 , i.e. an oriented Gabor-like ato
    having a scale and a central frequency
    ⇠0
    . Since the rotation and scale of the texton is handled
    the
    (✓, z)
    parameters, we can impose without loss of generality the normalization
    ⇠0 = (1, 0)
    .
    the special case where !
    0
    ,
    g
    is a grating of frequency
    ⇠0
    , and the image
    I
    is a dense mixt
    of drifting gratings, whose power-spectrum has a closed form expression detailed in Proposition
    Its proof can be found in the supplementary materials. We call this Gaussian field a Motion Clo
    (MC), and it is parameterized by the envelopes
    (
    P
    Z,
    P
    ⇥,
    P
    V )
    and has central frequency and sp
    (⇠0, v0)
    . Note that it is possible to consider any arbitrary textons
    g
    , which would give rise to m
    complicated parameterizations for the power spectrum
    ˆ
    g
    , but we decided here to stick to the sim
    case of gratings.
    Proposition 2.
    When
    g(x) = ei
    h
    x, ⇠0
    i, the image
    I
    defined in Proposition 1 is a stationary Gauss
    field of covariance having the power-spectrum
    8
    (⇠, ⌧)
    2 R2 ⇥ R
    , ˆ(⇠, ⌧) =
    P
    Z (
    ||

    ||
    )
    ||

    ||2
    P
    ⇥ (
    \
    ⇠)
    L
    (
    P
    ||
    V v0
    ||)


    ||

    ||
    ||
    v0
    ||
    cos(
    \
    v0
    \
    ⇠)

    76
    77
    78
    79
    80
    81
    82
    83
    84
    85
    86
    87
    88
    89
    90
    91
    92
    93
    94
    95
    96
    97
    98
    99
    00
    where
    R✓
    is the planar rotation of angle

    . We now give some physical and biological motiv
    underlying our particular choice for the distributions of the parameters. We assume that the dis
    tions P
    Z
    and P

    of spatial scales
    z
    and orientations

    , respectively (see Figure 1), are indepe
    and have densities, thus considering
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +,
    P
    A(a) =
    P
    Z(z)
    P
    ⇥(✓).
    The speed vector

    are assumed to be randomly fluctuating around a central speed
    v0
    , so that
    8

    2 R2
    ,
    P
    V (⌫) =
    P
    ||
    V v0
    ||(
    ||
    ⌫ v0
    ||
    ).
    In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen
    define the texton
    g
    to be equal to a standard receptive field of V1 , i.e. an oriented Gabor-like a
    having a scale and a central frequency
    ⇠0
    . Since the rotation and scale of the texton is handl
    the
    (✓, z)
    parameters, we can impose without loss of generality the normalization
    ⇠0 = (1, 0
    the special case where !
    0
    ,
    g
    is a grating of frequency
    ⇠0
    , and the image
    I
    is a dense mi
    of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositi
    Its proof can be found in the supplementary materials. We call this Gaussian field a Motion C
    (MC), and it is parameterized by the envelopes
    (
    P
    Z,
    P
    ⇥,
    P
    V )
    and has central frequency and s
    (⇠0, v0)
    . Note that it is possible to consider any arbitrary textons
    g
    , which would give rise to
    complicated parameterizations for the power spectrum
    ˆ
    g
    , but we decided here to stick to the si
    case of gratings.
    Proposition 2.
    When
    g(x) = ei
    h
    x, ⇠0
    i, the image
    I
    defined in Proposition 1 is a stationary Gau
    field of covariance having the power-spectrum
    8
    (⇠, ⌧)
    2 R2 ⇥ R
    , ˆ(⇠, ⌧) =
    P
    Z (
    ||

    ||
    )
    ||

    ||2
    P
    ⇥ (
    \
    ⇠)
    L
    (
    P
    ||
    V v0
    ||)


    ||

    ||
    ||
    v0
    ||
    cos(
    \
    v0
    \
    ⇠)

    R

    Rotations + zooms:
    Independency:
    Radial speed variations:

    View full-size slide

  44. Motion Clouds
    72
    73
    74
    75
    76
    77
    78
    79
    80
    81
    82
    83
    84
    85
    86
    87
    88
    89
    90
    91
    92
    93
    94
    95
    96
    We detail here this model where the warpings are rotations and scalings (see Figure 1). This al
    to account for the characteristic orientations and sizes (or spatial scales) in a scene with respe
    the observer
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +, 'a(x)
    def.
    = zR ✓(x),
    where
    R✓
    is the planar rotation of angle

    . We now give some physical and biological motiva
    underlying our particular choice for the distributions of the parameters. We assume that the dist
    tions P
    Z
    and P

    of spatial scales
    z
    and orientations

    , respectively (see Figure 1), are indepen
    and have densities, thus considering
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +,
    P
    A(a) =
    P
    Z(z)
    P
    ⇥(✓).
    The speed vector

    are assumed to be randomly fluctuating around a central speed
    v0
    , so that
    8

    2 R2
    ,
    P
    V (⌫) =
    P
    ||
    V v0
    ||(
    ||
    ⌫ v0
    ||
    ).
    In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen
    define the texton
    g
    to be equal to a standard receptive field of V1 , i.e. an oriented Gabor-like a
    having a scale and a central frequency
    ⇠0
    . Since the rotation and scale of the texton is handle
    the
    (✓, z)
    parameters, we can impose without loss of generality the normalization
    ⇠0 = (1, 0
    the special case where !
    0
    ,
    g
    is a grating of frequency
    ⇠0
    , and the image
    I
    is a dense mix
    of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositio
    Its proof can be found in the supplementary materials. We call this Gaussian field a Motion C
    (MC), and it is parameterized by the envelopes
    (
    P
    Z,
    P
    ⇥,
    P
    V )
    and has central frequency and s
    (⇠0, v0)
    . Note that it is possible to consider any arbitrary textons
    g
    , which would give rise to m
    complicated parameterizations for the power spectrum
    ˆ
    g
    , but we decided here to stick to the sim
    case of gratings.
    Proposition 2.
    When
    g(x) = ei
    h
    x, ⇠0
    i, the image
    I
    defined in Proposition 1 is a stationary Gaus
    + a ✓
    where
    R✓
    is the planar rotation of angle

    . We now give some physical and biological motivat
    underlying our particular choice for the distributions of the parameters. We assume that the distri
    tions P
    Z
    and P

    of spatial scales
    z
    and orientations

    , respectively (see Figure 1), are independ
    and have densities, thus considering
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +,
    P
    A(a) =
    P
    Z(z)
    P
    ⇥(✓).
    The speed vector

    are assumed to be randomly fluctuating around a central speed
    v0
    , so that
    8

    2 R2
    ,
    P
    V (⌫) =
    P
    ||
    V v0
    ||(
    ||
    ⌫ v0
    ||
    ).
    In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sense
    define the texton
    g
    to be equal to a standard receptive field of V1 , i.e. an oriented Gabor-like ato
    having a scale and a central frequency
    ⇠0
    . Since the rotation and scale of the texton is handled
    the
    (✓, z)
    parameters, we can impose without loss of generality the normalization
    ⇠0 = (1, 0)
    .
    the special case where !
    0
    ,
    g
    is a grating of frequency
    ⇠0
    , and the image
    I
    is a dense mixt
    of drifting gratings, whose power-spectrum has a closed form expression detailed in Proposition
    Its proof can be found in the supplementary materials. We call this Gaussian field a Motion Clo
    (MC), and it is parameterized by the envelopes
    (
    P
    Z,
    P
    ⇥,
    P
    V )
    and has central frequency and sp
    (⇠0, v0)
    . Note that it is possible to consider any arbitrary textons
    g
    , which would give rise to m
    complicated parameterizations for the power spectrum
    ˆ
    g
    , but we decided here to stick to the sim
    case of gratings.
    Proposition 2.
    When
    g(x) = ei
    h
    x, ⇠0
    i, the image
    I
    defined in Proposition 1 is a stationary Gauss
    field of covariance having the power-spectrum
    8
    (⇠, ⌧)
    2 R2 ⇥ R
    , ˆ(⇠, ⌧) =
    P
    Z (
    ||

    ||
    )
    ||

    ||2
    P
    ⇥ (
    \
    ⇠)
    L
    (
    P
    ||
    V v0
    ||)


    ||

    ||
    ||
    v0
    ||
    cos(
    \
    v0
    \
    ⇠)

    76
    77
    78
    79
    80
    81
    82
    83
    84
    85
    86
    87
    88
    89
    90
    91
    92
    93
    94
    95
    96
    97
    98
    99
    00
    where
    R✓
    is the planar rotation of angle

    . We now give some physical and biological motiv
    underlying our particular choice for the distributions of the parameters. We assume that the dis
    tions P
    Z
    and P

    of spatial scales
    z
    and orientations

    , respectively (see Figure 1), are indepe
    and have densities, thus considering
    8
    a = (✓, z)
    2
    [ ⇡, ⇡)
    ⇥ R⇤
    +,
    P
    A(a) =
    P
    Z(z)
    P
    ⇥(✓).
    The speed vector

    are assumed to be randomly fluctuating around a central speed
    v0
    , so that
    8

    2 R2
    ,
    P
    V (⌫) =
    P
    ||
    V v0
    ||(
    ||
    ⌫ v0
    ||
    ).
    In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen
    define the texton
    g
    to be equal to a standard receptive field of V1 , i.e. an oriented Gabor-like a
    having a scale and a central frequency
    ⇠0
    . Since the rotation and scale of the texton is handl
    the
    (✓, z)
    parameters, we can impose without loss of generality the normalization
    ⇠0 = (1, 0
    the special case where !
    0
    ,
    g
    is a grating of frequency
    ⇠0
    , and the image
    I
    is a dense mi
    of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositi
    Its proof can be found in the supplementary materials. We call this Gaussian field a Motion C
    (MC), and it is parameterized by the envelopes
    (
    P
    Z,
    P
    ⇥,
    P
    V )
    and has central frequency and s
    (⇠0, v0)
    . Note that it is possible to consider any arbitrary textons
    g
    , which would give rise to
    complicated parameterizations for the power spectrum
    ˆ
    g
    , but we decided here to stick to the si
    case of gratings.
    Proposition 2.
    When
    g(x) = ei
    h
    x, ⇠0
    i, the image
    I
    defined in Proposition 1 is a stationary Gau
    field of covariance having the power-spectrum
    8
    (⇠, ⌧)
    2 R2 ⇥ R
    , ˆ(⇠, ⌧) =
    P
    Z (
    ||

    ||
    )
    ||

    ||2
    P
    ⇥ (
    \
    ⇠)
    L
    (
    P
    ||
    V v0
    ||)


    ||

    ||
    ||
    v0
    ||
    cos(
    \
    v0
    \
    ⇠)

    R

    Rotations + zooms:
    Independency:
    Radial speed variations:
    Proposition: for g(x) = e
    i
    x1
    , I1 satisfies
    ⇠1
    ⇠2

    ⇠2
    ⌧ + hv0, ⇠i = 0
    Speed variations.
    Spatial angular variations.
    Spatial radial variations.
    8 (⇠, ⌧) 2 R2 ⇥ R, ˆ(⇠, ⌧) =
    P
    Z (||⇠||)
    ||⇠||2
    P
    ✓ (\⇠) L(P
    ||V v0
    ||
    )

    ⌧ + hv0, ⇠i
    ||⇠||

    where L
    (
    f
    )(
    u
    )
    def.
    =
    R ⇡/2
    ⇡/2
    f
    (
    u/
    cos(
    '
    ))d
    '

    View full-size slide

  45. Example of Parameterization
    ightly different bell-function (with a more complicated expression) should be used to obtain an
    ct equivalence with the sPDE discretization mentionned in Section 2.4.
    e distributions of the parameters are thus chosen as
    P
    Z(z)
    / z0
    z
    e
    ln(
    z
    z
    0
    )
    2
    2 ln(1+ 2
    Z ) ,
    P
    ⇥(✓)
    /
    e
    cos(2(
    ✓ ✓
    0))
    2

    and P
    ||
    V v0
    ||(r)
    /
    e
    r2
    2 2
    V
    .
    (6)
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    ope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    ure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    elopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    uds (right).
    gging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    ing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    a slightly different bell-function (with a more complicated expression) should be used to obtain
    exact equivalence with the sPDE discretization mentionned in Section 2.4.
    The distributions of the parameters are thus chosen as
    P
    Z(z)
    / z0
    z
    e
    ln(
    z
    z
    0
    )
    2
    2 ln(1+ 2
    Z ) ,
    P
    ⇥(✓)
    /
    e
    cos(2(
    ✓ ✓
    0))
    2

    and P
    ||
    V v0
    ||(r)
    /
    e
    r2
    2 2
    V
    .
    (
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencie
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of t
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Moti
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion clou
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The fo
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitu
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows
    exact equivalence with the sPDE discretization mentionned in Section 2.4.
    The distributions of the parameters are thus chosen as
    P
    Z(z)
    / z0
    z
    e
    ln(
    z
    z
    0
    )
    2
    2 ln(1+ 2
    Z ) ,
    P
    ⇥(✓)
    /
    e
    cos(2(
    ✓ ✓
    0))
    2

    and P
    ||
    V v0
    ||(r)
    /
    e
    r2
    2 2
    V
    .
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequenci
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Mot
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion clo
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The f
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitu
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 show
    232
    233
    234
    235
    236
    237
    238
    239
    240
    241
    242
    243
    244
    245
    246
    247
    248
    249
    250
    The distributions of the parameters are thus chosen as
    P
    Z(z)
    / z0
    z
    e
    ln(
    z
    z
    0
    )
    2
    2 ln(1+ 2
    Z ) ,
    P
    ⇥(✓)
    /
    e
    cos(2(
    ✓ ✓
    0))
    2

    and
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space MC of
    Figure 2: Graphical representation of the covariance (left) —
    envelopes– and an example of synthesized dynamics for narro
    Clouds (right).
    233
    234
    235
    236
    237
    238
    239
    240
    241
    242
    243
    244
    245
    246
    247
    248
    249
    250
    The distributions of the parameters are thus chosen as
    P
    Z(z)
    / z0
    z
    e
    ln(
    z
    z
    0
    )
    2
    2 ln(1+ 2
    Z ) ,
    P
    ⇥(✓)
    /
    e
    cos(2(
    ✓ ✓
    0))
    2

    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space MC
    Figure 2: Graphical representation of the covariance (lef
    envelopes– and an example of synthesized dynamics for n
    Clouds (right).
    z0
    Z
    ⇠1 ⇠1
    ✓0
    z0 Z
    lope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    gure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    velopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    ouds (right).
    ugging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    e obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    wing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    d orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a
    aphical display of the influence of these parameters.
    Speed Freq. orient. Freq. amplitude
    (mean, dispersion)
    (v0, V ) (✓0, ⇥) (z0, Z)
    mark 2. Note that the final envelope of
    ˆ
    is in agreement with the formulation that is used in [8].
    owever, this latter derivation was based on a heuristic following a trial-and-error strategy between
    odelers and psychophysicists. Herein, we justified these different points in a constructive manner.
    4 sPDE Formulation and Numerical Synthesis Algorithm

    View full-size slide

  46. Drifting Grating vs. Anisotropic Noise
    ⇠2
    ⇠2

    View full-size slide

  47. Drifting Grating vs. Anisotropic Noise
    ⇠2
    ⇠2

    View full-size slide

  48. Drifting Grating vs. Anisotropic Noise
    ⇠1
    ⇠2

    ⇠2
    ⇠2

    ⇠2

    View full-size slide

  49. Translating Anisotropic vs. Isotropic Noise
    ⇠2
    ⇠2

    View full-size slide

  50. Translating Anisotropic vs. Isotropic Noise
    ⇠2
    ⇠2

    View full-size slide

  51. Translating Anisotropic vs. Isotropic Noise
    ⇠1
    ⇠2

    ⇠2

    ⇠2
    ⇠2

    View full-size slide

  52. Power-law Spacial Frequency
    ⇠2
    ⇠2

    View full-size slide

  53. Power-law Spacial Frequency
    ⇠2
    ⇠2

    View full-size slide

  54. Power-law Spacial Frequency
    ⇠1
    ⇠2

    ⇠2

    ⇠2
    ⇠2

    View full-size slide

  55. Overview
    • Gaussian Texture Synthesis by Example
    • Spot Noise Models and Motion Clouds
    • Stochastic PDE Models
    • Bayesian Brain and Prior Estimation

    View full-size slide

  56. Stochastic PDE Models
    Dynamic Textures as Solutions of sPDE
    MC
    I
    with speed
    v0
    can be obtained from a MC
    I0
    with zero speed by the constant speed time
    rping
    I(x, t)
    def.
    = I0(x v0t, t).
    (2)
    now restrict our attention to
    I0
    .
    consider Gaussian random fields defined by a stochastic partial differential equation (sPDE) of
    form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    (3)
    is equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are stationary
    utions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. corresponding to
    temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in space and
    ?
    he spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at enforcing
    additional correlation in time of the model. Section 2.2 explains how to choose
    (↵, , ⌃W )
    so
    t the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    v0 = 0
    ),
    are motion clouds.
    is sPDE formulation is important since we aim to deal with dynamic stimulation, which should
    described by a causal equation which is local in time. This is crucial for numerical simulation
    explained in Section 2.4) but also to simplify the application of MC inside a bayesian model of
    ychophysical experiments (see Section 3).
    hile it is beyond the scope of this paper to study theoretically this equation, one can shows exis-
    ce and uniqueness results of stationary solutions for this class of sPDE under stability conditions
    the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfied in our
    I(x, t)
    def.
    = I0(x v0t, t).
    We now restrict our attention to
    I0
    .
    We consider Gaussian random fields defined by a stochastic partial differential equation (sPDE
    the form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    This equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are station
    solutions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. correspondin
    the temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in space an
    is the spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at enforc
    an additional correlation in time of the model. Section 2.2 explains how to choose
    (↵, , ⌃W )
    that the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    v0 =
    i.e. are motion clouds.
    This sPDE formulation is important since we aim to deal with dynamic stimulation, which sho
    be described by a causal equation which is local in time. This is crucial for numerical simulat
    (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian mode
    psychophysical experiments (see Section 3).
    While it is beyond the scope of this paper to study theoretically this equation, one can shows e
    tence and uniqueness results of stationary solutions for this class of sPDE under stability conditi
    on the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfied in
    simulations. Note also that one can show that in fact the stationary solutions to (3) all share
    same law. These solutions can be obtained by solving the sODE (4) forward for time
    t > t0
    w
    arbitrary boundary conditions at time
    t = t0
    , and letting
    t0
    ! 1. This is consistent with
    numerical scheme detailed in Section 2.4.
    063
    064
    065
    066
    067
    068
    069
    070
    071
    072
    073
    074
    075
    076
    077
    078
    079
    080
    081
    082
    083
    084
    085
    086
    087
    I(x, t)
    def.
    = I0(x v0t, t).
    We now restrict our attention to
    I0
    .
    We consider Gaussian random fields defined by a stochastic partial differential equation (
    the form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    This equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are s
    solutions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. corresp
    the temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in sp
    is the spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at
    an additional correlation in time of the model. Section 2.2 explains how to choose
    (↵,
    that the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    i.e. are motion clouds.
    This sPDE formulation is important since we aim to deal with dynamic stimulation, whic
    be described by a causal equation which is local in time. This is crucial for numerical s
    (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian
    psychophysical experiments (see Section 3).
    While it is beyond the scope of this paper to study theoretically this equation, one can sh
    tence and uniqueness results of stationary solutions for this class of sPDE under stability c
    on the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfi
    simulations. Note also that one can show that in fact the stationary solutions to (3) all
    same law. These solutions can be obtained by solving the sODE (4) forward for time
    t >
    arbitrary boundary conditions at time
    t = t0
    , and letting
    t0
    ! 1. This is consistent
    Constant speed motion:
    sPDE: where

