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
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.
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.
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.
g. 1. Some examples of micro-textures taken from a single image (water with sand, clouds, sand, waves with water ground, pebbles). The emplacements the original textures are displayed with red rectangles. Each micro-texture is displayed together with an outcome of the RPN algorithm to its right. These icro-textures are reasonably well emulated by RPN. Homogeneous regions that have lost their geometric details due to distance are often well simulated by
N1 N2 N3 Images Videos Gaussian model: m RN d, RNd Nd X µ = N(m, ) Texture synthesis: given (m, ), draw a realization f = X( ). highly under-determined problem. Factorize = AA (e.g. Cholesky). Compute f = m + Aw where w drawn from N(0, Id). Texture analysis: from f0 RN d, learn (m, ). f0 RN d Gaussian Texture Model N1 N2
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.]
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.]
covariance: i,j = 1 N x f0 (i + x) f0 (j + x) Rd d ˆ y( ) = ˆ( ) ˆ f( ) y = f computed as where ˆ f( ) = x f(x)e 2ix1⇥1 N1 + 2ix2⇥2 N2 = 0, ˆ( ) = ˆ f0 ( ) ˆ f0 ( ) Cd d is a spot noise = 0, ˆ( ) is rank-1. MLE of : Maximum likelihood estimate (MLE) of m from f0 : i, mi = 1 N x f0 (x) Rd Spot Noise Model [Galerne et al.]
Poisson point process of intensity , E (# {Xp 2 U} ) = |U| Spot noise: I ( x ) def. = 1 X p g ( x Xp) ( x ) = Z g ( y ) g ( x y )d y auto-correlation of g . Proposition: I is stationary of covariance ⌃(x, x 0 ) = (x x 0 )
Poisson point process of intensity , E (# {Xp 2 U} ) = |U| I !+1 ! I1 a stationary Gaussian ﬁeld of variance ⌃. Spot noise: EORY AND SYNTHESIS 3 rithms SN for I ( x ) def. = 1 X p g ( x Xp) ( x ) = Z g ( y ) g ( x y )d y auto-correlation of g . Proposition: I is stationary of covariance ⌃(x, x 0 ) = (x x 0 )
[17]. The simplicity of this allows for a ﬁne tuning of frequency-based (Fourier) parameterization, which is desira interpretation of psychophysical experiments. We deﬁne a random ﬁeld as I (x, t) def. = 1 p X p 2N g('Ap (x Xp Vpt)) where 'a : R2 ! R2 is a planar warping parameterized by a ﬁnite dimensional vector a . this model corresponds to a dense mixing of stereotyped, static textons as in [5]. The or two-fold. First, the components of this mixing are derived from the texton by visual trans 'Ap which may correspond to arbitrary transformations such as zooms or rotations, ill Figure 1. Second, we explicitly model the motion (position Xp and speed Vp ) of each texton. The parameters (Xp, Vp, Ap)p 2N are independent random vectors. They acco variability in the position of objects or observers and their speed, thus mimicking natural an ambient scene. The set of translations (Xp)p 2N is a 2-D Poisson point process of inten The following section instantiates this idea and proposes canonical choices for these v The warping parameters (Ap)p are distributed according to a distribution P A . The speed (Vp)p are distributed according to a distribution P V on R2. The following result show model (2) converges to a stationary Gaussian ﬁeld and gives the parameterization of the c Its proof follows from a specialization of [4, Theorem 3.1] to our setting. 3 “Texton” pattern g . ( Xp)p2N 2-D Poisson point process of intensity , E (# {Xp 2 U} ) = |U| ( Xp, Vp, Ap)p2N independent random vectors. Warpings 'a : R2 ! R2. Spot noise:
