Talk at Neuromathematics seminar, EITN, Paris
Dynamical Texture
Synthesis to Probe
Visual Perception
Gabriel Peyré
www.numericaltours.com
Joint work with:
Jonathan Vacher, Laurent Perrinet, Andrew Meso
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
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
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.
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.
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.
Natural Textures, Stationarity and Scales
IEEE TRANSACTIONS ON IMAGE PROCESSING
g. 1. Some examples of microtextures 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 microtexture is displayed together with an outcome of the RPN algorithm to its right. These
icrotextures are reasonably well emulated by RPN. Homogeneous regions that have lost their geometric details due to distance are often well simulated by
Overview
• Gaussian Texture Synthesis by Example
• Spot Noise Models and Motion Clouds
• Stochastic PDE Models
• Bayesian Brain and Prior Estimation
Exemplar f0
Texture Synthesis
Problem: given f0
, generate f
“random”
perceptually “similar”
analysis
Probability
distribution
µ = µ(p)
Exemplar f0
Texture Synthesis
Problem: given f0
, generate f
“random”
perceptually “similar”
analysis synthesis
Probability
distribution
µ = µ(p)
Exemplar f0
Outputs f µ(p)
Texture Synthesis
Problem: given f0
, generate f
“random”
perceptually “similar”
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”
Input exemplar:
d = 1 (grayscale), d = 3 (color)
N1
N2
N3
Images Videos
f0
RN d
Gaussian Texture Model
N1
N2
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
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 underdetermined problem.
Texture analysis: from f0
RN d, learn (m, ).
f0
RN d
Gaussian Texture Model
N1
N2
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 underdetermined 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
Stationarity hypothesis: X(· + ) X
(periodic BC)
Spot Noise Model [Galerne et al.]
Stationarity hypothesis: X(· + ) X
(periodic BC)
Blockdiagonal 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.]
Stationarity hypothesis: X(· + ) X
(periodic BC)
Blockdiagonal 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.]
Stationarity hypothesis: X(· + ) X
(periodic BC)
Blockdiagonal 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.]
Stationarity hypothesis: X(· + ) X
(periodic BC)
Blockdiagonal 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 rank1.
MLE of :
Maximum likelihood estimate (MLE) of m from f0
:
i, mi
=
1
N
x
f0
(x) Rd
Spot Noise Model [Galerne et al.]
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
)
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
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
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
).
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,
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,
Input
Spot Noise Barycenters
Input
Spot Noise Barycenters
Dynamic Textures Mixing
Dynamic Textures Mixing
Dynamic Textures Mixing
Dynamic Textures Mixing
Overview
• Gaussian Texture Synthesis by Example
• Spot Noise Models and Motion Clouds
• Stochastic PDE Models
• Bayesian Brain and Prior Estimation
Static Spot Noise
“Texton” pattern
g
.
(
Xp)p2N
2D Poisson point process of intensity ,
E
(#
{Xp
2 U}
) =
U
Spot noise: I
(
x
) def.
=
1 X
p
g
(
x Xp)
Static Spot Noise
“Texton” pattern
g
.
(
Xp)p2N
2D 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 autocorrelation of
g
.
Proposition: I is stationary of covariance ⌃(x, x
0
) = (x x
0
)
Static Spot Noise
“Texton” pattern
g
.
(
Xp)p2N
2D 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 autocorrelation of
g
.
Proposition: I is stationary of covariance ⌃(x, x
0
) = (x x
0
)
"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]
Dynamical Spot Noise
static microtextures [5] and dynamic natural phenomena [17]. The simplicity of this
allows for a ﬁne tuning of frequencybased (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
twofold. 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 2D 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
2D Poisson point process of intensity ,
E
(#
{Xp
2 U}
) =
U
(
Xp, Vp, Ap)p2N
independent random vectors.
Warpings 'a : R2 ! R2.
Spot noise:
Dynamical Spot Noise
static microtextures [5] and dynamic natural phenomena [17]. The simplicity of this
allows for a ﬁne tuning of frequencybased (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
twofold. 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 2D 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 autocorrelation 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
2D 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
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 Gaborlike 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 powerspectrum 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:
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 Gaborlike 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 powerspectrum 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 Gaborlike 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 powerspectrum 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 powerspectrum
8
(⇠, ⌧)
2 R2 ⇥ R
, ˆ(⇠, ⌧) =
P
Z (

⇠

)