    View full-size slide

  57. Stochastic PDE Models
    Dynamic Textures as Solutions of sPDE
    MC
    I
    with speed
    v0
    can be obtained from a MC
    I0
    with zero speed by the constant speed time
    rping
    I(x, t)
    def.
    = I0(x v0t, t).
    (2)
    now restrict our attention to
    I0
    .
    consider Gaussian random fields defined by a stochastic partial differential equation (sPDE) of
    form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    (3)
    is equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are stationary
    utions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. corresponding to
    temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in space and
    ?
    he spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at enforcing
    additional correlation in time of the model. Section 2.2 explains how to choose
    (↵, , ⌃W )
    so
    t the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    v0 = 0
    ),
    are motion clouds.
    is sPDE formulation is important since we aim to deal with dynamic stimulation, which should
    described by a causal equation which is local in time. This is crucial for numerical simulation
    explained in Section 2.4) but also to simplify the application of MC inside a bayesian model of
    ychophysical experiments (see Section 3).
    hile it is beyond the scope of this paper to study theoretically this equation, one can shows exis-
    ce and uniqueness results of stationary solutions for this class of sPDE under stability conditions
    the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfied in our
    I(x, t)
    def.
    = I0(x v0t, t).
    We now restrict our attention to
    I0
    .
    We consider Gaussian random fields defined by a stochastic partial differential equation (sPDE
    the form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    This equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are station
    solutions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. correspondin
    the temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in space an
    is the spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at enforc
    an additional correlation in time of the model. Section 2.2 explains how to choose
    (↵, , ⌃W )
    that the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    v0 =
    i.e. are motion clouds.
    This sPDE formulation is important since we aim to deal with dynamic stimulation, which sho
    be described by a causal equation which is local in time. This is crucial for numerical simulat
    (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian mode
    psychophysical experiments (see Section 3).
    While it is beyond the scope of this paper to study theoretically this equation, one can shows e
    tence and uniqueness results of stationary solutions for this class of sPDE under stability conditi
    on the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfied in
    simulations. Note also that one can show that in fact the stationary solutions to (3) all share
    same law. These solutions can be obtained by solving the sODE (4) forward for time
    t > t0
    w
    arbitrary boundary conditions at time
    t = t0
    , and letting
    t0
    ! 1. This is consistent with
    numerical scheme detailed in Section 2.4.
    063
    064
    065
    066
    067
    068
    069
    070
    071
    072
    073
    074
    075
    076
    077
    078
    079
    080
    081
    082
    083
    084
    085
    086
    087
    I(x, t)
    def.
    = I0(x v0t, t).
    We now restrict our attention to
    I0
    .
    We consider Gaussian random fields defined by a stochastic partial differential equation (
    the form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    This equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are s
    solutions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. corresp
    the temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in sp
    is the spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at
    an additional correlation in time of the model. Section 2.2 explains how to choose
    (↵,
    that the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    i.e. are motion clouds.
    This sPDE formulation is important since we aim to deal with dynamic stimulation, whic
    be described by a causal equation which is local in time. This is crucial for numerical s
    (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian
    psychophysical experiments (see Section 3).
    While it is beyond the scope of this paper to study theoretically this equation, one can sh
    tence and uniqueness results of stationary solutions for this class of sPDE under stability c
    on the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfi
    simulations. Note also that one can show that in fact the stationary solutions to (3) all
    same law. These solutions can be obtained by solving the sODE (4) forward for time
    t >
    arbitrary boundary conditions at time
    t = t0
    , and letting
    t0
    ! 1. This is consistent
    imulations. Note also that one can show that in fact the stationary solutions to (3) all share the
    ame law. These solutions can be obtained by solving the sODE (4) forward for time
    t > t0
    with
    rbitrary boundary conditions at time
    t = t0
    , and letting
    t0
    ! 1. This is consistent with the
    umerical scheme detailed in Section 2.4.
    .2 Equivalence Between Spectral and sPDE MC Formulations
    The sPDE equation (3) corresponds to a set of independent stochastic ODEs over the spatial Fourier
    omain, which reads, for each frequency

    ,
    8
    t
    2 R
    ,
    @
    2
    ˆ
    I0(⇠, t)
    @t
    2 + ˆ
    ↵(⇠)
    @ ˆ
    I0(⇠, t)
    @t
    + ˆ(⇠)ˆ
    I0(⇠, t) = ˆW (⇠) ˆ
    w(⇠, t)
    (4)
    where ˆ
    I0(⇠, t)
    denotes the Fourier transform with respect to the space variable
    x
    only. Here,
    ˆW (⇠)
    2
    s the spatial power spectrum of @W
    @t
    , which means that
    ⌃W (x, y) = c(x y)
    where
    ˆ
    c(⇠) = ˆ
    2
    W (⇠).
    (5)
    Here
    ˆ
    w(⇠, t)
    ⇠ N
    (0, 1)
    and
    w
    is a white noise in space and time. This formulation makes explicit
    hat

    ↵(⇠), ˆ(⇠))
    should be chosen in order to make the temporal covariance of the resulting process
    qual (or at least approximate) the temporal covariance appearing in (1) in the motion-less setting
    since we deal here with
    I0
    ), i.e. when
    v0 = 0
    . This covariance should be localized around 0 and
    on-oscillating. It thus make sense to constrain

    ↵(⇠), ˆ(⇠))
    for the corresponding ODE (4) to be
    ritically damped, which corresponds to imposing the following relationship
    8
    ⇠, ˆ
    ↵(⇠) =
    2
    ˆ
    ⌫(⇠)
    and ˆ(⇠) =
    1
    ˆ

    2
    (⇠)
    umerical scheme detailed in Section 2.4.
    .2 Equivalence Between Spectral and sPDE MC Formulations
    The sPDE equation (3) corresponds to a set of independent stochastic ODEs over the spatial Fourier
    omain, which reads, for each frequency

    ,
    8
    t
    2 R
    ,
    @
    2
    ˆ
    I0(⇠, t)
    @t
    2 + ˆ
    ↵(⇠)
    @ ˆ
    I0(⇠, t)
    @t
    + ˆ(⇠)ˆ
    I0(⇠, t) = ˆW (⇠) ˆ
    w(⇠, t)
    (4)
    where ˆ
    I0(⇠, t)
    denotes the Fourier transform with respect to the space variable
    x
    only. Here,
    ˆW (⇠)
    2
    s the spatial power spectrum of @W
    @t
    , which means that
    ⌃W (x, y) = c(x y)
    where
    ˆ
    c(⇠) = ˆ
    2
    W (⇠).
    (5)
    Here
    ˆ
    w(⇠, t)
    ⇠ N
    (0, 1)
    and
    w
    is a white noise in space and time. This formulation makes explicit
    hat

    ↵(⇠), ˆ(⇠))
    should be chosen in order to make the temporal covariance of the resulting process
    qual (or at least approximate) the temporal covariance appearing in (1) in the motion-less setting
    since we deal here with
    I0
    ), i.e. when
    v0 = 0
    . This covariance should be localized around 0 and
    on-oscillating. It thus make sense to constrain

    ↵(⇠), ˆ(⇠))
    for the corresponding ODE (4) to be
    ritically damped, which corresponds to imposing the following relationship
    8
    ⇠, ˆ
    ↵(⇠) =
    2
    ˆ
    ⌫(⇠)
    and ˆ(⇠) =
    1
    ˆ

    2
    (⇠)
    2
    Constant speed motion:
    sPDE: where
    Fourier in space:

    View full-size slide

  58. Stochastic PDE Models
    Dynamic Textures as Solutions of sPDE
    MC
    I
    with speed
    v0
    can be obtained from a MC
    I0
    with zero speed by the constant speed time
    rping
    I(x, t)
    def.
    = I0(x v0t, t).
    (2)
    now restrict our attention to
    I0
    .
    consider Gaussian random fields defined by a stochastic partial differential equation (sPDE) of
    form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    (3)
    is equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are stationary
    utions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. corresponding to
    temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in space and
    ?
    he spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at enforcing
    additional correlation in time of the model. Section 2.2 explains how to choose
    (↵, , ⌃W )
    so
    t the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    v0 = 0
    ),
    are motion clouds.
    is sPDE formulation is important since we aim to deal with dynamic stimulation, which should
    described by a causal equation which is local in time. This is crucial for numerical simulation
    explained in Section 2.4) but also to simplify the application of MC inside a bayesian model of
    ychophysical experiments (see Section 3).
    hile it is beyond the scope of this paper to study theoretically this equation, one can shows exis-
    ce and uniqueness results of stationary solutions for this class of sPDE under stability conditions
    the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfied in our
    I(x, t)
    def.
    = I0(x v0t, t).
    We now restrict our attention to
    I0
    .
    We consider Gaussian random fields defined by a stochastic partial differential equation (sPDE
    the form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    This equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are station
    solutions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. correspondin
    the temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in space an
    is the spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at enforc
    an additional correlation in time of the model. Section 2.2 explains how to choose
    (↵, , ⌃W )
    that the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    v0 =
    i.e. are motion clouds.
    This sPDE formulation is important since we aim to deal with dynamic stimulation, which sho
    be described by a causal equation which is local in time. This is crucial for numerical simulat
    (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian mode
    psychophysical experiments (see Section 3).
    While it is beyond the scope of this paper to study theoretically this equation, one can shows e
    tence and uniqueness results of stationary solutions for this class of sPDE under stability conditi
    on the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfied in
    simulations. Note also that one can show that in fact the stationary solutions to (3) all share
    same law. These solutions can be obtained by solving the sODE (4) forward for time
    t > t0
    w
    arbitrary boundary conditions at time
    t = t0
    , and letting
    t0
    ! 1. This is consistent with
    numerical scheme detailed in Section 2.4.
    063
    064
    065
    066
    067
    068
    069
    070
    071
    072
    073
    074
    075
    076
    077
    078
    079
    080
    081
    082
    083
    084
    085
    086
    087
    I(x, t)
    def.
    = I0(x v0t, t).
    We now restrict our attention to
    I0
    .
    We consider Gaussian random fields defined by a stochastic partial differential equation (
    the form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    This equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are s
    solutions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. corresp
    the temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in sp
    is the spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at
    an additional correlation in time of the model. Section 2.2 explains how to choose
    (↵,
    that the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    i.e. are motion clouds.
    This sPDE formulation is important since we aim to deal with dynamic stimulation, whic
    be described by a causal equation which is local in time. This is crucial for numerical s
    (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian
    psychophysical experiments (see Section 3).
    While it is beyond the scope of this paper to study theoretically this equation, one can sh
    tence and uniqueness results of stationary solutions for this class of sPDE under stability c
    on the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfi
    simulations. Note also that one can show that in fact the stationary solutions to (3) all
    same law. These solutions can be obtained by solving the sODE (4) forward for time
    t >
    arbitrary boundary conditions at time
    t = t0
    , and letting
    t0
    ! 1. This is consistent
    imulations. Note also that one can show that in fact the stationary solutions to (3) all share the
    ame law. These solutions can be obtained by solving the sODE (4) forward for time
    t > t0
    with
    rbitrary boundary conditions at time
    t = t0
    , and letting
    t0
    ! 1. This is consistent with the
    umerical scheme detailed in Section 2.4.
    .2 Equivalence Between Spectral and sPDE MC Formulations
    The sPDE equation (3) corresponds to a set of independent stochastic ODEs over the spatial Fourier
    omain, which reads, for each frequency

    ,
    8
    t
    2 R
    ,
    @
    2
    ˆ
    I0(⇠, t)
    @t
    2 + ˆ
    ↵(⇠)
    @ ˆ
    I0(⇠, t)
    @t
    + ˆ(⇠)ˆ
    I0(⇠, t) = ˆW (⇠) ˆ
    w(⇠, t)
    (4)
    where ˆ
    I0(⇠, t)
    denotes the Fourier transform with respect to the space variable
    x
    only. Here,
    ˆW (⇠)
    2
    s the spatial power spectrum of @W
    @t
    , which means that
    ⌃W (x, y) = c(x y)
    where
    ˆ
    c(⇠) = ˆ
    2
    W (⇠).
    (5)
    Here
    ˆ
    w(⇠, t)
    ⇠ N
    (0, 1)
    and
    w
    is a white noise in space and time. This formulation makes explicit
    hat

    ↵(⇠), ˆ(⇠))
    should be chosen in order to make the temporal covariance of the resulting process
    qual (or at least approximate) the temporal covariance appearing in (1) in the motion-less setting
    since we deal here with
    I0
    ), i.e. when
    v0 = 0
    . This covariance should be localized around 0 and
    on-oscillating. It thus make sense to constrain

    ↵(⇠), ˆ(⇠))
    for the corresponding ODE (4) to be
    ritically damped, which corresponds to imposing the following relationship
    8
    ⇠, ˆ
    ↵(⇠) =
    2
    ˆ
    ⌫(⇠)
    and ˆ(⇠) =
    1
    ˆ

    2
    (⇠)
    umerical scheme detailed in Section 2.4.
    .2 Equivalence Between Spectral and sPDE MC Formulations
    The sPDE equation (3) corresponds to a set of independent stochastic ODEs over the spatial Fourier
    omain, which reads, for each frequency

    ,
    8
    t
    2 R
    ,
    @
    2
    ˆ
    I0(⇠, t)
    @t
    2 + ˆ
    ↵(⇠)
    @ ˆ
    I0(⇠, t)
    @t
    + ˆ(⇠)ˆ
    I0(⇠, t) = ˆW (⇠) ˆ
    w(⇠, t)
    (4)
    where ˆ
    I0(⇠, t)
    denotes the Fourier transform with respect to the space variable
    x
    only. Here,
    ˆW (⇠)
    2
    s the spatial power spectrum of @W
    @t
    , which means that
    ⌃W (x, y) = c(x y)
    where
    ˆ
    c(⇠) = ˆ
    2
    W (⇠).
    (5)
    Here
    ˆ
    w(⇠, t)
    ⇠ N
    (0, 1)
    and
    w
    is a white noise in space and time. This formulation makes explicit
    hat

    ↵(⇠), ˆ(⇠))
    should be chosen in order to make the temporal covariance of the resulting process
    qual (or at least approximate) the temporal covariance appearing in (1) in the motion-less setting
    since we deal here with
    I0
    ), i.e. when
    v0 = 0
    . This covariance should be localized around 0 and
    on-oscillating. It thus make sense to constrain

    ↵(⇠), ˆ(⇠))
    for the corresponding ODE (4) to be
    ritically damped, which corresponds to imposing the following relationship
    8
    ⇠, ˆ
    ↵(⇠) =
    2
    ˆ
    ⌫(⇠)
    and ˆ(⇠) =
    1
    ˆ

    2
    (⇠)
    2
    093
    094
    095
    096
    097
    098
    099
    100
    101
    102
    103
    104
    105
    106
    107
    8
    t
    2 R
    ,
    @
    2
    ˆ
    I0(⇠, t)
    @t
    2 + ˆ
    ↵(⇠)
    @ ˆ
    I0(⇠, t)
    @t
    + ˆ(⇠)ˆ
    I0(⇠, t) = ˆW (⇠)
    where ˆ
    I0(⇠, t)
    denotes the Fourier transform with respect to the space variable
    x
    is the spatial power spectrum of @W
    @t
    , which means that
    ⌃W (x, y) = c(x y)
    where
    ˆ
    c(⇠) = ˆ
    2
    W (⇠).
    Here
    ˆ
    w(⇠, t)
    ⇠ N
    (0, 1)
    and
    w
    is a white noise in space and time. This formu
    that

    ↵(⇠), ˆ(⇠))
    should be chosen in order to make the temporal covariance of
    equal (or at least approximate) the temporal covariance appearing in (1) in th
    (since we deal here with
    I0
    ), i.e. when
    v0 = 0
    . This covariance should be lo
    non-oscillating. It thus make sense to constrain

    ↵(⇠), ˆ(⇠))
    for the correspo
    critically damped, which corresponds to imposing the following relationship
    8
    ⇠, ˆ
    ↵(⇠) =
    2
    ˆ
    ⌫(⇠)
    and ˆ(⇠) =
    1
    ˆ

    2
    (⇠)
    2
    Constant speed motion:
    sPDE: where
    Fourier in space:
    Optimal damping parameter choice:
    ˆ

    (

    ) : controls time correlation of frequency

    .

    View full-size slide

  59. Stochastic PDE Models
    Dynamic Textures as Solutions of sPDE
    MC
    I
    with speed
    v0
    can be obtained from a MC
    I0
    with zero speed by the constant speed time
    rping
    I(x, t)
    def.
    = I0(x v0t, t).
    (2)
    now restrict our attention to
    I0
    .
    consider Gaussian random fields defined by a stochastic partial differential equation (sPDE) of
    form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    (3)
    is equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are stationary
    utions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. corresponding to
    temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in space and
    ?
    he spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at enforcing
    additional correlation in time of the model. Section 2.2 explains how to choose
    (↵, , ⌃W )
    so
    t the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    v0 = 0
    ),
    are motion clouds.
    is sPDE formulation is important since we aim to deal with dynamic stimulation, which should
    described by a causal equation which is local in time. This is crucial for numerical simulation
    explained in Section 2.4) but also to simplify the application of MC inside a bayesian model of
    ychophysical experiments (see Section 3).
    hile it is beyond the scope of this paper to study theoretically this equation, one can shows exis-
    ce and uniqueness results of stationary solutions for this class of sPDE under stability conditions
    the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfied in our
    I(x, t)
    def.
    = I0(x v0t, t).
    We now restrict our attention to
    I0
    .
    We consider Gaussian random fields defined by a stochastic partial differential equation (sPDE
    the form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    This equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are station
    solutions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. correspondin
    the temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in space an
    is the spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at enforc
    an additional correlation in time of the model. Section 2.2 explains how to choose
    (↵, , ⌃W )
    that the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    v0 =
    i.e. are motion clouds.
    This sPDE formulation is important since we aim to deal with dynamic stimulation, which sho
    be described by a causal equation which is local in time. This is crucial for numerical simulat
    (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian mode
    psychophysical experiments (see Section 3).
    While it is beyond the scope of this paper to study theoretically this equation, one can shows e
    tence and uniqueness results of stationary solutions for this class of sPDE under stability conditi
    on the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfied in
    simulations. Note also that one can show that in fact the stationary solutions to (3) all share
    same law. These solutions can be obtained by solving the sODE (4) forward for time
    t > t0
    w
    arbitrary boundary conditions at time
    t = t0
    , and letting
    t0
    ! 1. This is consistent with
    numerical scheme detailed in Section 2.4.
    063
    064
    065
    066
    067
    068
    069
    070
    071
    072
    073
    074
    075
    076
    077
    078
    079
    080
    081
    082
    083
    084
    085
    086
    087
    I(x, t)
    def.
    = I0(x v0t, t).
    We now restrict our attention to
    I0
    .
    We consider Gaussian random fields defined by a stochastic partial differential equation (
    the form
    D
    (I0) =
    @W
    @t
    (x)
    where D
    (I0)
    def.
    =
    @
    2
    I0
    @t
    2 (x) + ↵ ?
    @I0
    @t
    (x) + ? I0(x)
    This equation should be satisfied for all
    (x, t)
    , and we look for Gaussian fields that are s
    solutions of this equation. In this sPDE, the driving noise @W
    @t
    is white in time (i.e. corresp
    the temporal derivative of a Brownian motion in time) and has a 2-D covariance
    ⌃W
    in sp
    is the spatial convolution operator. The parameters
    (↵, )
    are 2-D spatial filters that aim at
    an additional correlation in time of the model. Section 2.2 explains how to choose
    (↵,
    that the stationary solutions of (3) have the power spectrum given in (1) (in the case that
    i.e. are motion clouds.
    This sPDE formulation is important since we aim to deal with dynamic stimulation, whic
    be described by a causal equation which is local in time. This is crucial for numerical s
    (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian
    psychophysical experiments (see Section 3).
    While it is beyond the scope of this paper to study theoretically this equation, one can sh
    tence and uniqueness results of stationary solutions for this class of sPDE under stability c
    on the filers
    (↵, )
    (see for instance [8]) that we found numerically to be always satisfi
    simulations. Note also that one can show that in fact the stationary solutions to (3) all
    same law. These solutions can be obtained by solving the sODE (4) forward for time
    t >
    arbitrary boundary conditions at time
    t = t0
    , and letting
    t0
    ! 1. This is consistent
    imulations. Note also that one can show that in fact the stationary solutions to (3) all share the
    ame law. These solutions can be obtained by solving the sODE (4) forward for time
    t > t0
    with
    rbitrary boundary conditions at time
    t = t0
    , and letting
    t0
    ! 1. This is consistent with the
    umerical scheme detailed in Section 2.4.
    .2 Equivalence Between Spectral and sPDE MC Formulations
    The sPDE equation (3) corresponds to a set of independent stochastic ODEs over the spatial Fourier
    omain, which reads, for each frequency

    ,
    8
    t
    2 R
    ,
    @
    2
    ˆ
    I0(⇠, t)
    @t
    2 + ˆ
    ↵(⇠)
    @ ˆ
    I0(⇠, t)
    @t
    + ˆ(⇠)ˆ
    I0(⇠, t) = ˆW (⇠) ˆ
    w(⇠, t)
    (4)
    where ˆ
    I0(⇠, t)
    denotes the Fourier transform with respect to the space variable
    x
    only. Here,
    ˆW (⇠)
    2
    s the spatial power spectrum of @W
    @t
    , which means that
    ⌃W (x, y) = c(x y)
    where
    ˆ
    c(⇠) = ˆ
    2
    W (⇠).
    (5)
    Here
    ˆ
    w(⇠, t)
    ⇠ N
    (0, 1)
    and
    w
    is a white noise in space and time. This formulation makes explicit
    hat

    ↵(⇠), ˆ(⇠))
    should be chosen in order to make the temporal covariance of the resulting process
    qual (or at least approximate) the temporal covariance appearing in (1) in the motion-less setting
    since we deal here with
    I0
    ), i.e. when
    v0 = 0
    . This covariance should be localized around 0 and
    on-oscillating. It thus make sense to constrain

    ↵(⇠), ˆ(⇠))
    for the corresponding ODE (4) to be
    ritically damped, which corresponds to imposing the following relationship
    8
    ⇠, ˆ
    ↵(⇠) =
    2
    ˆ
    ⌫(⇠)
    and ˆ(⇠) =
    1
    ˆ

    2
    (⇠)
    umerical scheme detailed in Section 2.4.
    .2 Equivalence Between Spectral and sPDE MC Formulations
    The sPDE equation (3) corresponds to a set of independent stochastic ODEs over the spatial Fourier
    omain, which reads, for each frequency

    ,
    8
    t
    2 R
    ,
    @
    2
    ˆ
    I0(⇠, t)
    @t
    2 + ˆ
    ↵(⇠)
    @ ˆ
    I0(⇠, t)
    @t
    + ˆ(⇠)ˆ
    I0(⇠, t) = ˆW (⇠) ˆ
    w(⇠, t)
    (4)
    where ˆ
    I0(⇠, t)
    denotes the Fourier transform with respect to the space variable
    x
    only. Here,
    ˆW (⇠)
    2
    s the spatial power spectrum of @W
    @t
    , which means that
    ⌃W (x, y) = c(x y)
    where
    ˆ
    c(⇠) = ˆ
    2
    W (⇠).
    (5)
    Here
    ˆ
    w(⇠, t)
    ⇠ N
    (0, 1)
    and
    w
    is a white noise in space and time. This formulation makes explicit
    hat

    ↵(⇠), ˆ(⇠))
    should be chosen in order to make the temporal covariance of the resulting process
    qual (or at least approximate) the temporal covariance appearing in (1) in the motion-less setting
    since we deal here with
    I0
    ), i.e. when
    v0 = 0
    . This covariance should be localized around 0 and
    on-oscillating. It thus make sense to constrain

    ↵(⇠), ˆ(⇠))
    for the corresponding ODE (4) to be
    ritically damped, which corresponds to imposing the following relationship
    8
    ⇠, ˆ
    ↵(⇠) =
    2
    ˆ
    ⌫(⇠)
    and ˆ(⇠) =
    1
    ˆ

    2
    (⇠)
    2
    093
    094
    095
    096
    097
    098
    099
    100
    101
    102
    103
    104
    105
    106
    107
    8
    t
    2 R
    ,
    @
    2
    ˆ
    I0(⇠, t)
    @t
    2 + ˆ
    ↵(⇠)
    @ ˆ
    I0(⇠, t)
    @t
    + ˆ(⇠)ˆ
    I0(⇠, t) = ˆW (⇠)
    where ˆ
    I0(⇠, t)
    denotes the Fourier transform with respect to the space variable
    x
    is the spatial power spectrum of @W
    @t
    , which means that
    ⌃W (x, y) = c(x y)
    where
    ˆ
    c(⇠) = ˆ
    2
    W (⇠).
    Here
    ˆ
    w(⇠, t)
    ⇠ N
    (0, 1)
    and
    w
    is a white noise in space and time. This formu
    that

    ↵(⇠), ˆ(⇠))
    should be chosen in order to make the temporal covariance of
    equal (or at least approximate) the temporal covariance appearing in (1) in th
    (since we deal here with
    I0
    ), i.e. when
    v0 = 0
    . This covariance should be lo
    non-oscillating. It thus make sense to constrain

    ↵(⇠), ˆ(⇠))
    for the correspo
    critically damped, which corresponds to imposing the following relationship
    8
    ⇠, ˆ
    ↵(⇠) =
    2
    ˆ
    ⌫(⇠)
    and ˆ(⇠) =
    1
    ˆ

    2
    (⇠)
    2
    Constant speed motion:
    sPDE: where
    Fourier in space:
    Optimal damping parameter choice:
    ˆ

    (

    ) : controls time correlation of frequency

    .
    H(t) def.
    = e |t| (|t| + 1)
    Proposition:
    t
    ˆ
    ⌫(⇠)
    cov(ˆ
    I0(
    ⇠, ·
    ))t
    cov(ˆ
    I0(
    ⇠, ·
    ))t =
    ˆW (

    )
    2
    ˆ

    (

    )
    4
    H

    t
    ˆ

    (


    View full-size slide

  60. Equivalence with Motion Clouds
    115
    116
    117
    118
    119
    120
    121
    122
    123
    124
    125
    126
    127
    128
    129
    130
    131
    132
    133
    134
    135
    136
    137
    138
    139
    ||
    V v0
    ||
    V
    where L is defined in (1), equation (4) admits a solution
    I
    which is a stationary Gaussian
    power spectrum (1) when setting
    ˆ
    2
    W (⇠) =
    1
    ˆ
    ⌫(⇠)
    ||

    ||2
    P
    Z(
    ||

    ||
    )
    P
    ⇥(
    \
    ⇠),
    and
    ˆ
    ⌫(⇠) =
    1
    V
    ||

    ||.
    Proof. For this proof, we denote
    I
    MC the motion cloud defined by (1), and
    I
    a stationa
    of the sPDE defined by (3). We aim at showing that under the specification (7), they hav
    covariance. This is equivalent to showing that
    I
    MC
    0 (x, t) = I
    MC
    (x+ct, t)
    has the same cov
    I0
    . One shows that for any fixed

    , equation (4) admits a unique (in law) stationary soluti
    which is a stationary Gaussian process of zero mean and with a covariance which is
    ˆ
    where
    r
    is the impulse response (i.e. taking formally
    a =
    ) of the ODE
    r
    00
    + 2r
    0
    /u + r
    where we denoted
    u = ˆ
    ⌫(⇠)
    . This impulse response is easily shown to be
    r(t) = te
    The covariance of ˆ
    I0(⇠,
    ·
    )
    is thus, after some computation, equal to
    ˆ
    2
    W (⇠)r ? ¯
    r = ˆ
    2
    W
    where
    h(t)
    /
    (1 +
    |
    t
    |
    )e
    |
    t
    |. Taking the Fourier transform of this equality, the power spec
    I0
    thus reads
    ˆ0(⇠, ⌧) = ˆ
    2
    W (⇠)ˆ
    ⌫(⇠)h(ˆ
    ⌫(⇠)⌧)
    where
    h(u) =
    1
    (1 + u
    2
    )
    2
    and where it should be noted that this
    h
    function is the same as the one introduced in
    covariance MC of
    I
    MC and MC
    0
    of
    I
    MC
    0
    are related by the relation
    ˆ
    MC
    0 (⇠, ⌧) = ˆ
    MC
    (⇠, ⌧
    h
    ⇠, v0
    i
    ) =
    1
    ||

    ||2
    P
    Z(
    ||

    ||
    )
    P
    ⇥ (
    \
    ⇠) h


    V
    h
    ⇠, v0
    i

    .
    where we used the expression (1) for
    ˆ
    MC and the value of L
    (
    P
    ||
    V v0
    ||)
    given by (6). Co
    MC
    Theorem:
    selecting
    stationary solutions of the sPDE have covariance
    8 (⇠, ⌧) 2 R2 ⇥ R, ˆsPDE
    (⇠, ⌧) =
    P
    Z (||⇠||)
    ||⇠||2
    P
    ✓ (\⇠) h

    ⌧ + hv0, ⇠i
    V
    ||⇠||

    where h(u) = (1 + u2) 2

    View full-size slide

  61. Equivalence with Motion Clouds
    115
    116
    117
    118
    119
    120
    121
    122
    123
    124
    125
    126
    127
    128
    129
    130
    131
    132
    133
    134
    135
    136
    137
    138
    139
    ||
    V v0
    ||
    V
    where L is defined in (1), equation (4) admits a solution
    I
    which is a stationary Gaussian
    power spectrum (1) when setting
    ˆ
    2
    W (⇠) =
    1
    ˆ
    ⌫(⇠)
    ||

    ||2
    P
    Z(
    ||

    ||
    )
    P
    ⇥(
    \
    ⇠),
    and
    ˆ
    ⌫(⇠) =
    1
    V
    ||

    ||.
    Proof. For this proof, we denote
    I
    MC the motion cloud defined by (1), and
    I
    a stationa
    of the sPDE defined by (3). We aim at showing that under the specification (7), they hav
    covariance. This is equivalent to showing that
    I
    MC
    0 (x, t) = I
    MC
    (x+ct, t)
    has the same cov
    I0
    . One shows that for any fixed

    , equation (4) admits a unique (in law) stationary soluti
    which is a stationary Gaussian process of zero mean and with a covariance which is
    ˆ
    where
    r
    is the impulse response (i.e. taking formally
    a =
    ) of the ODE
    r
    00
    + 2r
    0
    /u + r
    where we denoted
    u = ˆ
    ⌫(⇠)
    . This impulse response is easily shown to be
    r(t) = te
    The covariance of ˆ
    I0(⇠,
    ·
    )
    is thus, after some computation, equal to
    ˆ
    2
    W (⇠)r ? ¯
    r = ˆ
    2
    W
    where
    h(t)
    /
    (1 +
    |
    t
    |
    )e
    |
    t
    |. Taking the Fourier transform of this equality, the power spec
    I0
    thus reads
    ˆ0(⇠, ⌧) = ˆ
    2
    W (⇠)ˆ
    ⌫(⇠)h(ˆ
    ⌫(⇠)⌧)
    where
    h(u) =
    1
    (1 + u
    2
    )
    2
    and where it should be noted that this
    h
    function is the same as the one introduced in
    covariance MC of
    I
    MC and MC
    0
    of
    I
    MC
    0
    are related by the relation
    ˆ
    MC
    0 (⇠, ⌧) = ˆ
    MC
    (⇠, ⌧
    h
    ⇠, v0
    i
    ) =
    1
    ||

    ||2
    P
    Z(
    ||

    ||
    )
    P
    ⇥ (
    \
    ⇠) h


    V
    h
    ⇠, v0
    i

    .
    where we used the expression (1) for
    ˆ
    MC and the value of L
    (
    P
    ||
    V v0
    ||)
    given by (6). Co
    MC
    Theorem:
    selecting
    stationary solutions of the sPDE have covariance
    8 (⇠, ⌧) 2 R2 ⇥ R, ˆ(⇠, ⌧) =
    P
    Z (||⇠||)
    ||⇠||2
    P
    ✓ (\⇠) L(P
    ||V v0
    ||
    )

    ⌧ + hv0, ⇠i
    ||⇠||

    Motion cloud covariance:
    8 (⇠, ⌧) 2 R2 ⇥ R, ˆsPDE
    (⇠, ⌧) =
    P
    Z (||⇠||)
    ||⇠||2
    P
    ✓ (\⇠) h

    ⌧ + hv0, ⇠i
    V
    ||⇠||

    where h(u) = (1 + u2) 2
    L
    (
    f
    )(
    u
    )
    def.
    =
    R ⇡/2
    ⇡/2
    f
    (
    u/
    cos(
    '
    ))d
    '

    View full-size slide

  62. Equivalence with Motion Clouds
    162
    163
    164
    165
    166
    167
    168
    169
    170
    171
    172
    173
    174
    175
    176
    177
    denoted
    u = ˆ
    ⌫(⇠)
    . This impulse response is easily shown to be
    r(t) = te t/u
    1R+
    (t)
    .
    iance of ˆ
    I0(⇠,
    ·
    )
    is thus, after some computation, equal to
    ˆ
    2
    W (⇠)r ? ¯
    r = ˆ
    2
    W (⇠)h(
    ·
    /u)
    )
    /
    (1 +
    |
    t
    |
    )e
    |
    t
    |. Taking the Fourier transform of this equality, the power spectrum
    ˆ0
    of
    ads
    ˆ0(⇠, ⌧) = ˆ
    2
    W (⇠)ˆ
    ⌫(⇠)h(ˆ
    ⌫(⇠)⌧)
    where
    h(u) =
    1
    (1 + u
    2
    )
    2
    e it should be noted that this
    h
    function is the same as the one introduced in (6). The
    e MC of
    I
    MC and MC
    0
    of
    I
    MC
    0
    are related by the relation
    ˆ
    MC
    0 (⇠, ⌧) = ˆ
    MC
    (⇠, ⌧
    h
    ⇠, v0
    i
    ) =
    1
    ||

    ||2
    P
    Z(
    ||

    ||
    )
    P
    ⇥ (
    \
    ⇠) h


    V
    h
    ⇠, v0
    i

    .
    used the expression (1) for
    ˆ
    MC and the value of L
    (
    P
    ||
    V v0
    ||)
    given by (6). Condition (7)
    s that expression (2.2) and (2.2) coincide, and thus
    ˆ0 = ˆ
    MC
    0
    .
    ression for
    P
    ||
    V v0
    ||
    (6) states that in order to obtain a perfect equivalence between the MC defined by (1) and
    e function has L 1
    (h)
    to be well-defined. It means we need to compute the inverse of the
    of the linear operator L
    8
    u
    2 R
    ,
    L
    (f)(u) =
    Z


    f( u/ cos('))d'.
    ction
    h
    . Manipulation of the integral defining L shows that one can actually write in closed
    (h)
    . The variable substitution
    x = cos(')
    allows to rewrite L as a Mellin convolution
    ld be then inverted using Mellin transform, see Figure 2. One obtains
    L 1
    (h)(u) =
    2 u
    2
    ⇡(1 + u
    2
    )
    2
    u
    2
    (u
    2
    + 4)(log(u) log(
    p
    u
    2
    + 1 + 1))
    ⇡(u
    2
    + 1)
    5
    /
    2 .
    115
    116
    117
    118
    119
    120
    121
    122
    123
    124
    125
    126
    127
    128
    129
    130
    131
    132
    133
    134
    135
    136
    137
    138
    139
    ||
    V v0
    ||
    V
    where L is defined in (1), equation (4) admits a solution
    I
    which is a stationary Gaussian
    power spectrum (1) when setting
    ˆ
    2
    W (⇠) =
    1
    ˆ
    ⌫(⇠)
    ||

    ||2
    P
    Z(
    ||

    ||
    )
    P
    ⇥(
    \
    ⇠),
    and
    ˆ
    ⌫(⇠) =
    1
    V
    ||

    ||.
    Proof. For this proof, we denote
    I
    MC the motion cloud defined by (1), and
    I
    a stationa
    of the sPDE defined by (3). We aim at showing that under the specification (7), they hav
    covariance. This is equivalent to showing that
    I
    MC
    0 (x, t) = I
    MC
    (x+ct, t)
    has the same cov
    I0
    . One shows that for any fixed