⇠
2
P
⇥ (
\
⇠)
L
(
P

V v0
)
✓
⌧

⇠


v0

cos(
\
v0
\
⇠)
◆
Rotations + zooms:
Independency:
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 Gaborlike 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 powerspectrum 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 Gaborlike 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 powerspectrum 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 powerspectrum
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 Gaborlike 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 powerspectrum 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 powerspectrum
8
(⇠, ⌧)
2 R2 ⇥ R
, ˆ(⇠, ⌧) =
P
Z (

⇠

)

⇠
2
P
⇥ (
\
⇠)
L
(
P

V v0
)
✓
⌧

⇠


v0

cos(
\
v0
\
⇠)
◆
R
⇡
Rotations + zooms:
Independency:
Radial speed variations:
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 Gaborlike 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 powerspectrum 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 Gaborlike 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 powerspectrum 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 powerspectrum
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 Gaborlike 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 powerspectrum 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 powerspectrum
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
'
Example of Parameterization
ightly different bellfunction (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 conelike shape of the
elopes– and an example of synthesized dynamics for narrowband and broadband 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 bellfunction (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 conelike shape of t
envelopes– and an example of synthesized dynamics for narrowband and broadband 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 conelike shape of
envelopes– and an example of synthesized dynamics for narrowband and broadband 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 conelike shape of the
velopes– and an example of synthesized dynamics for narrowband and broadband 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 trialanderror strategy between
odelers and psychophysicists. Herein, we justiﬁed these different points in a constructive manner.
4 sPDE Formulation and Numerical Synthesis Algorithm
Drifting Grating vs. Anisotropic Noise
⇠2
⇠2
⌧
Drifting Grating vs. Anisotropic Noise
⇠2
⇠2
⌧
Drifting Grating vs. Anisotropic Noise
⇠1
⇠2
⌧
⇠2
⇠2
⌧
⇠2
Translating Anisotropic vs. Isotropic Noise
⇠2
⇠2
⌧
Translating Anisotropic vs. Isotropic Noise
⇠2
⇠2
⌧
Translating Anisotropic vs. Isotropic Noise
⇠1
⇠2
⌧
⇠2
⌧
⇠2
⇠2
Powerlaw Spacial Frequency
⇠2
⇠2
⌧
Powerlaw Spacial Frequency
⇠2
⇠2
⌧
Powerlaw Spacial Frequency
⇠1
⇠2
⌧
⇠2
⌧
⇠2
⇠2
Overview
• Gaussian Texture Synthesis by Example
• Spot Noise Models and Motion Clouds
• Stochastic PDE Models
• Bayesian Brain and Prior Estimation
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 2D covariance
⌃W
in space and
?
he spatial convolution operator. The parameters
(↵, )
are 2D 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 2D covariance
⌃W
in space an
is the spatial convolution operator. The parameters
(↵, )
are 2D 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 2D covariance
⌃W
in sp
is the spatial convolution operator. The parameters
(↵, )
are 2D 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
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 2D covariance
⌃W
in space and
?
he spatial convolution operator. The parameters
(↵, )
are 2D 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 2D covariance
⌃W
in space an
is the spatial convolution operator. The parameters
(↵, )
are 2D 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 2D covariance
⌃W
in sp
is the spatial convolution operator. The parameters
(↵, )
are 2D 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
↵(⇠), 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 motionless setting
since we deal here with
I0
), i.e. when
v0 = 0
. This covariance should be localized around 0 and
onoscillating. It thus make sense to constrain
(ˆ
↵(⇠), 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
↵(⇠), 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 motionless setting
since we deal here with
I0
), i.e. when
v0 = 0
. This covariance should be localized around 0 and
onoscillating. It thus make sense to constrain
(ˆ
↵(⇠), 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:
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 2D covariance
⌃W
in space and
?
he spatial convolution operator. The parameters
(↵, )
are 2D 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 2D covariance
⌃W
in space an
is the spatial convolution operator. The parameters
(↵, )
are 2D 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 2D covariance
⌃W
in sp
is the spatial convolution operator. The parameters
(↵, )
are 2D 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
↵(⇠), 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 motionless setting
since we deal here with
I0
), i.e. when
v0 = 0
. This covariance should be localized around 0 and
onoscillating. It thus make sense to constrain
(ˆ
↵(⇠), 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
↵(⇠), 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 motionless setting
since we deal here with
I0
), i.e. when
v0 = 0
. This covariance should be localized around 0 and
onoscillating. It thus make sense to constrain
(ˆ
↵(⇠), 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
↵(⇠), 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
nonoscillating. It thus make sense to constrain
(ˆ
↵(⇠), 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
⇠
.
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 2D covariance
⌃W
in space and
?
he spatial convolution operator. The parameters
(↵, )
are 2D 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 2D covariance
⌃W
in space an
is the spatial convolution operator. The parameters
(↵, )
are 2D 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 2D covariance
⌃W
in sp
is the spatial convolution operator. The parameters
(↵, )
are 2D 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
↵(⇠), 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 motionless setting
since we deal here with
I0
), i.e. when
v0 = 0
. This covariance should be localized around 0 and
onoscillating. It thus make sense to constrain
(ˆ
↵(⇠), 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
↵(⇠), 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 motionless setting
since we deal here with
I0
), i.e. when
v0 = 0
. This covariance should be localized around 0 and
onoscillating. It thus make sense to constrain