    , equation (4) admits a unique (in law) stationary soluti
    which is a stationary Gaussian process of zero mean and with a covariance which is
    ˆ
    where
    r
    is the impulse response (i.e. taking formally
    a =
    ) of the ODE
    r
    00
    + 2r
    0
    /u + r
    where we denoted
    u = ˆ
    ⌫(⇠)
    . This impulse response is easily shown to be
    r(t) = te
    The covariance of ˆ
    I0(⇠,
    ·
    )
    is thus, after some computation, equal to
    ˆ
    2
    W (⇠)r ? ¯
    r = ˆ
    2
    W
    where
    h(t)
    /
    (1 +
    |
    t
    |
    )e
    |
    t
    |. Taking the Fourier transform of this equality, the power spec
    I0
    thus reads
    ˆ0(⇠, ⌧) = ˆ
    2
    W (⇠)ˆ
    ⌫(⇠)h(ˆ
    ⌫(⇠)⌧)
    where
    h(u) =
    1
    (1 + u
    2
    )
    2
    and where it should be noted that this
    h
    function is the same as the one introduced in
    covariance MC of
    I
    MC and MC
    0
    of
    I
    MC
    0
    are related by the relation
    ˆ
    MC
    0 (⇠, ⌧) = ˆ
    MC
    (⇠, ⌧
    h
    ⇠, v0
    i
    ) =
    1
    ||

    ||2
    P
    Z(
    ||

    ||
    )
    P
    ⇥ (
    \
    ⇠) h


    V
    h
    ⇠, v0
    i

    .
    where we used the expression (1) for
    ˆ
    MC and the value of L
    (
    P
    ||
    V v0
    ||)
    given by (6). Co
    MC
    Theorem:
    selecting
    stationary solutions of the sPDE have covariance
    8 (⇠, ⌧) 2 R2 ⇥ R, ˆ(⇠, ⌧) =
    P
    Z (||⇠||)
    ||⇠||2
    P
    ✓ (\⇠) L(P
    ||V v0
    ||
    )

    ⌧ + hv0, ⇠i
    ||⇠||

    Motion cloud covariance:
    8 (⇠, ⌧) 2 R2 ⇥ R, ˆsPDE
    (⇠, ⌧) =
    P
    Z (||⇠||)
    ||⇠||2
    P
    ✓ (\⇠) h

    ⌧ + hv0, ⇠i
    V
    ||⇠||

    where h(u) = (1 + u2) 2
    Proposition: perfect equivalence for
    P
    ||V v0
    ||
    = L 1(h)(·/ V )
    L
    (
    f
    )(
    u
    )
    def.
    =
    R ⇡/2
    ⇡/2
    f
    (
    u/
    cos(
    '
    ))d
    '

    View full-size slide

  63. Numerical Implementation
    187
    188
    189
    190
    191
    192
    193
    194
    195
    196
    197
    198
    199
    200
    201
    202
    203
    204
    205
    206
    207
    208
    209
    210
    211
    fields typically decays too fast in time. The detailed derivation of the AR(2) implementation
    can be found in the supplementary materials.
    The discretization computes a (possibly infinite) discrete set of 2-D frames
    (I
    (
    `
    )
    0 )`
    >
    `0
    separa
    a time step , and we approach at time
    t = `
    the derivatives as
    @I0(
    ·
    , t)
    @t
    ⇡ 1
    (I
    (
    `
    )
    0 I
    (
    `
    1)
    0 )
    and @
    2
    I0(
    ·
    , t)
    @t
    2
    ⇡ 2
    (I
    (
    `
    +1)
    0 + I
    (
    `
    1)
    0 2I
    (
    `
    )
    0 ),
    which leads to the following explicit recursion
    8
    `
    >
    `0, I
    (
    `
    +1)
    0 = (2 ↵
    2
    ) ? I
    (
    `
    )
    0 + ( + ↵) ? I
    (
    `
    1)
    0 +
    2
    W
    (
    `
    )
    ,
    where is the 2-D Dirac distribution and where
    (W
    (
    `
    )
    )`
    are i.i.d. 2-D Gaussian field with di
    tion N
    (0, ⌃W )
    , and
    (I
    (
    `0 1)
    0 , I
    (
    `0 1)
    0 )
    can be arbitrary initialized.
    One can show that when
    `0
    ! 1 (to allow for a long enough “warmup” phase to reach a
    imate time-stationarity) and !
    0
    , then
    I0
    defined by interpolating
    I0 (
    ·
    , `) = I
    (
    `
    ) con
    (in the sense of finite dimensional distributions) toward a solution
    I0
    of the sPDE (3). W
    to [9] for a similar result in the 1-D case (stochastic ODE). We implemented the recursion
    computing the 2-D convolutions with FFT’s on a GPU, which allows us to generate high res
    videos in real time, without the need to explicitly store the synthesized video.
    3 Experimental Likelihood vs. the MC Model
    In our paper, we propose to directly fit the likelihood P
    M
    |
    V,Z(m
    |
    v, z)
    from the experiment
    chophysical curve. While this makes sense from a data-analysis point of view, this required
    modeling hypothesis, in particular, that the likelihood is Gaussian with a variance 2
    z
    indep
    of the parameter
    v
    to be estimated by the observer.
    s formulation (3). Indeed, numerical simulations show that AR(1)
    mporal artifacts: in particular, the time correlation of AR(1) random
    time. The detailed derivation of the AR(2) implementation of MC
    y materials.
    ossibly infinite) discrete set of 2-D frames
    (I
    (
    `
    )
    0 )`
    >
    `0
    separated by
    at time
    t = `
    the derivatives as
    (
    `
    1)
    0 )
    and @
    2
    I0(
    ·
    , t)
    @t
    2
    ⇡ 2
    (I
    (
    `
    +1)
    0 + I
    (
    `
    1)
    0 2I
    (
    `
    )
    0 ),
    licit recursion

    2
    ) ? I
    (
    `
    )
    0 + ( + ↵) ? I
    (
    `
    1)
    0 +
    2
    W
    (
    `
    )
    ,
    (8)
    tion and where
    (W
    (
    `
    )
    )`
    are i.i.d. 2-D Gaussian field with distribu-
    0 1)
    )
    can be arbitrary initialized.
    1 (to allow for a long enough “warmup” phase to reach approx-
    !
    0
    , then
    I0
    defined by interpolating
    I0 (
    ·
    , `) = I
    (
    `
    ) converges
    nal distributions) toward a solution
    I0
    of the sPDE (3). We refer
    1-D case (stochastic ODE). We implemented the recursion (8) by
    with FFT’s on a GPU, which allows us to generate high resolution
    eed to explicitly store the synthesized video.
    ood vs. the MC Model
    ctly fit the likelihood P
    M
    |
    V,Z(m
    |
    v, z)
    from the experimental psy-
    Discretization:
    finite di↵erence in time

    View full-size slide

  64. Numerical Implementation
    187
    188
    189
    190
    191
    192
    193
    194
    195
    196
    197
    198
    199
    200
    201
    202
    203
    204
    205
    206
    207
    208
    209
    210
    211
    fields typically decays too fast in time. The detailed derivation of the AR(2) implementation
    can be found in the supplementary materials.
    The discretization computes a (possibly infinite) discrete set of 2-D frames
    (I
    (
    `
    )
    0 )`
    >
    `0
    separa
    a time step , and we approach at time
    t = `
    the derivatives as
    @I0(
    ·
    , t)
    @t
    ⇡ 1
    (I
    (
    `
    )
    0 I
    (
    `
    1)
    0 )
    and @
    2
    I0(
    ·
    , t)
    @t
    2
    ⇡ 2
    (I
    (
    `
    +1)
    0 + I
    (
    `
    1)
    0 2I
    (
    `
    )
    0 ),
    which leads to the following explicit recursion
    8
    `
    >
    `0, I
    (
    `
    +1)
    0 = (2 ↵
    2
    ) ? I
    (
    `
    )
    0 + ( + ↵) ? I
    (
    `
    1)
    0 +
    2
    W
    (
    `
    )
    ,
    where is the 2-D Dirac distribution and where
    (W
    (
    `
    )
    )`
    are i.i.d. 2-D Gaussian field with di
    tion N
    (0, ⌃W )
    , and
    (I
    (
    `0 1)
    0 , I
    (
    `0 1)
    0 )
    can be arbitrary initialized.
    One can show that when
    `0
    ! 1 (to allow for a long enough “warmup” phase to reach a
    imate time-stationarity) and !
    0
    , then
    I0
    defined by interpolating
    I0 (
    ·
    , `) = I
    (
    `
    ) con
    (in the sense of finite dimensional distributions) toward a solution
    I0
    of the sPDE (3). W
    to [9] for a similar result in the 1-D case (stochastic ODE). We implemented the recursion
    computing the 2-D convolutions with FFT’s on a GPU, which allows us to generate high res
    videos in real time, without the need to explicitly store the synthesized video.
    3 Experimental Likelihood vs. the MC Model
    In our paper, we propose to directly fit the likelihood P
    M
    |
    V,Z(m
    |
    v, z)
    from the experiment
    chophysical curve. While this makes sense from a data-analysis point of view, this required
    modeling hypothesis, in particular, that the likelihood is Gaussian with a variance 2
    z
    indep
    of the parameter
    v
    to be estimated by the observer.
    s formulation (3). Indeed, numerical simulations show that AR(1)
    mporal artifacts: in particular, the time correlation of AR(1) random
    time. The detailed derivation of the AR(2) implementation of MC
    y materials.
    ossibly infinite) discrete set of 2-D frames
    (I
    (
    `
    )
    0 )`
    >
    `0
    separated by
    at time
    t = `
    the derivatives as
    (
    `
    1)
    0 )
    and @
    2
    I0(
    ·
    , t)
    @t
    2
    ⇡ 2
    (I
    (
    `
    +1)
    0 + I
    (
    `
    1)
    0 2I
    (
    `
    )
    0 ),
    licit recursion

    2
    ) ? I
    (
    `
    )
    0 + ( + ↵) ? I
    (
    `
    1)
    0 +
    2
    W
    (
    `
    )
    ,
    (8)
    tion and where
    (W
    (
    `
    )
    )`
    are i.i.d. 2-D Gaussian field with distribu-
    0 1)
    )
    can be arbitrary initialized.
    1 (to allow for a long enough “warmup” phase to reach approx-
    !
    0
    , then
    I0
    defined by interpolating
    I0 (
    ·
    , `) = I
    (
    `
    ) converges
    nal distributions) toward a solution
    I0
    of the sPDE (3). We refer
    1-D case (stochastic ODE). We implemented the recursion (8) by
    with FFT’s on a GPU, which allows us to generate high resolution
    eed to explicitly store the synthesized video.
    ood vs. the MC Model
    ctly fit the likelihood P
    M
    |
    V,Z(m
    |
    v, z)
    from the experimental psy-
    AR(2) regression (in place of a first order AR(1) model). Using higher order recursions is crucia
    be consistent with the continuous formulation (3). Indeed, numerical simulations show that AR
    iterations lead to unacceptable temporal artifacts: in particular, the time correlation of AR(1) rand
    fields typically decays too fast in time. The detailed derivation of the AR(2) implementation of M
    can be found in the supplementary materials.
    The discretization computes a (possibly infinite) discrete set of 2-D frames
    (I
    (
    `
    )
    0 )`
    >
    `0
    separated
    a time step , and we approach at time
    t = `
    the derivatives as
    @I0(
    ·
    , t)
    @t
    ⇡ 1
    (I
    (
    `
    )
    0 I
    (
    `
    1)
    0 )
    and @
    2
    I0(
    ·
    , t)
    @t
    2
    ⇡ 2
    (I
    (
    `
    +1)
    0 + I
    (
    `
    1)
    0 2I
    (
    `
    )
    0 ),
    which leads to the following explicit recursion
    8
    `
    >
    `0, I
    (
    `
    +1)
    0 = (2 ↵
    2
    ) ? I
    (
    `
    )
    0 + ( + ↵) ? I
    (
    `
    1)
    0 +
    2
    W
    (
    `
    )
    ,
    where is the 2-D Dirac distribution and where
    (W
    (
    `
    )
    )`
    are i.i.d. 2-D Gaussian field with distri
    tion N
    (0, ⌃W )
    , and
    (I
    (
    `0 1)
    0 , I
    (
    `0 1)
    0 )
    can be arbitrary initialized.
    One can show that when
    `0
    ! 1 (to allow for a long enough “warmup” phase to reach appr
    imate time-stationarity) and !
    0
    , then
    I0
    defined by interpolating
    I0 (
    ·
    , `) = I
    (
    `
    ) conver
    (in the sense of finite dimensional distributions) toward a solution
    I0
    of the sPDE (3). We re
    to [9] for a similar result in the 1-D case (stochastic ODE). We implemented the recursion (8)
    computing the 2-D convolutions with FFT’s on a GPU, which allows us to generate high resolut
    videos in real time, without the need to explicitly store the synthesized video.
    3 Experimental Likelihood vs. the MC Model
    Discretization:
    finite di↵erence in time
    Auto-regressive model:
    AR(2)

    View full-size slide

  65. Numerical Implementation
    187
    188
    189
    190
    191
    192
    193
    194
    195
    196
    197
    198
    199
    200
    201
    202
    203
    204
    205
    206
    207
    208
    209
    210
    211
    fields typically decays too fast in time. The detailed derivation of the AR(2) implementation
    can be found in the supplementary materials.
    The discretization computes a (possibly infinite) discrete set of 2-D frames
    (I
    (
    `
    )
    0 )`
    >
    `0
    separa
    a time step , and we approach at time
    t = `
    the derivatives as
    @I0(
    ·
    , t)
    @t
    ⇡ 1
    (I
    (
    `
    )
    0 I
    (
    `
    1)
    0 )
    and @
    2
    I0(
    ·
    , t)
    @t
    2
    ⇡ 2
    (I
    (
    `
    +1)
    0 + I
    (
    `
    1)
    0 2I
    (
    `
    )
    0 ),
    which leads to the following explicit recursion
    8
    `
    >
    `0, I
    (
    `
    +1)
    0 = (2 ↵
    2
    ) ? I
    (
    `
    )
    0 + ( + ↵) ? I
    (
    `
    1)
    0 +
    2
    W
    (
    `
    )
    ,
    where is the 2-D Dirac distribution and where
    (W
    (
    `
    )
    )`
    are i.i.d. 2-D Gaussian field with di
    tion N
    (0, ⌃W )
    , and
    (I
    (
    `0 1)
    0 , I
    (
    `0 1)
    0 )
    can be arbitrary initialized.
    One can show that when
    `0
    ! 1 (to allow for a long enough “warmup” phase to reach a
    imate time-stationarity) and !
    0
    , then
    I0
    defined by interpolating
    I0 (
    ·
    , `) = I
    (
    `
    ) con
    (in the sense of finite dimensional distributions) toward a solution
    I0
    of the sPDE (3). W
    to [9] for a similar result in the 1-D case (stochastic ODE). We implemented the recursion
    computing the 2-D convolutions with FFT’s on a GPU, which allows us to generate high res
    videos in real time, without the need to explicitly store the synthesized video.
    3 Experimental Likelihood vs. the MC Model
    In our paper, we propose to directly fit the likelihood P
    M
    |
    V,Z(m
    |
    v, z)
    from the experiment
    chophysical curve. While this makes sense from a data-analysis point of view, this required
    modeling hypothesis, in particular, that the likelihood is Gaussian with a variance 2
    z
    indep
    of the parameter
    v
    to be estimated by the observer.
    s formulation (3). Indeed, numerical simulations show that AR(1)
    mporal artifacts: in particular, the time correlation of AR(1) random
    time. The detailed derivation of the AR(2) implementation of MC
    y materials.
    ossibly infinite) discrete set of 2-D frames
    (I
    (
    `
    )
    0 )`
    >
    `0
    separated by
    at time
    t = `
    the derivatives as
    (
    `
    1)
    0 )
    and @
    2
    I0(
    ·
    , t)
    @t
    2
    ⇡ 2
    (I
    (
    `
    +1)
    0 + I
    (
    `
    1)
    0 2I
    (
    `
    )
    0 ),
    licit recursion

    2
    ) ? I
    (
    `
    )
    0 + ( + ↵) ? I
    (
    `
    1)
    0 +
    2
    W
    (
    `
    )
    ,
    (8)
    tion and where
    (W
    (
    `
    )
    )`
    are i.i.d. 2-D Gaussian field with distribu-
    0 1)
    )
    can be arbitrary initialized.
    1 (to allow for a long enough “warmup” phase to reach approx-
    !
    0
    , then
    I0
    defined by interpolating
    I0 (
    ·
    , `) = I
    (
    `
    ) converges
    nal distributions) toward a solution
    I0
    of the sPDE (3). We refer
    1-D case (stochastic ODE). We implemented the recursion (8) by
    with FFT’s on a GPU, which allows us to generate high resolution
    eed to explicitly store the synthesized video.
    ood vs. the MC Model
    ctly fit the likelihood P
    M
    |
    V,Z(m
    |
    v, z)
    from the experimental psy-
    AR(2) regression (in place of a first order AR(1) model). Using higher order recursions is crucia
    be consistent with the continuous formulation (3). Indeed, numerical simulations show that AR
    iterations lead to unacceptable temporal artifacts: in particular, the time correlation of AR(1) rand
    fields typically decays too fast in time. The detailed derivation of the AR(2) implementation of M
    can be found in the supplementary materials.
    The discretization computes a (possibly infinite) discrete set of 2-D frames
    (I
    (
    `
    )
    0 )`
    >
    `0
    separated
    a time step , and we approach at time
    t = `
    the derivatives as
    @I0(
    ·
    , t)
    @t
    ⇡ 1
    (I
    (
    `
    )
    0 I
    (
    `
    1)
    0 )
    and @
    2
    I0(
    ·
    , t)
    @t
    2
    ⇡ 2
    (I
    (
    `
    +1)
    0 + I
    (
    `
    1)
    0 2I
    (
    `
    )
    0 ),
    which leads to the following explicit recursion
    8
    `
    >
    `0, I
    (
    `
    +1)
    0 = (2 ↵
    2
    ) ? I
    (
    `
    )
    0 + ( + ↵) ? I
    (
    `
    1)
    0 +
    2
    W
    (
    `
    )
    ,
    where is the 2-D Dirac distribution and where
    (W
    (
    `
    )
    )`
    are i.i.d. 2-D Gaussian field with distri
    tion N
    (0, ⌃W )
    , and
    (I
    (
    `0 1)
    0 , I
    (
    `0 1)
    0 )
    can be arbitrary initialized.
    One can show that when
    `0
    ! 1 (to allow for a long enough “warmup” phase to reach appr
    imate time-stationarity) and !
    0
    , then
    I0
    defined by interpolating
    I0 (
    ·
    , `) = I
    (
    `
    ) conver
    (in the sense of finite dimensional distributions) toward a solution
    I0
    of the sPDE (3). We re
    to [9] for a similar result in the 1-D case (stochastic ODE). We implemented the recursion (8)
    computing the 2-D convolutions with FFT’s on a GPU, which allows us to generate high resolut
    videos in real time, without the need to explicitly store the synthesized video.
    3 Experimental Likelihood vs. the MC Model
    Discretization:
    finite di↵erence in time
    Auto-regressive model:
    AR(2)

    View full-size slide

  66. Slow-fast Morphing
    Slow translating grating morphs into
    fast translating isotropic noise with
    orthogonal direction
    Non-stationarity in time: change (
    ↵, , W ) with
    `
    , rotations/zooms, etc.
    Translating anisotropic noise
    morphs into isotropic noise

    View full-size slide

  67. Slow-fast Morphing
    Slow translating grating morphs into
    fast translating isotropic noise with
    orthogonal direction
    Non-stationarity in time: change (
    ↵, , W ) with
    `
    , rotations/zooms, etc.
    Translating anisotropic noise
    morphs into isotropic noise

    View full-size slide

  68. Natural Parameter Morphing
    Natural noise morphing from
    rough to smooth
    Natural noise morphing from
    anisotropic to isotropic
    Natural noise with
    rotating anisotropy

    View full-size slide

  69. Natural Parameter Morphing
    Natural noise morphing from
    rough to smooth
    Natural noise morphing from
    anisotropic to isotropic
    Natural noise with
    rotating anisotropy

    View full-size slide

  70. Rotations / Zooms

    View full-size slide

  71. Rotations / Zooms

    View full-size slide

  72. Overview
    • Gaussian Texture Synthesis by Example
    • Spot Noise Models and Motion Clouds
    • Stochastic PDE Models
    • Bayesian Brain and Prior Estimation

    View full-size slide

  73. (v1, z1)
    Psychophysic Experiment
    Discrimination task: horizontal speed
    v0 = (
    v,
    0).
    Two alternative forced choice (2AFC)
    Nuisance: spatial frequency z def.
    = z0.
    232
    233
    234
    235
    236
    237
    238
    239
    240
    241
    242
    243
    244
    245
    246
    247
    248
    249
    250
    251
    252
    253
    254
    255
    256
    257
    258
    259
    260
    The distributions of the parameters are thus chosen as
    P
    Z(z)
    / z0
    z
    e
    ln(
    z
    z
    0
    )
    2
    2 ln(1+ 2
    Z ) ,
    P
    ⇥(✓)
    /
    e
    cos(2(
    ✓ ✓
    0))
    2

    and P
    ||
    V v0
    ||(r)
    /
    e
    r2
    2 2
    V
    .
    (6)
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a
    graphical display of the influence of these parameters.
    Speed Freq. orient. Freq. amplitude
    (mean, dispersion)
    (v0, V ) (✓0, ⇥) (z0, Z)
    Remark 2. Note that the final envelope of
    ˆ
    is in agreement with the formulation that is used in [8].
    232
    233
    234
    235
    236
    237
    238
    239
    240
    241
    242
    243
    244
    245
    246
    247
    248
    249
    250
    251
    252
    253
    254
    255
    256
    257
    258
    259
    260
    The distributions of the parameters are thus chosen as
    P
    Z(z)
    / z0
    z
    e
    ln(
    z
    z
    0
    )
    2
    2 ln(1+ 2
    Z ) ,
    P
    ⇥(✓)
    /
    e
    cos(2(
    ✓ ✓
    0))
    2

    and P
    ||
    V v0
    ||(r)
    /
    e
    r2
    2 2
    V
    .
    (6)
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a
    graphical display of the influence of these parameters.
    Speed Freq. orient. Freq. amplitude
    (mean, dispersion)
    (v0, V ) (✓0, ⇥) (z0, Z)
    Remark 2. Note that the final envelope of
    ˆ
    is in agreement with the formulation that is used in [8].
    (v2, z2)
    (v1, z1) (v2, z2)

    View full-size slide

  74. (v1, z1)
    Psychophysic Experiment
    Discrimination task: horizontal speed
    v0 = (
    v,
    0).
    Two alternative forced choice (2AFC)
    Nuisance: spatial frequency z def.
    = z0.
    232
    233
    234
    235
    236
    237
    238
    239
    240
    241
    242
    243
    244
    245
    246
    247
    248
    249
    250
    251
    252
    253
    254
    255
    256
    257
    258
    259
    260
    The distributions of the parameters are thus chosen as
    P
    Z(z)
    / z0
    z
    e
    ln(
    z
    z
    0
    )
    2
    2 ln(1+ 2
    Z ) ,
    P
    ⇥(✓)
    /
    e
    cos(2(
    ✓ ✓
    0))
    2

    and P
    ||
    V v0
    ||(r)
    /
    e
    r2
    2 2
    V
    .
    (6)
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a
    graphical display of the influence of these parameters.
    Speed Freq. orient. Freq. amplitude
    (mean, dispersion)
    (v0, V ) (✓0, ⇥) (z0, Z)
    Remark 2. Note that the final envelope of
    ˆ
    is in agreement with the formulation that is used in [8].
    232
    233
    234
    235
    236
    237
    238
    239
    240
    241
    242
    243
    244
    245
    246
    247
    248
    249
    250
    251
    252
    253
    254
    255
    256
    257
    258
    259
    260
    The distributions of the parameters are thus chosen as
    P
    Z(z)
    / z0
    z
    e
    ln(
    z
    z
    0
    )
    2
    2 ln(1+ 2
    Z ) ,
    P
    ⇥(✓)
    /
    e
    cos(2(
    ✓ ✓
    0))
    2

    and P
    ||
    V v0
    ||(r)
    /
    e
    r2
    2 2
    V
    .
    (6)
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a
    graphical display of the influence of these parameters.
    Speed Freq. orient. Freq. amplitude
    (mean, dispersion)
    (v0, V ) (✓0, ⇥) (z0, Z)
    Remark 2. Note that the final envelope of
    ˆ
    is in agreement with the formulation that is used in [8].
    (v2, z2)
    (v1, z1) (v2, z2)

    View full-size slide

  75. Empirical Psychometric Curves
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the participant to report
    which one of the two intervals was perceived as moving faster by pressing one of two buttons, that
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected so that

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16, 0.47
    }
    ,
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across trials), and where
    z
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitions of each of the 25
    possible combinations of these parameters are made per block of 250 trials and at least four such
    blocks were collected per condition tested. The outcome of these experiments are summarized by
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)
    is the
    empirical probability (averaged over the 40 trials) that a stimulus generated with parameters
    (v?
    , z)
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    To asses the validity of our model, we tested four different scenarios by considering all possible
    choices among
    z?
    = 0.78
    c/
    , v? 2 {
    5
    /s
    , 10
    /s}
    , ⌧? 2 {
    0.1s, 0.2s
    }
    ,
    which corresponds to combinations of low/high speeds and temporal frequency bandwidths. Stimuli
    were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic p227f monitor with
    resolution
    1024

    768
    at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9
    controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three observers
    with normal or corrected to normal vision were used. They gave their informed consent and the
    experiments received ethical approval from the Aix-Marseille Ethics Committee in accordance with
    the declaration of Helsinki.
    a small dark fixation spot was followed by two stimulus intervals of
    250
    ms each, separated by a
    grey 250 ms inter-stimulus interval. The first stimulus has parameters
    (v1, z1)
    and the second one
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the participant to report
    which one of the two intervals was perceived as moving faster by pressing one of two buttons, that
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected so that

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16, 0.47
    }
    ,
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across trials), and where
    z
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitions of each of the 25
    possible combinations of these parameters are made per block of 250 trials and at least four such
    blocks were collected per condition tested. The outcome of these experiments are summarized by
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)
    is the
    empirical probability (averaged over the 40 trials) that a stimulus generated with parameters
    (v?
    , z)
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    To asses the validity of our model, we tested four different scenarios by considering all possible
    choices among
    z?
    = 0.78
    c/
    , v? 2 {
    5
    /s
    , 10
    /s}
    , ⌧? 2 {
    0.1s, 0.2s
    }
    ,
    which corresponds to combinations of low/high speeds and temporal frequency bandwidths. Stimuli
    were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic p227f monitor with
    resolution
    1024

    768
    at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9
    controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three observers
    with normal or corrected to normal vision were used. They gave their informed consent and the
    experiments received ethical approval from the Aix-Marseille Ethics Committee in accordance with
    the declaration of Helsinki.
    0 V z0
    frequency bandwidth, as illustrated on the left of Figure 2.
    oice (2AFC) paradigm. In each trial a grey fixation screen with
    wed by two stimulus intervals of
    250
    ms each, separated by a
    The first stimulus has parameters
    (v1, z1)
    and the second one
    of the trial, a grey screen appears asking the participant to report
    perceived as moving faster by pressing one of two buttons, that
    r each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected so that
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16, 0.47
    }
    ,
    , 1)
    (i.e. the ordering is randomized across trials), and where
    z
    egree (c/ ) and
    v
    values in /s. Ten repetitions of each of the 25
    ameters are made per block of 250 trials and at least four such
    tested. The outcome of these experiments are summarized by
    or all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)
    is the
    r the 40 trials) that a stimulus generated with parameters
    (v?
    , z)
    h parameters
    (v, z?
    )
    .
    we tested four different scenarios by considering all possible
    v? 2 {
    5
    /s
    , 10
    /s}
    , ⌧? 2 {
    0.1s, 0.2s
    }
    ,
    of low/high speeds and temporal frequency bandwidths. Stimuli
    S 10.6.8 and displayed on a 20” Viewsonic p227f monitor with
    utines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9
    ervers sat 57 cm from the screen in a dark room. Three observers
    v
    z
    (v?, z?)

    View full-size slide

  76. Empirical Psychometric Curves
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the participant to report
    which one of the two intervals was perceived as moving faster by pressing one of two buttons, that
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected so that

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16, 0.47
    }
    ,
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across trials), and where
    z
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitions of each of the 25
    possible combinations of these parameters are made per block of 250 trials and at least four such
    blocks were collected per condition tested. The outcome of these experiments are summarized by
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)
    is the
    empirical probability (averaged over the 40 trials) that a stimulus generated with parameters
    (v?
    , z)
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    To asses the validity of our model, we tested four different scenarios by considering all possible
    choices among
    z?
    = 0.78
    c/
    , v? 2 {
    5
    /s
    , 10
    /s}
    , ⌧? 2 {
    0.1s, 0.2s
    }
    ,
    which corresponds to combinations of low/high speeds and temporal frequency bandwidths. Stimuli
    were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic p227f monitor with
    resolution
    1024

    768
    at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9
    controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three observers
    with normal or corrected to normal vision were used. They gave their informed consent and the
    experiments received ethical approval from the Aix-Marseille Ethics Committee in accordance with
    the declaration of Helsinki.
    a small dark fixation spot was followed by two stimulus intervals of
    250
    ms each, separated by a
    grey 250 ms inter-stimulus interval. The first stimulus has parameters
    (v1, z1)
    and the second one
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the participant to report
    which one of the two intervals was perceived as moving faster by pressing one of two buttons, that
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected so that

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16, 0.47
    }
    ,
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across trials), and where
    z
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitions of each of the 25
    possible combinations of these parameters are made per block of 250 trials and at least four such
    blocks were collected per condition tested. The outcome of these experiments are summarized by
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)
    is the
    empirical probability (averaged over the 40 trials) that a stimulus generated with parameters
    (v?
    , z)
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    To asses the validity of our model, we tested four different scenarios by considering all possible
    choices among
    z?
    = 0.78
    c/
    , v? 2 {
    5
    /s
    , 10
    /s}
    , ⌧? 2 {
    0.1s, 0.2s
    }
    ,
    which corresponds to combinations of low/high speeds and temporal frequency bandwidths. Stimuli
    were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic p227f monitor with
    resolution
    1024

    768
    at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9
    controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three observers
    with normal or corrected to normal vision were used. They gave their informed consent and the
    experiments received ethical approval from the Aix-Marseille Ethics Committee in accordance with
    the declaration of Helsinki.
    Output:
    psychometric curves
    ˆ
    'v?,z?
    (·, z?)
    0
    1
    v
    v?
    ˆ
    'v?,z?
    (·, z 6= z?)
    0
    1
    v
    v?
    Bias
    0 V z0
    frequency bandwidth, as illustrated on the left of Figure 2.
    oice (2AFC) paradigm. In each trial a grey fixation screen with
    wed by two stimulus intervals of
    250
    ms each, separated by a
    The first stimulus has parameters
    (v1, z1)
    and the second one
    of the trial, a grey screen appears asking the participant to report
    perceived as moving faster by pressing one of two buttons, that
    r each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected so that
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16, 0.47
    }
    ,
    , 1)
    (i.e. the ordering is randomized across trials), and where
    z
    egree (c/ ) and
    v
    values in /s. Ten repetitions of each of the 25
    ameters are made per block of 250 trials and at least four such
    tested. The outcome of these experiments are summarized by
    or all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)
    is the
    r the 40 trials) that a stimulus generated with parameters
    (v?
    , z)
    h parameters
    (v, z?
    )
    .
    we tested four different scenarios by considering all possible
    v? 2 {
    5
    /s
    , 10
    /s}
    , ⌧? 2 {
    0.1s, 0.2s
    }
    ,
    of low/high speeds and temporal frequency bandwidths. Stimuli
    S 10.6.8 and displayed on a 20” Viewsonic p227f monitor with
    utines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9
    ervers sat 57 cm from the screen in a dark room. Three observers
    v
    z
    (v?, z?)
    ˆ
    'v?,z?
    (v, z) def.
    = “empirical probability that MC with parameter (
    v?, z
    )
    is perceived faster than MC with parameter (v, z?)”

    View full-size slide

  77. MAP Decision Process
    v
    z
    P
    M|V,Z
    random
    noise
    stimuli
    likelihood
    nuisance
    parameter m
    internal
    representation

    View full-size slide

  78. MAP Decision Process
    v
    z
    P
    M|V,Z
    P
    V |Z
    ˆ
    v = ˆ
    vz(m)
    random
    noise
    stimuli
    deterministic
    prior
    likelihood
    nuisance
    parameter m
    internal
    representation

    View full-size slide

  79. MAP Decision Process
    313
    314
    315
    316
    317
    318
    319
    320
    321
    322
    323
    resolution
    1024

    768
    at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolb
    controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three o
    with normal or corrected to normal vision were used. They gave their informed consent
    experiments received ethical approval from the Aix-Marseille Ethics Committee in accorda
    the declaration of Helsinki.
    3.2 Bayesian modeling
    To make full use of our MC paradigm in analyzing the obtained results, we follow the meth
    of the Bayesian observer used for instance in [12]. We assume the observer makes its decisi
    a Maximum A Posteriori (MAP) estimator
    ˆ
    vz(m) = argmin
    v
    [ log(
    P
    M
    |
    V,Z(m
    |
    v, z)) log(
    P
    V
    |
    Z(v
    |
    z))]
    6
    v
    z
    P
    M|V,Z
    P
    V |Z
    ˆ
    v = ˆ
    vz(m)
    random
    noise
    stimuli
    deterministic
    prior
    likelihood
    nuisance
    parameter
    MAP estimator:
    m
    internal
    representation

    View full-size slide

  80. MAP Decision Process
    313
    314
    315
    316
    317
    318
    319
    320
    321
    322
    323
    resolution
    1024

    768
    at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolb
    controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three o
    with normal or corrected to normal vision were used. They gave their informed consent
    experiments received ethical approval from the Aix-Marseille Ethics Committee in accorda
    the declaration of Helsinki.
    3.2 Bayesian modeling
    To make full use of our MC paradigm in analyzing the obtained results, we follow the meth
    of the Bayesian observer used for instance in [12]. We assume the observer makes its decisi
    a Maximum A Posteriori (MAP) estimator
    ˆ
    vz(m) = argmin
    v
    [ log(
    P
    M
    |
    V,Z(m
    |
    v, z)) log(
    P
    V
    |
    Z(v
    |
    z))]
    6
    v
    z
    P
    M|V,Z
    P
    V |Z
    ˆ
    v = ˆ
    vz(m)
    random
    noise
    stimuli
    deterministic
    prior
    likelihood
    nuisance
    parameter
    MAP estimator:
    Mv,z
    ⇠ P
    M|V,Z
    (·, v, z) ˆ
    Vv,z
    def.
    = ˆ
    vz(Mv,z)
    (z, v)
    Response model:
    m
    internal
    representation

    View full-size slide

  81. MAP Decision Process
    313
    314
    315
    316
    317
    318
    319
    320
    321
    322
    323
    resolution
    1024

    768
    at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolb
    controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three o
    with normal or corrected to normal vision were used. They gave their informed consent
    experiments received ethical approval from the Aix-Marseille Ethics Committee in accorda
    the declaration of Helsinki.
    3.2 Bayesian modeling
    To make full use of our MC paradigm in analyzing the obtained results, we follow the meth
    of the Bayesian observer used for instance in [12]. We assume the observer makes its decisi
    a Maximum A Posteriori (MAP) estimator
    ˆ
    vz(m) = argmin
    v
    [ log(
    P
    M
    |
    V,Z(m
    |
    v, z)) log(
    P
    V
    |
    Z(v
    |
    z))]
    6
    v
    z
    P
    M|V,Z
    P
    V |Z
    ˆ
    v = ˆ
    vz(m)
    random
    noise
    stimuli
    deterministic
    prior
    likelihood
    nuisance
    parameter
    MAP estimator:
    Theoretical psychophysical curve:
    Mv,z
    ⇠ P
    M|V,Z
    (·, v, z) ˆ
    Vv,z
    def.
    = ˆ
    vz(Mv,z)
    (z, v)
    Response model:
    'v?,z?
    def.
    = E( ˆ
    Vv?,z > ˆ
    Vv,z?
    )
    m
    internal
    representation
    0
    1
    v
    v?
    ˆ
    'v?,z?
    (·, z) 'v?,z?
    (·, z)

    View full-size slide

  82. MAP Decision Process
    313
    314
    315
    316
    317
    318
    319
    320
    321
    322
    323
    resolution
    1024

    768
    at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolb
    controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three o
    with normal or corrected to normal vision were used. They gave their informed consent
    experiments received ethical approval from the Aix-Marseille Ethics Committee in accorda
    the declaration of Helsinki.
    3.2 Bayesian modeling
    To make full use of our MC paradigm in analyzing the obtained results, we follow the meth
    of the Bayesian observer used for instance in [12]. We assume the observer makes its decisi
    a Maximum A Posteriori (MAP) estimator
    ˆ
    vz(m) = argmin
    v
    [ log(
    P
    M
    |
    V,Z(m
    |
    v, z)) log(
    P
    V
    |
    Z(v
    |
    z))]
    6
    v
    z
    P
    M|V,Z
    P
    V |Z
    ˆ
    v = ˆ
    vz(m)
    random
    noise
    stimuli
    deterministic
    prior
    likelihood
    nuisance
    parameter
    MAP estimator:
    Theoretical psychophysical curve:
    Mv,z
    ⇠ P
    M|V,Z
    (·, v, z) ˆ
    Vv,z
    def.
    = ˆ
    vz(Mv,z)
    (z, v)
    Response model:
    'v?,z?
    def.
    = E( ˆ
    Vv?,z > ˆ
    Vv,z?
    )
    m
    internal
    representation
    0
    1
    v
    v?
    ˆ
    'v?,z?
    (·, z) 'v?,z?
    (·, z)
    Inverse Bayesian estimation
    Compute (
    P
    M|V,Z
    , P
    V |Z) by
    comparing ˆ
    '
    and
    '
    .