(ˆ
↵(⇠), 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
↵(⇠), 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
nonoscillating. It thus make sense to constrain
(ˆ
↵(⇠), 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
⌫
(
⇠
◆
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
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
'
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 welldeﬁned. 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
'
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 2D 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 2D Dirac distribution and where
(W
(
`
)
)`
are i.i.d. 2D 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 timestationarity) 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 1D case (stochastic ODE). We implemented the recursion
computing the 2D 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 dataanalysis 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 2D 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. 2D 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
1D 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
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 2D 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 2D Dirac distribution and where
(W
(
`
)
)`
are i.i.d. 2D 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 timestationarity) 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 1D case (stochastic ODE). We implemented the recursion
computing the 2D 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 dataanalysis 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 2D 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. 2D 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
1D 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 2D 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 2D Dirac distribution and where
(W
(
`
)
)`
are i.i.d. 2D 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 timestationarity) 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 1D case (stochastic ODE). We implemented the recursion (8)
computing the 2D 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
Autoregressive model:
AR(2)
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 2D 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 2D Dirac distribution and where
(W
(
`
)
)`
are i.i.d. 2D 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 timestationarity) 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 1D case (stochastic ODE). We implemented the recursion
computing the 2D 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 dataanalysis 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 2D 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. 2D 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
1D 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 2D 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 2D Dirac distribution and where
(W
(
`
)
)`
are i.i.d. 2D 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 timestationarity) 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 1D case (stochastic ODE). We implemented the recursion (8)
computing the 2D 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
Autoregressive model:
AR(2)
Slowfast Morphing
Slow translating grating morphs into
fast translating isotropic noise with
orthogonal direction
Nonstationarity in time: change (
↵, , W ) with
`
, rotations/zooms, etc.
Translating anisotropic noise
morphs into isotropic noise
Slowfast Morphing
Slow translating grating morphs into
fast translating isotropic noise with
orthogonal direction
Nonstationarity in time: change (
↵, , W ) with
`
, rotations/zooms, etc.
Translating anisotropic noise
morphs into isotropic noise
Natural Parameter Morphing
Natural noise morphing from
rough to smooth
Natural noise morphing from
anisotropic to isotropic
Natural noise with
rotating anisotropy
Natural Parameter Morphing
Natural noise morphing from
rough to smooth
Natural noise morphing from
anisotropic to isotropic
Natural noise with
rotating anisotropy
Rotations / Zooms
Rotations / Zooms
Overview
• Gaussian Texture Synthesis by Example
• Spot Noise Models and Motion Clouds
• Stochastic PDE Models
• Bayesian Brain and Prior Estimation
(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 conelike shape of the
envelopes– and an example of synthesized dynamics for narrowband and broadband 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 conelike shape of the
envelopes– and an example of synthesized dynamics for narrowband and broadband 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)
(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 conelike shape of the
envelopes– and an example of synthesized dynamics for narrowband and broadband 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 conelike shape of the
envelopes– and an example of synthesized dynamics for narrowband and broadband 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)
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 AixMarseille 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 interstimulus 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 AixMarseille 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?)
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 AixMarseille 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 interstimulus 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 AixMarseille 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?)”
MAP Decision Process
v
z
P
MV,Z
random
noise
stimuli
likelihood
nuisance
parameter m
internal
representation
MAP Decision Process
v
z
P
MV,Z
P
V Z
ˆ
v = ˆ
vz(m)
random
noise
stimuli
deterministic
prior
likelihood
nuisance
parameter m
internal
representation
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 AixMarseille 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
MV,Z
P
V Z
ˆ
v = ˆ
vz(m)
random
noise
stimuli
deterministic
prior
likelihood
nuisance
parameter
MAP estimator:
m
internal
representation
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 AixMarseille 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
MV,Z
P
V Z
ˆ
v = ˆ
vz(m)
random
noise
stimuli
deterministic
prior
likelihood
nuisance
parameter
MAP estimator:
Mv,z
⇠ P
MV,Z
(·, v, z) ˆ
Vv,z
def.
= ˆ
vz(Mv,z)
(z, v)
Response model:
m
internal
representation
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 AixMarseille 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
MV,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
MV,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)
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 AixMarseille 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