    View full-size slide

  83. Low Noise Estimation Process
    Prior/likelihood estimation: impossible problem
    prior/likelihood decomposition
    PV |M
    ⇠ PM|V
    PV is ambiguous
    low likelihood variance hypothesis (low noise)

    View full-size slide

  84. Low Noise Estimation Process
    omputed from some internal representation
    m
    2 R of the observed stimulus. For simplicity, we
    ssume that the observer estimates
    z
    from
    m
    without bias.
    o simplify both the exposition and the numerical analysis, we assume that the likelihood is Gaus-
    an, with a variance independent of
    v
    . Furthermore, we assume that the prior is Laplacian as this
    ives a good description of the a priori statistics of speeds in natural images [2]:
    P
    M
    |
    V,Z(m
    |
    v, z) =
    1
    p
    2⇡ z
    e
    |m v|2
    2 2
    z and P
    V
    |
    Z(v
    |
    z)
    /
    eazv
    1[0,vmax](v).
    (8)
    here
    vmax > 0
    is a cutoff speed ensuring that P
    V
    |
    Z
    is a well defined density even if
    az > 0
    .
    oth
    az
    and
    z
    are unknown parameters of the model, and are obtained from the outcome of the
    xperiments by a fitting process we now explain.
    .3 Likelihood and Prior Estimation
    ollowing for instance [12], the theoretical psychophysical curve obtained by a Bayesian decision
    model is
    'v
    ?
    ,z
    ?
    (v, z)
    def.
    =
    E

    vz
    ?
    (Mv,z
    ?
    ) > ˆ
    vz(Mv
    ?
    ,z)).
    he following proposition shows that in our special case of Gaussian prior and Laplacian likelihood,
    can be computed in closed form. Its proof follows closely the derivation of [10, Appendix A], and
    an be found in the supplementary materials.
    roposition 3.
    In the special case of the estimator (7) with a parameterization (8), one has
    computed from some internal representation
    m
    2 R of the observed stimulus. For simplicity, we
    assume that the observer estimates
    z
    from
    m
    without bias.
    To simplify both the exposition and the numerical analysis, we assume that the likelihood is Gaus-
    sian, with a variance independent of
    v
    . Furthermore, we assume that the prior is Laplacian as this
    gives a good description of the a priori statistics of speeds in natural images [2]:
    P
    M
    |
    V,Z(m
    |
    v, z) =
    1
    p
    2⇡ z
    e
    |m v|2
    2 2
    z and P
    V
    |
    Z(v
    |
    z)
    /
    eazv
    1[0,vmax](v).
    (8)
    where
    vmax > 0
    is a cutoff speed ensuring that P
    V
    |
    Z
    is a well defined density even if
    az > 0
    .
    Both
    az
    and
    z
    are unknown parameters of the model, and are obtained from the outcome of the
    experiments by a fitting process we now explain.
    3.3 Likelihood and Prior Estimation
    Following for instance [12], the theoretical psychophysical curve obtained by a Bayesian decision
    model is
    'v
    ?
    ,z
    ?
    (v, z)
    def.
    =
    E

    vz
    ?
    (Mv,z
    ?
    ) > ˆ
    vz(Mv
    ?
    ,z)).
    The following proposition shows that in our special case of Gaussian prior and Laplacian likelihood,
    it can be computed in closed form. Its proof follows closely the derivation of [10, Appendix A], and
    can be found in the supplementary materials.
    Proposition 3.
    In the special case of the estimator (7) with a parameterization (8), one has
    !
    Prior/likelihood estimation: impossible problem
    prior/likelihood decomposition
    PV |M
    ⇠ PM|V
    PV is ambiguous
    too many things to estimate
    low likelihood variance hypothesis (low noise)
    low dimensional parameterization:

    View full-size slide

  85. Low Noise Estimation Process
    omputed from some internal representation
    m
    2 R of the observed stimulus. For simplicity, we
    ssume that the observer estimates
    z
    from
    m
    without bias.
    o simplify both the exposition and the numerical analysis, we assume that the likelihood is Gaus-
    an, with a variance independent of
    v
    . Furthermore, we assume that the prior is Laplacian as this
    ives a good description of the a priori statistics of speeds in natural images [2]:
    P
    M
    |
    V,Z(m
    |
    v, z) =
    1
    p
    2⇡ z
    e
    |m v|2
    2 2
    z and P
    V
    |
    Z(v
    |
    z)
    /
    eazv
    1[0,vmax](v).
    (8)
    here
    vmax > 0
    is a cutoff speed ensuring that P
    V
    |
    Z
    is a well defined density even if
    az > 0
    .
    oth
    az
    and
    z
    are unknown parameters of the model, and are obtained from the outcome of the
    xperiments by a fitting process we now explain.
    .3 Likelihood and Prior Estimation
    ollowing for instance [12], the theoretical psychophysical curve obtained by a Bayesian decision
    model is
    'v
    ?
    ,z
    ?
    (v, z)
    def.
    =
    E

    vz
    ?
    (Mv,z
    ?
    ) > ˆ
    vz(Mv
    ?
    ,z)).
    he following proposition shows that in our special case of Gaussian prior and Laplacian likelihood,
    can be computed in closed form. Its proof follows closely the derivation of [10, Appendix A], and
    an be found in the supplementary materials.
    roposition 3.
    In the special case of the estimator (7) with a parameterization (8), one has
    computed from some internal representation
    m
    2 R of the observed stimulus. For simplicity, we
    assume that the observer estimates
    z
    from
    m
    without bias.
    To simplify both the exposition and the numerical analysis, we assume that the likelihood is Gaus-
    sian, with a variance independent of
    v
    . Furthermore, we assume that the prior is Laplacian as this
    gives a good description of the a priori statistics of speeds in natural images [2]:
    P
    M
    |
    V,Z(m
    |
    v, z) =
    1
    p
    2⇡ z
    e
    |m v|2
    2 2
    z and P
    V
    |
    Z(v
    |
    z)
    /
    eazv
    1[0,vmax](v).
    (8)
    where
    vmax > 0
    is a cutoff speed ensuring that P
    V
    |
    Z
    is a well defined density even if
    az > 0
    .
    Both
    az
    and
    z
    are unknown parameters of the model, and are obtained from the outcome of the
    experiments by a fitting process we now explain.
    3.3 Likelihood and Prior Estimation
    Following for instance [12], the theoretical psychophysical curve obtained by a Bayesian decision
    model is
    'v
    ?
    ,z
    ?
    (v, z)
    def.
    =
    E

    vz
    ?
    (Mv,z
    ?
    ) > ˆ
    vz(Mv
    ?
    ,z)).
    The following proposition shows that in our special case of Gaussian prior and Laplacian likelihood,
    it can be computed in closed form. Its proof follows closely the derivation of [10, Appendix A], and
    can be found in the supplementary materials.
    Proposition 3.
    In the special case of the estimator (7) with a parameterization (8), one has
    !
    |
    ˆ
    g z
    1
    R✓(⇠)
    |2
    Q(v0 + r(cos('), sin(')))
    P
    ⇥(✓)
    P
    Z(z)
    P
    ||
    V v0
    ||(r) d✓ dz dr d'.
    se of
    g
    being a grating, i.e. |
    ˆ
    g
    |2
    = ⇠0
    , one has in the sense of distributions
    ˆ
    g z
    1
    R✓(⇠)
    |2
    = B(✓, z)
    where B
    = (✓, z) ; z
    1
    R✓(⇠) = ⇠0 .
    Q(⌫) B(✓, z) = C(✓, z, r)
    where
    , z, r) ; z =
    ||

    ||
    , ✓ =
    \
    ⇠, r =

    ||

    ||
    cos(
    \
    ⇠ ')
    ||
    v0
    ||
    cos(
    \

    \
    v0)
    cos(
    \
    ⇠ ')
    desired formula.
    roposition 3
    sed form expression for the MAP estimator
    ˆ
    vz(m) = m az
    2
    z,
    ting N
    (µ,
    2
    )
    the Gaussian distribution of mean
    µ
    and variance 2,
    ˆ
    vz(Mv,z)
    ⇠ N
    (v az
    2
    z,
    2
    z)
    equality of distributions. One thus has
    z
    ?
    (Mv,z
    ?
    ) ˆ
    vz(Mv
    ?
    ,z)
    ⇠ N
    (v v?
    az
    ?
    2
    z
    ?
    + az
    2
    z,
    2
    z
    ?
    +
    2
    z),
    he results by taking expectation.
    Prior/likelihood estimation: impossible problem
    prior/likelihood decomposition
    PV |M
    ⇠ PM|V
    PV is ambiguous
    too many things to estimate
    low likelihood variance hypothesis (low noise)
    low dimensional parameterization:
    Proposition:
    Mvz
    ⇠ N(v, 2
    z
    ) ˆ
    Vvz
    ⇠ N(v az
    2
    z
    , 2
    z
    )
    Mv,z
    ⇠ P
    M|V,Z
    (·, v, z) ˆ
    Vv,z
    def.
    = ˆ
    vz(Mv,z)
    (z, v)
    Response model:
    v
    P
    Mv,z

    Vv,z
    v az
    2
    z
    Bias

    View full-size slide

  86. Theoretical Psychometric Curve
    Mv,z
    ⇠ P
    M|V,Z
    (·, v, z) ˆ
    Vv,z
    def.
    = ˆ
    vz(Mv,z)
    (z, v)
    Response model:
    Theoretical psychophysical curve:
    'v?,z?
    def.
    = E( ˆ
    Vv?,z > ˆ
    Vv,z?
    )
    v az?
    2
    z?
    v? az
    2
    z
    v?
    v

    Vv,z?

    Vv?,z
    P
    Mv?,z
    P
    Mv,z?

    View full-size slide

  87. Theoretical Psychometric Curve
    Mv,z
    ⇠ P
    M|V,Z
    (·, v, z) ˆ
    Vv,z
    def.
    = ˆ
    vz(Mv,z)
    (z, v)
    Response model:
    Theoretical psychophysical curve:
    'v?,z?
    def.
    = E( ˆ
    Vv?,z > ˆ
    Vv,z?
    )
    Z(m
    |
    v, z) =
    p
    2⇡ z
    e 2 2
    z and P
    V
    |
    Z(v
    |
    z)
    /
    eazv
    1[0,vmax](v).
    (8)
    is a cutoff speed ensuring that P
    V
    |
    Z
    is a well defined density even if
    az > 0
    .
    e unknown parameters of the model, and are obtained from the outcome of the
    fitting process we now explain.
    and Prior Estimation
    ance [12], the theoretical psychophysical curve obtained by a Bayesian decision
    'v
    ?
    ,z
    ?
    (v, z)
    def.
    =
    E

    vz
    ?
    (Mv,z
    ?
    ) > ˆ
    vz(Mv
    ?
    ,z)).
    position shows that in our special case of Gaussian prior and Laplacian likelihood,
    d in closed form. Its proof follows closely the derivation of [10, Appendix A], and
    e supplementary materials.
    the special case of the estimator (7) with a parameterization (8), one has
    'v
    ?
    ,z
    ?
    (v, z) =
    v v?
    az
    ? 2
    z
    ?
    + az
    2
    z
    p
    2
    z
    ?
    + 2
    z
    !
    (9)
    R
    t
    1 e s2
    /2
    ds
    is a sigmoid function.
    , one can thus fit the experimental psychometric function to compute the percep-
    ?
    2 R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    Proposition:
    where (t) =
    1
    p
    2⇡
    Z t
    1
    e t2
    2 dt
    v az?
    2
    z?
    v? az
    2
    z
    v?
    v

    Vv,z?

    Vv?,z
    P
    Mv?,z
    P
    Mv,z?
    0
    1
    v
    v?
    'v?,z?
    (·, z)
    az?
    2
    z?
    az
    2
    z
    q
    2
    z?
    + 2
    z

    View full-size slide

  88. Prior/Likelihood Identification
    0
    1
    v
    v?
    Sigmoid fit:
    e can thus fit the experimental psychometric function to compute the percep-
    R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    retical and experimental psychopysical curves (9) and (10), one thus obtains
    ons
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    known is
    az
    ?
    , that can be set as any negative number knowing the previous
    or or determined by test another central spatial frequency
    z?.
    esults
    ummarized in Figure 3 showing the parameters
    µz,z
    ?
    in Figure 3.3 and the
    re 3.3. The conclusion are
    [ToDo: Gab: why “both”?]
    both – spatial fre-
    ffect on perceived speed meaning that speed is perceived faster when spatial
    – this shift cannot be explained by a increasing in the likelihood width (Fig-
    patial frequency as for the case of contrast [12, 10]. Therefore the positive
    ned by a negative effect in prior slopes
    az
    as the spatial frequency grows.
    ve any explanation for the observed constant likelihood width as it is not con-
    width of the stimuli
    V = 1

    ?
    z0
    which is decreasing with spatial frequency.
    increase of noise in observer measurement of speed at high spatial frequency.
    µz,z?
    ˆ
    'v?,z?
    (·, z) z,z?

    View full-size slide

  89. Prior/Likelihood Identification
    0
    1
    v
    v?
    Sigmoid fit:
    e can thus fit the experimental psychometric function to compute the percep-
    R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    retical and experimental psychopysical curves (9) and (10), one thus obtains
    ons
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    known is
    az
    ?
    , that can be set as any negative number knowing the previous
    or or determined by test another central spatial frequency
    z?.
    esults
    ummarized in Figure 3 showing the parameters
    µz,z
    ?
    in Figure 3.3 and the
    re 3.3. The conclusion are
    [ToDo: Gab: why “both”?]
    both – spatial fre-
    ffect on perceived speed meaning that speed is perceived faster when spatial
    – this shift cannot be explained by a increasing in the likelihood width (Fig-
    patial frequency as for the case of contrast [12, 10]. Therefore the positive
    ned by a negative effect in prior slopes
    az
    as the spatial frequency grows.
    ve any explanation for the observed constant likelihood width as it is not con-
    width of the stimuli
    V = 1

    ?
    z0
    which is decreasing with spatial frequency.
    increase of noise in observer measurement of speed at high spatial frequency.
    µz,z?
    ˆ
    'v?,z?
    (·, z) z,z?
    ance [12], the theoretical psychophysical curve obtained by a Bayesian decision
    'v
    ?
    ,z
    ?
    (v, z)
    def.
    =
    E

    vz
    ?
    (Mv,z
    ?
    ) > ˆ
    vz(Mv
    ?
    ,z)).
    position shows that in our special case of Gaussian prior and Laplacian likelihood,
    d in closed form. Its proof follows closely the derivation of [10, Appendix A], and
    e supplementary materials.
    the special case of the estimator (7) with a parameterization (8), one has
    'v
    ?
    ,z
    ?
    (v, z) =
    v v?
    az
    ? 2
    z
    ?
    + az
    2
    z
    p
    2
    z
    ?
    + 2
    z
    !
    (9)
    R
    t
    1 e s2
    /2
    ds
    is a sigmoid function.
    , one can thus fit the experimental psychometric function to compute the percep-
    ?
    2 R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    theoretical and experimental psychopysical curves (9) and (10), one thus obtains
    essions
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    g unknown is
    az
    ?
    , that can be set as any negative number knowing the previous
    ?
    Theoretical prediction:

    View full-size slide

  90. Prior/Likelihood Identification
    0
    1
    v
    v?
    Sigmoid fit:
    e can thus fit the experimental psychometric function to compute the percep-
    R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    retical and experimental psychopysical curves (9) and (10), one thus obtains
    ons
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    known is
    az
    ?
    , that can be set as any negative number knowing the previous
    or or determined by test another central spatial frequency
    z?.
    esults
    ummarized in Figure 3 showing the parameters
    µz,z
    ?
    in Figure 3.3 and the
    re 3.3. The conclusion are
    [ToDo: Gab: why “both”?]
    both – spatial fre-
    ffect on perceived speed meaning that speed is perceived faster when spatial
    – this shift cannot be explained by a increasing in the likelihood width (Fig-
    patial frequency as for the case of contrast [12, 10]. Therefore the positive
    ned by a negative effect in prior slopes
    az
    as the spatial frequency grows.
    ve any explanation for the observed constant likelihood width as it is not con-
    width of the stimuli
    V = 1

    ?
    z0
    which is decreasing with spatial frequency.
    increase of noise in observer measurement of speed at high spatial frequency.
    µz,z?
    ˆ
    'v?,z?
    (·, z) z,z?
    ance [12], the theoretical psychophysical curve obtained by a Bayesian decision
    'v
    ?
    ,z
    ?
    (v, z)
    def.
    =
    E

    vz
    ?
    (Mv,z
    ?
    ) > ˆ
    vz(Mv
    ?
    ,z)).
    position shows that in our special case of Gaussian prior and Laplacian likelihood,
    d in closed form. Its proof follows closely the derivation of [10, Appendix A], and
    e supplementary materials.
    the special case of the estimator (7) with a parameterization (8), one has
    'v
    ?
    ,z
    ?
    (v, z) =
    v v?
    az
    ? 2
    z
    ?
    + az
    2
    z
    p
    2
    z
    ?
    + 2
    z
    !
    (9)
    R
    t
    1 e s2
    /2
    ds
    is a sigmoid function.
    , one can thus fit the experimental psychometric function to compute the percep-
    ?
    2 R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    theoretical and experimental psychopysical curves (9) and (10), one thus obtains
    essions
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    g unknown is
    az
    ?
    , that can be set as any negative number knowing the previous
    ?
    Theoretical prediction:
    ound in the supplementary materials.
    tion 3.
    In the special case of the estimator (7) with a parameterization (8), one has
    'v
    ?
    ,z
    ?
    (v, z) =
    v v?
    az
    ? 2
    z
    ?
    + az
    2
    z
    p
    2
    z
    ?
    + 2
    z
    !
    (9)
    (t) = 1
    p
    2⇡
    R
    t
    1 e s2
    /2
    ds
    is a sigmoid function.
    is known, one can thus fit the experimental psychometric function to compute the percep-
    term
    µz,z
    ?
    2 R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    paring the theoretical and experimental psychopysical curves (9) and (10), one thus obtains
    wing expressions
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    y remaining unknown is
    az
    ?
    , that can be set as any negative number knowing the previous
    n low speed prior or determined by test another central spatial frequency
    z?.
    ychophysic Results
    n results are summarized in Figure 3 showing the parameters
    µz,z
    ?
    in Figure 3.3 and the
    ers
    z
    in Figure 3.3. The conclusion are
    [ToDo: Gab: why “both”?]
    both – spatial fre-
    has a positive effect on perceived speed meaning that speed is perceived faster when spatial
    d in the supplementary materials.
    3.
    In the special case of the estimator (7) with a parameterization (8), one has
    'v
    ?
    ,z
    ?
    (v, z) =
    v v?
    az
    ? 2
    z
    ?
    + az
    2
    z
    p
    2
    z
    ?
    + 2
    z
    !
    (9)
    = 1
    p
    2⇡
    R
    t
    1 e s2
    /2
    ds
    is a sigmoid function.
    known, one can thus fit the experimental psychometric function to compute the percep-
    m
    µz,z
    ?
    2 R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    ng the theoretical and experimental psychopysical curves (9) and (10), one thus obtains
    g expressions
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    maining unknown is
    az
    ?
    , that can be set as any negative number knowing the previous
    w speed prior or determined by test another central spatial frequency
    z?.
    ophysic Results
    sults are summarized in Figure 3 showing the parameters
    µz,z
    ?
    in Figure 3.3 and the
    z
    in Figure 3.3. The conclusion are
    [ToDo: Gab: why “both”?]
    both – spatial fre-
    a positive effect on perceived speed meaning that speed is perceived faster when spatial
    “Fundamental” relations of psychophysic:

    View full-size slide

  91. Prior/Likelihood Identification
    0
    1
    v
    v?
    Sigmoid fit:
    e can thus fit the experimental psychometric function to compute the percep-
    R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    retical and experimental psychopysical curves (9) and (10), one thus obtains
    ons
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    known is
    az
    ?
    , that can be set as any negative number knowing the previous
    or or determined by test another central spatial frequency
    z?.
    esults
    ummarized in Figure 3 showing the parameters
    µz,z
    ?
    in Figure 3.3 and the
    re 3.3. The conclusion are
    [ToDo: Gab: why “both”?]
    both – spatial fre-
    ffect on perceived speed meaning that speed is perceived faster when spatial
    – this shift cannot be explained by a increasing in the likelihood width (Fig-
    patial frequency as for the case of contrast [12, 10]. Therefore the positive
    ned by a negative effect in prior slopes
    az
    as the spatial frequency grows.
    ve any explanation for the observed constant likelihood width as it is not con-
    width of the stimuli
    V = 1

    ?
    z0
    which is decreasing with spatial frequency.
    increase of noise in observer measurement of speed at high spatial frequency.
    µz,z?
    ˆ
    'v?,z?
    (·, z) z,z?
    ance [12], the theoretical psychophysical curve obtained by a Bayesian decision
    'v
    ?
    ,z
    ?
    (v, z)
    def.
    =
    E

    vz
    ?
    (Mv,z
    ?
    ) > ˆ
    vz(Mv
    ?
    ,z)).
    position shows that in our special case of Gaussian prior and Laplacian likelihood,
    d in closed form. Its proof follows closely the derivation of [10, Appendix A], and
    e supplementary materials.
    the special case of the estimator (7) with a parameterization (8), one has
    'v
    ?
    ,z
    ?
    (v, z) =
    v v?
    az
    ? 2
    z
    ?
    + az
    2
    z
    p
    2
    z
    ?
    + 2
    z
    !
    (9)
    R
    t
    1 e s2
    /2
    ds
    is a sigmoid function.
    , one can thus fit the experimental psychometric function to compute the percep-
    ?
    2 R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    theoretical and experimental psychopysical curves (9) and (10), one thus obtains
    essions
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    g unknown is
    az
    ?
    , that can be set as any negative number knowing the previous
    ?
    Theoretical prediction:
    ound in the supplementary materials.
    tion 3.
    In the special case of the estimator (7) with a parameterization (8), one has
    'v
    ?
    ,z
    ?
    (v, z) =
    v v?
    az
    ? 2
    z
    ?
    + az
    2
    z
    p
    2
    z
    ?
    + 2
    z
    !
    (9)
    (t) = 1
    p
    2⇡
    R
    t
    1 e s2
    /2
    ds
    is a sigmoid function.
    is known, one can thus fit the experimental psychometric function to compute the percep-
    term
    µz,z
    ?
    2 R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    paring the theoretical and experimental psychopysical curves (9) and (10), one thus obtains
    wing expressions
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    y remaining unknown is
    az
    ?
    , that can be set as any negative number knowing the previous
    n low speed prior or determined by test another central spatial frequency
    z?.
    ychophysic Results
    n results are summarized in Figure 3 showing the parameters
    µz,z
    ?
    in Figure 3.3 and the
    ers
    z
    in Figure 3.3. The conclusion are
    [ToDo: Gab: why “both”?]
    both – spatial fre-
    has a positive effect on perceived speed meaning that speed is perceived faster when spatial
    d in the supplementary materials.
    3.
    In the special case of the estimator (7) with a parameterization (8), one has
    'v
    ?
    ,z
    ?
    (v, z) =
    v v?
    az
    ? 2
    z
    ?
    + az
    2
    z
    p
    2
    z
    ?
    + 2
    z
    !
    (9)
    = 1
    p
    2⇡
    R
    t
    1 e s2
    /2
    ds
    is a sigmoid function.
    known, one can thus fit the experimental psychometric function to compute the percep-
    m
    µz,z
    ?
    2 R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    ng the theoretical and experimental psychopysical curves (9) and (10), one thus obtains
    g expressions
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    maining unknown is
    az
    ?
    , that can be set as any negative number knowing the previous
    w speed prior or determined by test another central spatial frequency
    z?.
    ophysic Results
    sults are summarized in Figure 3 showing the parameters
    µz,z
    ?
    in Figure 3.3 and the
    z
    in Figure 3.3. The conclusion are
    [ToDo: Gab: why “both”?]
    both – spatial fre-
    a positive effect on perceived speed meaning that speed is perceived faster when spatial
    “Fundamental” relations of psychophysic:
    ! parameters are identifiable . . .
    . . . up to the value of
    az?

    View full-size slide

  92. Prior/Likelihood Identification
    0
    1
    v
    v?
    Sigmoid fit:
    e can thus fit the experimental psychometric function to compute the percep-
    R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    retical and experimental psychopysical curves (9) and (10), one thus obtains
    ons
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    known is
    az
    ?
    , that can be set as any negative number knowing the previous
    or or determined by test another central spatial frequency
    z?.
    esults
    ummarized in Figure 3 showing the parameters
    µz,z
    ?
    in Figure 3.3 and the
    re 3.3. The conclusion are
    [ToDo: Gab: why “both”?]
    both – spatial fre-
    ffect on perceived speed meaning that speed is perceived faster when spatial
    – this shift cannot be explained by a increasing in the likelihood width (Fig-
    patial frequency as for the case of contrast [12, 10]. Therefore the positive
    ned by a negative effect in prior slopes
    az
    as the spatial frequency grows.
    ve any explanation for the observed constant likelihood width as it is not con-
    width of the stimuli
    V = 1

    ?
    z0
    which is decreasing with spatial frequency.
    increase of noise in observer measurement of speed at high spatial frequency.
    µz,z?
    ˆ
    'v?,z?
    (·, z) z,z?
    ance [12], the theoretical psychophysical curve obtained by a Bayesian decision
    'v
    ?
    ,z
    ?
    (v, z)
    def.
    =
    E

    vz
    ?
    (Mv,z
    ?
    ) > ˆ
    vz(Mv
    ?
    ,z)).
    position shows that in our special case of Gaussian prior and Laplacian likelihood,
    d in closed form. Its proof follows closely the derivation of [10, Appendix A], and
    e supplementary materials.
    the special case of the estimator (7) with a parameterization (8), one has
    'v
    ?
    ,z
    ?
    (v, z) =
    v v?
    az
    ? 2
    z
    ?
    + az
    2
    z
    p
    2
    z
    ?
    + 2
    z
    !
    (9)
    R
    t
    1 e s2
    /2
    ds
    is a sigmoid function.
    , one can thus fit the experimental psychometric function to compute the percep-
    ?
    2 R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    theoretical and experimental psychopysical curves (9) and (10), one thus obtains
    essions
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    g unknown is
    az
    ?
    , that can be set as any negative number knowing the previous
    ?
    Theoretical prediction:
    ound in the supplementary materials.
    tion 3.
    In the special case of the estimator (7) with a parameterization (8), one has
    'v
    ?
    ,z
    ?
    (v, z) =
    v v?
    az
    ? 2
    z
    ?
    + az
    2
    z
    p
    2
    z
    ?
    + 2
    z
    !
    (9)
    (t) = 1
    p
    2⇡
    R
    t
    1 e s2
    /2
    ds
    is a sigmoid function.
    is known, one can thus fit the experimental psychometric function to compute the percep-
    term
    µz,z
    ?
    2 R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    paring the theoretical and experimental psychopysical curves (9) and (10), one thus obtains
    wing expressions
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    y remaining unknown is
    az
    ?
    , that can be set as any negative number knowing the previous
    n low speed prior or determined by test another central spatial frequency
    z?.
    ychophysic Results
    n results are summarized in Figure 3 showing the parameters
    µz,z
    ?
    in Figure 3.3 and the
    ers
    z
    in Figure 3.3. The conclusion are
    [ToDo: Gab: why “both”?]
    both – spatial fre-
    has a positive effect on perceived speed meaning that speed is perceived faster when spatial
    d in the supplementary materials.
    3.
    In the special case of the estimator (7) with a parameterization (8), one has
    'v
    ?
    ,z
    ?
    (v, z) =
    v v?
    az
    ? 2
    z
    ?
    + az
    2
    z
    p
    2
    z
    ?
    + 2
    z
    !
    (9)
    = 1
    p
    2⇡
    R
    t
    1 e s2
    /2
    ds
    is a sigmoid function.
    known, one can thus fit the experimental psychometric function to compute the percep-
    m
    µz,z
    ?
    2 R and an uncertainty
    z,z
    ?
    such that
    ˆ
    'v
    ?
    ,z
    ?
    (v, z)


    v v?
    µz,z
    ?
    z,z
    ?

    .
    (10)
    ng the theoretical and experimental psychopysical curves (9) and (10), one thus obtains
    g expressions
    2
    z = 2
    z,z
    ?
    1
    2
    2
    z
    ?
    ,z
    ?
    and
    az = az
    ?
    2
    z
    ?
    2
    z
    µz,z
    ?
    2
    z
    .
    maining unknown is
    az
    ?
    , that can be set as any negative number knowing the previous
    w speed prior or determined by test another central spatial frequency
    z?.
    ophysic Results
    sults are summarized in Figure 3 showing the parameters
    µz,z
    ?
    in Figure 3.3 and the
    z
    in Figure 3.3. The conclusion are
    [ToDo: Gab: why “both”?]
    both – spatial fre-
    a positive effect on perceived speed meaning that speed is perceived faster when spatial
    “Fundamental” relations of psychophysic:
    v
    z
    (v?, z?)
    Workaround: use additional
    z?
    .
    ! parameters are identifiable . . .
    . . . up to the value of
    az?

    View full-size slide

  93. Experimental Findings
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    323
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    To asses the validity of our model, we tested four different scenarios by cons
    choices among
    z?
    = 0.78
    c/
    , v? 2 {
    5
    /s
    , 10
    /s}
    , ⌧? 2 {
    0.1s, 0.2s
    }
    ,
    which corresponds to combinations of low/high speeds and temporal frequency b
    were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic
    resolution
    1024

    768
    at 100 Hz. Routines were written using Matlab 7.10.0 and
    controlled the stimulus display. Observers sat 57 cm from the screen in a dark roo
    with normal or corrected to normal vision were used. They gave their inform
    experiments received ethical approval from the Aix-Marseille Ethics Committee
    the declaration of Helsinki.
    3.2 Bayesian modeling
    To make full use of our MC paradigm in analyzing the obtained results, we follo
    of the Bayesian observer used for instance in [12]. We assume the observer mak
    a Maximum A Posteriori (MAP) estimator
    ˆ
    vz(m) = argmin
    v
    [ log(
    P
    M
    |
    V,Z(m
    |
    v, z)) log(
    P
    V
    |
    Z(v
    |
    z))
    6
    281
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    286
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    288
    289
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    295
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    301
    302
    303
    304
    305
    3.1 Methods
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving w
    v0 = (v, 0)
    . We assign as independent experimental variable the aver
    we denote in the following
    z
    (we drop the index
    0
    to ease readabilit
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated
    We used a two alternative forced choice (2AFC) paradigm. In each tri
    a small dark fixation spot was followed by two stimulus intervals of
    grey 250 ms inter-stimulus interval. The first stimulus has parameter
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears a
    which one of the two intervals was perceived as moving faster by pres
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    ar

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0
    Z =
    {
    0.31,
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomize
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten
    possible combinations of these parameters are made per block of 250
    blocks were collected per condition tested. The outcome of these exp
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    283
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    306
    307
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving with a horizontal
    v0 = (v, 0)
    . We assign as independent experimental variable the average spatial freq
    we denote in the following
    z
    (we drop the index
    0
    to ease readability in this sectio
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.78
    c/
    .
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    = 1
    V z0
    stays
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated on the left of F
    We used a two alternative forced choice (2AFC) paradigm. In each trial a grey fixatio
    a small dark fixation spot was followed by two stimulus intervals of
    250
    ms each, s
    grey 250 ms inter-stimulus interval. The first stimulus has parameters
    (v1, z1)
    and t
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the partic
    which one of the two intervals was perceived as moving faster by pressing one of tw
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected so th

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16, 0.
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across trials)
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitions of
    possible combinations of these parameters are made per block of 250 trials and at l
    blocks were collected per condition tested. The outcome of these experiments are s
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    ˆ
    'v
    ?
    empirical probability (averaged over the 40 trials) that a stimulus generated with para
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    283
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    302
    303
    304
    305
    306
    307
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving with a horizo
    v0 = (v, 0)
    . We assign as independent experimental variable the average spatial f
    we denote in the following
    z
    (we drop the index
    0
    to ease readability in this se
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.78
    c/
    .
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    = 1
    V z0
    sta
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated on the left
    We used a two alternative forced choice (2AFC) paradigm. In each trial a grey fix
    a small dark fixation spot was followed by two stimulus intervals of
    250
    ms eac
    grey 250 ms inter-stimulus interval. The first stimulus has parameters
    (v1, z1)
    an
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the pa
    which one of the two intervals was perceived as moving faster by pressing one of
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected s

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across tri
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitions
    possible combinations of these parameters are made per block of 250 trials and
    blocks were collected per condition tested. The outcome of these experiments ar
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    '
    empirical probability (averaged over the 40 trials) that a stimulus generated with p
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    283
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    287
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    302
    303
    304
    305
    306
    307
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving with a hori
    v0 = (v, 0)
    . We assign as independent experimental variable the average spatia
    we denote in the following
    z
    (we drop the index
    0
    to ease readability in this
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.78
    c/
    .
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    = 1
    V z0
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated on the le
    We used a two alternative forced choice (2AFC) paradigm. In each trial a grey
    a small dark fixation spot was followed by two stimulus intervals of
    250
    ms e
    grey 250 ms inter-stimulus interval. The first stimulus has parameters
    (v1, z1)
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the
    which one of the two intervals was perceived as moving faster by pressing one
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitio
    possible combinations of these parameters are made per block of 250 trials an
    blocks were collected per condition tested. The outcome of these experiments
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the valu
    empirical probability (averaged over the 40 trials) that a stimulus generated wit
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    P
    Z(z)
    /
    z
    e ,
    P
    ⇥(✓)
    /
    e
    and P
    ||
    V v0
    ||(r)
    /
    e .
    (6)
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a
    graphical display of the influence of these parameters.
    Speed Freq. orient. Freq. amplitude
    (mean, dispersion)
    (v0, V ) (✓0, ⇥) (z0, Z)
    Remark 2. Note that the final envelope of
    ˆ
    is in agreement with the formulation that is used in [8].
    However, this latter derivation was based on a heuristic following a trial-and-error strategy between
    modelers and psychophysicists. Herein, we justified these different points in a constructive manner.
    P
    Z(z)
    /
    z
    e ,
    P
    ⇥(✓)
    /
    e
    and P
    ||
    V v0
    ||(r)
    /
    e .
    (6)
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a
    graphical display of the influence of these parameters.
    Speed Freq. orient. Freq. amplitude
    (mean, dispersion)
    (v0, V ) (✓0, ⇥) (z0, Z)
    Remark 2. Note that the final envelope of
    ˆ
    is in agreement with the formulation that is used in [8].
    However, this latter derivation was based on a heuristic following a trial-and-error strategy between
    modelers and psychophysicists. Herein, we justified these different points in a constructive manner.