MV,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
MV,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
MV,Z
, P
V Z) by
comparing ˆ
'
and
'
.
Low Noise Estimation Process
Prior/likelihood estimation: impossible problem
prior/likelihood decomposition
PV M
⇠ PMV
PV is ambiguous
low likelihood variance hypothesis (low noise)
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 v2
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
(ˆ
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 v2
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
(ˆ
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
⇠ PMV
PV is ambiguous
too many things to estimate
low likelihood variance hypothesis (low noise)
low dimensional parameterization:
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 v2
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
(ˆ
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 v2
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
(ˆ
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
⇠ PMV
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
MV,Z
(·, v, z) ˆ
Vv,z
def.
= ˆ
vz(Mv,z)
(z, v)
Response model:
v
P
Mv,z
Pˆ
Vv,z
v az
2
z
Bias
Theoretical Psychometric Curve
Mv,z
⇠ P
MV,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
Pˆ
Vv,z?
Pˆ
Vv?,z
P
Mv?,z
P
Mv,z?
Theoretical Psychometric Curve
Mv,z
⇠ P
MV,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
(ˆ
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
Pˆ
Vv,z?
Pˆ
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
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?
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
(ˆ
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:
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
(ˆ
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:
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
(ˆ
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?
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
(ˆ
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?
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 AixMarseille 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 interstimulus 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
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 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 interstimulus 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 interstimulus 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 interstimulus 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 conelike shape of the
envelopes– and an example of synthesized dynamics for narrowband and broadband 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 trialanderror 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 conelike shape of the
envelopes– and an example of synthesized dynamics for narrowband and broadband 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 trialanderror strategy between
modelers and psychophysicists. Herein, we justiﬁed these different points in a constructive manner.
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 AixMarseille 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 interstimulus 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
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 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 interstimulus 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 interstimulus 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 interstimulus 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 conelike shape of the
envelopes– and an example of synthesized dynamics for narrowband and broadband 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 trialanderror 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 conelike shape of the
envelopes– and an example of synthesized dynamics for narrowband and broadband 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 trialanderror strategy between
modelers and psychophysicists. Herein, we justiﬁed these different points in a constructive manner.
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 AixMarseille 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 interstimulus 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
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 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 interstimulus 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 interstimulus 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 interstimulus 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 conelike shape of the
envelopes– and an example of synthesized dynamics for narrowband and broadband 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 trialanderror 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 conelike shape of the
envelopes– and an example of synthesized dynamics for narrowband and broadband 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 trialanderror strategy between
modelers and psychophysicists. Herein, we justiﬁed these different points in a constructive manner.
µz,z?
z
z
µz,z?
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
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}
Conclusion
Gaussian texture models:
! surprisingly e cient.
!
simple to estimate/manipulate.
!
equivalent to spot noise.
Conclusion
Gaussian texture models:
! surprisingly e cient.
!
simple to estimate/manipulate.
!
equivalent to spot noise.
Conclusion
Gaussian texture models:
! surprisingly e cient.
!
simple to estimate/manipulate.
!
equivalent to spot noise.
!
subclass of Gaussian spotnoises.
!
meaningful textonbased generative model.
!
equivalent to a dynamic sPDE model.
Motion clouds:
Conclusion
Gaussian texture models:
! surprisingly e cient.
!
simple to estimate/manipulate.
!
equivalent to spot noise.
!
subclass of Gaussian spotnoises.
!
meaningful textonbased generative model.
!
equivalent to a dynamic sPDE model.
Motion clouds:
Psychophysics:
!
Interpretable through inverse Bayesian estimation.
!
Making use of MC parameters (not yet fully).
Conclusion
Gaussian texture models:
! surprisingly e cient.
!
simple to estimate/manipulate.
!
equivalent to spot noise.
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subclass of Gaussian spotnoises.
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meaningful textonbased generative model.
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equivalent to a dynamic sPDE model.
Motion clouds:
Psychophysics:
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Interpretable through inverse Bayesian estimation.
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Making use of MC parameters (not yet fully).
Future work: application to VSD invivo imaging.