    View full-size slide

  94. Experimental Findings
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    310
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    315
    316
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    318
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    321
    322
    323
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    To asses the validity of our model, we tested four different scenarios by cons
    choices among
    z?
    = 0.78
    c/
    , v? 2 {
    5
    /s
    , 10
    /s}
    , ⌧? 2 {
    0.1s, 0.2s
    }
    ,
    which corresponds to combinations of low/high speeds and temporal frequency b
    were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic
    resolution
    1024

    768
    at 100 Hz. Routines were written using Matlab 7.10.0 and
    controlled the stimulus display. Observers sat 57 cm from the screen in a dark roo
    with normal or corrected to normal vision were used. They gave their inform
    experiments received ethical approval from the Aix-Marseille Ethics Committee
    the declaration of Helsinki.
    3.2 Bayesian modeling
    To make full use of our MC paradigm in analyzing the obtained results, we follo
    of the Bayesian observer used for instance in [12]. We assume the observer mak
    a Maximum A Posteriori (MAP) estimator
    ˆ
    vz(m) = argmin
    v
    [ log(
    P
    M
    |
    V,Z(m
    |
    v, z)) log(
    P
    V
    |
    Z(v
    |
    z))
    6
    281
    282
    283
    284
    285
    286
    287
    288
    289
    290
    291
    292
    293
    294
    295
    296
    297
    298
    299
    300
    301
    302
    303
    304
    305
    3.1 Methods
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving w
    v0 = (v, 0)
    . We assign as independent experimental variable the aver
    we denote in the following
    z
    (we drop the index
    0
    to ease readabilit
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated
    We used a two alternative forced choice (2AFC) paradigm. In each tri
    a small dark fixation spot was followed by two stimulus intervals of
    grey 250 ms inter-stimulus interval. The first stimulus has parameter
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears a
    which one of the two intervals was perceived as moving faster by pres
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    ar

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0
    Z =
    {
    0.31,
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomize
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten
    possible combinations of these parameters are made per block of 250
    blocks were collected per condition tested. The outcome of these exp
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    283
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    302
    303
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    305
    306
    307
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving with a horizontal
    v0 = (v, 0)
    . We assign as independent experimental variable the average spatial freq
    we denote in the following
    z
    (we drop the index
    0
    to ease readability in this sectio
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.78
    c/
    .
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    = 1
    V z0
    stays
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated on the left of F
    We used a two alternative forced choice (2AFC) paradigm. In each trial a grey fixatio
    a small dark fixation spot was followed by two stimulus intervals of
    250
    ms each, s
    grey 250 ms inter-stimulus interval. The first stimulus has parameters
    (v1, z1)
    and t
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the partic
    which one of the two intervals was perceived as moving faster by pressing one of tw
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected so th

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16, 0.
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across trials)
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitions of
    possible combinations of these parameters are made per block of 250 trials and at l
    blocks were collected per condition tested. The outcome of these experiments are s
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    ˆ
    'v
    ?
    empirical probability (averaged over the 40 trials) that a stimulus generated with para
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    283
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    287
    288
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    297
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    302
    303
    304
    305
    306
    307
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving with a horizo
    v0 = (v, 0)
    . We assign as independent experimental variable the average spatial f
    we denote in the following
    z
    (we drop the index
    0
    to ease readability in this se
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.78
    c/
    .
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    = 1
    V z0
    sta
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated on the left
    We used a two alternative forced choice (2AFC) paradigm. In each trial a grey fix
    a small dark fixation spot was followed by two stimulus intervals of
    250
    ms eac
    grey 250 ms inter-stimulus interval. The first stimulus has parameters
    (v1, z1)
    an
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the pa
    which one of the two intervals was perceived as moving faster by pressing one of
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected s

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across tri
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitions
    possible combinations of these parameters are made per block of 250 trials and
    blocks were collected per condition tested. The outcome of these experiments ar
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    '
    empirical probability (averaged over the 40 trials) that a stimulus generated with p
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    283
    284
    285
    286
    287
    288
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    292
    293
    294
    295
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    297
    298
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    300
    301
    302
    303
    304
    305
    306
    307
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving with a hori
    v0 = (v, 0)
    . We assign as independent experimental variable the average spatia
    we denote in the following
    z
    (we drop the index
    0
    to ease readability in this
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.78
    c/
    .
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    = 1
    V z0
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated on the le
    We used a two alternative forced choice (2AFC) paradigm. In each trial a grey
    a small dark fixation spot was followed by two stimulus intervals of
    250
    ms e
    grey 250 ms inter-stimulus interval. The first stimulus has parameters
    (v1, z1)
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the
    which one of the two intervals was perceived as moving faster by pressing one
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitio
    possible combinations of these parameters are made per block of 250 trials an
    blocks were collected per condition tested. The outcome of these experiments
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the valu
    empirical probability (averaged over the 40 trials) that a stimulus generated wit
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    P
    Z(z)
    /
    z
    e ,
    P
    ⇥(✓)
    /
    e
    and P
    ||
    V v0
    ||(r)
    /
    e .
    (6)
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a
    graphical display of the influence of these parameters.
    Speed Freq. orient. Freq. amplitude
    (mean, dispersion)
    (v0, V ) (✓0, ⇥) (z0, Z)
    Remark 2. Note that the final envelope of
    ˆ
    is in agreement with the formulation that is used in [8].
    However, this latter derivation was based on a heuristic following a trial-and-error strategy between
    modelers and psychophysicists. Herein, we justified these different points in a constructive manner.
    P
    Z(z)
    /
    z
    e ,
    P
    ⇥(✓)
    /
    e
    and P
    ||
    V v0
    ||(r)
    /
    e .
    (6)
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a
    graphical display of the influence of these parameters.
    Speed Freq. orient. Freq. amplitude
    (mean, dispersion)
    (v0, V ) (✓0, ⇥) (z0, Z)
    Remark 2. Note that the final envelope of
    ˆ
    is in agreement with the formulation that is used in [8].
    However, this latter derivation was based on a heuristic following a trial-and-error strategy between
    modelers and psychophysicists. Herein, we justified these different points in a constructive manner.

    View full-size slide

  95. 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
    0.3
    0.2
    0.1
    0.0
    0.1
    0.2
    0.3
    0.4
    PSE bias (µz,z⇤
    )
    Subject 1
    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
    0.15
    0.10
    0.05
    0.00
    0.05
    0.10
    0.15
    Subject 2
    v⇤ = 5, ⌧ = 100
    v⇤ = 5, ⌧ = 200
    v⇤ = 10, ⌧ = 100
    v⇤ = 10, ⌧ = 200
    1.0 1.1 1.2 1.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
    0.15
    0.10
    0.05
    0.00
    0.05
    0.10
    0.15
    Subject 2
    v⇤ = 5, ⌧ = 100
    v⇤ = 5, ⌧ = 200
    v⇤ = 10, ⌧ = 100
    v⇤ = 10, ⌧ = 200
    Experimental Findings
    307
    308
    309
    310
    311
    312
    313
    314
    315
    316
    317
    318
    319
    320
    321
    322
    323
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    To asses the validity of our model, we tested four different scenarios by cons
    choices among
    z?
    = 0.78
    c/
    , v? 2 {
    5
    /s
    , 10
    /s}
    , ⌧? 2 {
    0.1s, 0.2s
    }
    ,
    which corresponds to combinations of low/high speeds and temporal frequency b
    were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic
    resolution
    1024

    768
    at 100 Hz. Routines were written using Matlab 7.10.0 and
    controlled the stimulus display. Observers sat 57 cm from the screen in a dark roo
    with normal or corrected to normal vision were used. They gave their inform
    experiments received ethical approval from the Aix-Marseille Ethics Committee
    the declaration of Helsinki.
    3.2 Bayesian modeling
    To make full use of our MC paradigm in analyzing the obtained results, we follo
    of the Bayesian observer used for instance in [12]. We assume the observer mak
    a Maximum A Posteriori (MAP) estimator
    ˆ
    vz(m) = argmin
    v
    [ log(
    P
    M
    |
    V,Z(m
    |
    v, z)) log(
    P
    V
    |
    Z(v
    |
    z))
    6
    281
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    284
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    294
    295
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    298
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    301
    302
    303
    304
    305
    3.1 Methods
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving w
    v0 = (v, 0)
    . We assign as independent experimental variable the aver
    we denote in the following
    z
    (we drop the index
    0
    to ease readabilit
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated
    We used a two alternative forced choice (2AFC) paradigm. In each tri
    a small dark fixation spot was followed by two stimulus intervals of
    grey 250 ms inter-stimulus interval. The first stimulus has parameter
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears a
    which one of the two intervals was perceived as moving faster by pres
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    ar

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0
    Z =
    {
    0.31,
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomize
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten
    possible combinations of these parameters are made per block of 250
    blocks were collected per condition tested. The outcome of these exp
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    283
    284
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    287
    288
    289
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    297
    298
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    300
    301
    302
    303
    304
    305
    306
    307
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving with a horizontal
    v0 = (v, 0)
    . We assign as independent experimental variable the average spatial freq
    we denote in the following
    z
    (we drop the index
    0
    to ease readability in this sectio
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.78
    c/
    .
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    = 1
    V z0
    stays
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated on the left of F
    We used a two alternative forced choice (2AFC) paradigm. In each trial a grey fixatio
    a small dark fixation spot was followed by two stimulus intervals of
    250
    ms each, s
    grey 250 ms inter-stimulus interval. The first stimulus has parameters
    (v1, z1)
    and t
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the partic
    which one of the two intervals was perceived as moving faster by pressing one of tw
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected so th

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16, 0.
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across trials)
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitions of
    possible combinations of these parameters are made per block of 250 trials and at l
    blocks were collected per condition tested. The outcome of these experiments are s
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    ˆ
    'v
    ?
    empirical probability (averaged over the 40 trials) that a stimulus generated with para
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    283
    284
    285
    286
    287
    288
    289
    290
    291
    292
    293
    294
    295
    296
    297
    298
    299
    300
    301
    302
    303
    304
    305
    306
    307
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving with a horizo
    v0 = (v, 0)
    . We assign as independent experimental variable the average spatial f
    we denote in the following
    z
    (we drop the index
    0
    to ease readability in this se
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.78
    c/
    .
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    = 1
    V z0
    sta
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated on the left
    We used a two alternative forced choice (2AFC) paradigm. In each trial a grey fix
    a small dark fixation spot was followed by two stimulus intervals of
    250
    ms eac
    grey 250 ms inter-stimulus interval. The first stimulus has parameters
    (v1, z1)
    an
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the pa
    which one of the two intervals was perceived as moving faster by pressing one of
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected s

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0.16
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across tri
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitions
    possible combinations of these parameters are made per block of 250 trials and
    blocks were collected per condition tested. The outcome of these experiments ar
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the value
    '
    empirical probability (averaged over the 40 trials) that a stimulus generated with p
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    283
    284
    285
    286
    287
    288
    289
    290
    291
    292
    293
    294
    295
    296
    297
    298
    299
    300
    301
    302
    303
    304
    305
    306
    307
    The task is to discriminate the speed
    v
    2 R of MC stimuli moving with a hori
    v0 = (v, 0)
    . We assign as independent experimental variable the average spatia
    we denote in the following
    z
    (we drop the index
    0
    to ease readability in this
    parameters are set to the following values
    V =
    1
    ⌧?z0
    , ✓0 =

    2
    , ⇥ =

    6
    , Z = 0.78
    c/
    .
    Note that
    V
    is thus dependent of the value of
    z = z0
    to ensure that
    ⌧?
    = 1
    V z0
    parameter
    ⌧? controls the temporal frequency bandwidth, as illustrated on the le
    We used a two alternative forced choice (2AFC) paradigm. In each trial a grey
    a small dark fixation spot was followed by two stimulus intervals of
    250
    ms e
    grey 250 ms inter-stimulus interval. The first stimulus has parameters
    (v1, z1)
    has parameters
    (v2, z2)
    . At the end of the trial, a grey screen appears asking the
    which one of the two intervals was perceived as moving faster by pressing one
    is whether
    v1 > v2
    or
    v2 > v1
    .
    Given references values
    (v?
    , z?
    )
    , for each trial,
    (v1, z1)
    and
    (v2, z2)
    are selected

    vi = v?
    , zi
    2
    z?
    + Z
    vj
    2
    v?
    + V , zj = z?
    where

    V =
    {
    2, 1, 0, 1, 2
    }
    ,
    Z =
    {
    0.31, 0.15, 0, 0
    where
    (i, j) = (1, 2)
    or
    (i, j) = (2, 1)
    (i.e. the ordering is randomized across
    values are expressed in cycles per degree (c/ ) and
    v
    values in /s. Ten repetitio
    possible combinations of these parameters are made per block of 250 trials an
    blocks were collected per condition tested. The outcome of these experiments
    psychometric curves
    ˆ
    'v
    ?
    ,z
    ?
    , where for all
    (v v?
    , z z?
    )
    2
    V

    Z
    , the valu
    empirical probability (averaged over the 40 trials) that a stimulus generated wit
    is moving faster than a stimulus with parameters
    (v, z?
    )
    .
    P
    Z(z)
    /
    z
    e ,
    P
    ⇥(✓)
    /
    e
    and P
    ||
    V v0
    ||(r)
    /
    e .
    (6)
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a
    graphical display of the influence of these parameters.
    Speed Freq. orient. Freq. amplitude
    (mean, dispersion)
    (v0, V ) (✓0, ⇥) (z0, Z)
    Remark 2. Note that the final envelope of
    ˆ
    is in agreement with the formulation that is used in [8].
    However, this latter derivation was based on a heuristic following a trial-and-error strategy between
    modelers and psychophysicists. Herein, we justified these different points in a constructive manner.
    P
    Z(z)
    /
    z
    e ,
    P
    ⇥(✓)
    /
    e
    and P
    ||
    V v0
    ||(r)
    /
    e .
    (6)
    z0
    Z
    V
    ⇠1
    ⌧ ⇠2
    ⇠1
    ✓0
    z0

    Z
    Slope: \
    v0
    Two different projections of
    ˆ
    in Fourier space t
    MC of two different spatial frequencies
    z
    Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the
    envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion
    Clouds (right).
    Plugging these expressions (6) into the definition (5) of the power spectrum of the motion cloud,
    one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol-
    lowing table articulates the speed
    v0
    and frequency
    (✓0, z0)
    central parameters in term of amplitude
    and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a
    graphical display of the influence of these parameters.
    Speed Freq. orient. Freq. amplitude
    (mean, dispersion)
    (v0, V ) (✓0, ⇥) (z0, Z)
    Remark 2. Note that the final envelope of
    ˆ
    is in agreement with the formulation that is used in [8].
    However, this latter derivation was based on a heuristic following a trial-and-error strategy between
    modelers and psychophysicists. Herein, we justified these different points in a constructive manner.
    µz,z?
    z
    z
    µz,z?

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  96. Experimental Findings
    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
    Spatial frequency (z) in cycles/deg
    0.1
    0.0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    Likehood width ( z
    )
    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
    0.05
    0.00
    0.05
    0.10
    0.15
    0.20
    0.25
    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
    Spatial frequency (z) in cycles/deg
    0.1
    0.0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    Likehood width ( z
    )
    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
    0.05
    0.00
    0.05
    0.10
    0.15
    0.20
    0.25
    Subject 1 Subject 2
    z z
    z z

    View full-size slide

  97. Experimental Findings
    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
    Spatial frequency (z) in cycles/deg
    0.1
    0.0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    Likehood width ( z
    )
    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
    0.05
    0.00
    0.05
    0.10
    0.15
    0.20
    0.25
    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
    Spatial frequency (z) in cycles/deg
    0.1
    0.0
    0.1
    0.2
    0.3
    0.4
    0.5
    0.6
    Likehood width ( z
    )
    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
    0.05
    0.00
    0.05
    0.10
    0.15
    0.20
    0.25
    Subject 1 Subject 2
    z z
    z z
    0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
    Spatial freqency z in cycles/deg
    25
    20
    15
    10
    5
    0
    5
    The slope az
    ⌧ = 0.2, v = 10, az⇤
    2 { 8, 10, 12, 13}
    ⌧ = 0.1, v = 10, az⇤
    2 { 8, 10, 12, 13}

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  98. Conclusion
    Gaussian texture models:
    ! surprisingly e cient.
    !
    simple to estimate/manipulate.
    !
    equivalent to spot noise.

    View full-size slide

  99. Conclusion
    Gaussian texture models:
    ! surprisingly e cient.
    !
    simple to estimate/manipulate.
    !
    equivalent to spot noise.

    View full-size slide

  100. Conclusion
    Gaussian texture models:
    ! surprisingly e cient.
    !
    simple to estimate/manipulate.
    !
    equivalent to spot noise.
    !
    sub-class of Gaussian spot-noises.
    !
    meaningful texton-based generative model.
    !
    equivalent to a dynamic s-PDE model.
    Motion clouds:

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  101. Conclusion
    Gaussian texture models:
    ! surprisingly e cient.
    !
    simple to estimate/manipulate.
    !
    equivalent to spot noise.
    !
    sub-class of Gaussian spot-noises.
    !
    meaningful texton-based generative model.
    !
    equivalent to a dynamic s-PDE model.
    Motion clouds:
    Psychophysics:
    !
    Interpretable through inverse Bayesian estimation.
    !
    Making use of MC parameters (not yet fully).

    View full-size slide

  102. Conclusion
    Gaussian texture models:
    ! surprisingly e cient.
    !
    simple to estimate/manipulate.
    !
    equivalent to spot noise.
    !
    sub-class of Gaussian spot-noises.
    !
    meaningful texton-based generative model.
    !
    equivalent to a dynamic s-PDE model.
    Motion clouds:
    Psychophysics:
    !
    Interpretable through inverse Bayesian estimation.
    !
    Making use of MC parameters (not yet fully).
    Future work: application to VSD in-vivo imaging.

    View full-size slide