[17]. The simplicity of this allows for a ﬁne tuning of frequency-based (Fourier) parameterization, which is desira interpretation of psychophysical experiments. We deﬁne a random ﬁeld as I (x, t) def. = 1 p X p 2N g('Ap (x Xp Vpt)) where 'a : R2 ! R2 is a planar warping parameterized by a ﬁnite dimensional vector a . this model corresponds to a dense mixing of stereotyped, static textons as in [5]. The or two-fold. First, the components of this mixing are derived from the texton by visual trans 'Ap which may correspond to arbitrary transformations such as zooms or rotations, ill Figure 1. Second, we explicitly model the motion (position Xp and speed Vp ) of each texton. The parameters (Xp, Vp, Ap)p 2N are independent random vectors. They acco variability in the position of objects or observers and their speed, thus mimicking natural an ambient scene. The set of translations (Xp)p 2N is a 2-D Poisson point process of inten The following section instantiates this idea and proposes canonical choices for these v The warping parameters (Ap)p are distributed according to a distribution P A . The speed (Vp)p are distributed according to a distribution P V on R2. The following result show model (2) converges to a stationary Gaussian ﬁeld and gives the parameterization of the c Its proof follows from a specialization of [4, Theorem 3.1] to our setting. 3 on 1. I is stationary with bounded second order moments. Its covariance is t 0 ) = (x x 0 , t t 0 ) where satisﬁes 8 (x, t) 2 R3 , (x, t) = Z Z R 2 cg('a(x ⌫t)) P V (⌫) P A(a)d⌫da (3) = g ? ¯ g is the auto-correlation of g . When ! + 1, it converges (in the sense of ﬁnite al distributions) toward a stationary Gaussian ﬁeld I of zero mean and covariance ⌃ . nition of “Motion Clouds” here this model where the warpings are rotations and scalings (see Figure 1). This allows t for the characteristic orientations and sizes (or spatial scales) in a scene with respect to er 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, 'a(x) def. = zR ✓(x), “Texton” pattern g . ( Xp)p2N 2-D Poisson point process of intensity , E (# {Xp 2 U} ) = |U| Proposition: I is stationary of covariance ⌃(x, t, x 0 , t 0 ) = (x x 0 , t t 0 ) where cg = g ? ¯ g. I !+1 ! I1 a stationary Gaussian ﬁeld of variance ⌃. ( Xp, Vp, Ap)p2N independent random vectors. Warpings 'a : R2 ! R2. Spot noise: EORY AND SYNTHESIS 3 rithms SN for
80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 We detail here this model where the warpings are rotations and scalings (see Figure 1). This al to account for the characteristic orientations and sizes (or spatial scales) in a scene with respe the observer 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, 'a(x) def. = zR ✓(x), where R✓ is the planar rotation of angle ✓ . We now give some physical and biological motiva underlying our particular choice for the distributions of the parameters. We assume that the dist tions P Z and P ⇥ of spatial scales z and orientations ✓ , respectively (see Figure 1), are indepen and have densities, thus considering 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, P A(a) = P Z(z) P ⇥(✓). The speed vector ⌫ are assumed to be randomly ﬂuctuating around a central speed v0 , so that 8 ⌫ 2 R2 , P V (⌫) = P || V v0 ||( || ⌫ v0 || ). In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen deﬁne the texton g to be equal to a standard receptive ﬁeld of V1 , i.e. an oriented Gabor-like a having a scale and a central frequency ⇠0 . Since the rotation and scale of the texton is handle the (✓, z) parameters, we can impose without loss of generality the normalization ⇠0 = (1, 0 the special case where ! 0 , g is a grating of frequency ⇠0 , and the image I is a dense mix of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositio Its proof can be found in the supplementary materials. We call this Gaussian ﬁeld a Motion C (MC), and it is parameterized by the envelopes ( P Z, P ⇥, P V ) and has central frequency and s (⇠0, v0) . Note that it is possible to consider any arbitrary textons g , which would give rise to m complicated parameterizations for the power spectrum ˆ g , but we decided here to stick to the sim case of gratings. Proposition 2. When g(x) = ei h x, ⇠0 i, the image I deﬁned in Proposition 1 is a stationary Gaus Rotations + zooms:
80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 We detail here this model where the warpings are rotations and scalings (see Figure 1). This al to account for the characteristic orientations and sizes (or spatial scales) in a scene with respe the observer 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, 'a(x) def. = zR ✓(x), where R✓ is the planar rotation of angle ✓ . We now give some physical and biological motiva underlying our particular choice for the distributions of the parameters. We assume that the dist tions P Z and P ⇥ of spatial scales z and orientations ✓ , respectively (see Figure 1), are indepen and have densities, thus considering 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, P A(a) = P Z(z) P ⇥(✓). The speed vector ⌫ are assumed to be randomly ﬂuctuating around a central speed v0 , so that 8 ⌫ 2 R2 , P V (⌫) = P || V v0 ||( || ⌫ v0 || ). In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen deﬁne the texton g to be equal to a standard receptive ﬁeld of V1 , i.e. an oriented Gabor-like a having a scale and a central frequency ⇠0 . Since the rotation and scale of the texton is handle the (✓, z) parameters, we can impose without loss of generality the normalization ⇠0 = (1, 0 the special case where ! 0 , g is a grating of frequency ⇠0 , and the image I is a dense mix of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositio Its proof can be found in the supplementary materials. We call this Gaussian ﬁeld a Motion C (MC), and it is parameterized by the envelopes ( P Z, P ⇥, P V ) and has central frequency and s (⇠0, v0) . Note that it is possible to consider any arbitrary textons g , which would give rise to m complicated parameterizations for the power spectrum ˆ g , but we decided here to stick to the sim case of gratings. Proposition 2. When g(x) = ei h x, ⇠0 i, the image I deﬁned in Proposition 1 is a stationary Gaus + a ✓ where R✓ is the planar rotation of angle ✓ . We now give some physical and biological motivat underlying our particular choice for the distributions of the parameters. We assume that the distri tions P Z and P ⇥ of spatial scales z and orientations ✓ , respectively (see Figure 1), are independ and have densities, thus considering 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, P A(a) = P Z(z) P ⇥(✓). The speed vector ⌫ are assumed to be randomly ﬂuctuating around a central speed v0 , so that 8 ⌫ 2 R2 , P V (⌫) = P || V v0 ||( || ⌫ v0 || ). In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sense deﬁne the texton g to be equal to a standard receptive ﬁeld of V1 , i.e. an oriented Gabor-like ato having a scale and a central frequency ⇠0 . Since the rotation and scale of the texton is handled the (✓, z) parameters, we can impose without loss of generality the normalization ⇠0 = (1, 0) . the special case where ! 0 , g is a grating of frequency ⇠0 , and the image I is a dense mixt of drifting gratings, whose power-spectrum has a closed form expression detailed in Proposition Its proof can be found in the supplementary materials. We call this Gaussian ﬁeld a Motion Clo (MC), and it is parameterized by the envelopes ( P Z, P ⇥, P V ) and has central frequency and sp (⇠0, v0) . Note that it is possible to consider any arbitrary textons g , which would give rise to m complicated parameterizations for the power spectrum ˆ g , but we decided here to stick to the sim case of gratings. Proposition 2. When g(x) = ei h x, ⇠0 i, the image I deﬁned in Proposition 1 is a stationary Gauss ﬁeld of covariance having the power-spectrum 8 (⇠, ⌧) 2 R2 ⇥ R , ˆ(⇠, ⌧) = P Z ( || ⇠ || ) || ⇠ ||2 P ⇥ ( \ ⇠) L ( P || V v0 ||) ✓ ⌧ || ⇠ || || v0 || cos( \ v0 \ ⇠) ◆ Rotations + zooms: Independency:
80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 We detail here this model where the warpings are rotations and scalings (see Figure 1). This al to account for the characteristic orientations and sizes (or spatial scales) in a scene with respe the observer 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, 'a(x) def. = zR ✓(x), where R✓ is the planar rotation of angle ✓ . We now give some physical and biological motiva underlying our particular choice for the distributions of the parameters. We assume that the dist tions P Z and P ⇥ of spatial scales z and orientations ✓ , respectively (see Figure 1), are indepen and have densities, thus considering 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, P A(a) = P Z(z) P ⇥(✓). The speed vector ⌫ are assumed to be randomly ﬂuctuating around a central speed v0 , so that 8 ⌫ 2 R2 , P V (⌫) = P || V v0 ||( || ⌫ v0 || ). In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen deﬁne the texton g to be equal to a standard receptive ﬁeld of V1 , i.e. an oriented Gabor-like a having a scale and a central frequency ⇠0 . Since the rotation and scale of the texton is handle the (✓, z) parameters, we can impose without loss of generality the normalization ⇠0 = (1, 0 the special case where ! 0 , g is a grating of frequency ⇠0 , and the image I is a dense mix of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositio Its proof can be found in the supplementary materials. We call this Gaussian ﬁeld a Motion C (MC), and it is parameterized by the envelopes ( P Z, P ⇥, P V ) and has central frequency and s (⇠0, v0) . Note that it is possible to consider any arbitrary textons g , which would give rise to m complicated parameterizations for the power spectrum ˆ g , but we decided here to stick to the sim case of gratings. Proposition 2. When g(x) = ei h x, ⇠0 i, the image I deﬁned in Proposition 1 is a stationary Gaus + a ✓ where R✓ is the planar rotation of angle ✓ . We now give some physical and biological motivat underlying our particular choice for the distributions of the parameters. We assume that the distri tions P Z and P ⇥ of spatial scales z and orientations ✓ , respectively (see Figure 1), are independ and have densities, thus considering 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, P A(a) = P Z(z) P ⇥(✓). The speed vector ⌫ are assumed to be randomly ﬂuctuating around a central speed v0 , so that 8 ⌫ 2 R2 , P V (⌫) = P || V v0 ||( || ⌫ v0 || ). In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sense deﬁne the texton g to be equal to a standard receptive ﬁeld of V1 , i.e. an oriented Gabor-like ato having a scale and a central frequency ⇠0 . Since the rotation and scale of the texton is handled the (✓, z) parameters, we can impose without loss of generality the normalization ⇠0 = (1, 0) . the special case where ! 0 , g is a grating of frequency ⇠0 , and the image I is a dense mixt of drifting gratings, whose power-spectrum has a closed form expression detailed in Proposition Its proof can be found in the supplementary materials. We call this Gaussian ﬁeld a Motion Clo (MC), and it is parameterized by the envelopes ( P Z, P ⇥, P V ) and has central frequency and sp (⇠0, v0) . Note that it is possible to consider any arbitrary textons g , which would give rise to m complicated parameterizations for the power spectrum ˆ g , but we decided here to stick to the sim case of gratings. Proposition 2. When g(x) = ei h x, ⇠0 i, the image I deﬁned in Proposition 1 is a stationary Gauss ﬁeld of covariance having the power-spectrum 8 (⇠, ⌧) 2 R2 ⇥ R , ˆ(⇠, ⌧) = P Z ( || ⇠ || ) || ⇠ ||2 P ⇥ ( \ ⇠) L ( P || V v0 ||) ✓ ⌧ || ⇠ || || v0 || cos( \ v0 \ ⇠) ◆ 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 where R✓ is the planar rotation of angle ✓ . We now give some physical and biological motiv underlying our particular choice for the distributions of the parameters. We assume that the dis tions P Z and P ⇥ of spatial scales z and orientations ✓ , respectively (see Figure 1), are indepe and have densities, thus considering 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, P A(a) = P Z(z) P ⇥(✓). The speed vector ⌫ are assumed to be randomly ﬂuctuating around a central speed v0 , so that 8 ⌫ 2 R2 , P V (⌫) = P || V v0 ||( || ⌫ v0 || ). In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen deﬁne the texton g to be equal to a standard receptive ﬁeld of V1 , i.e. an oriented Gabor-like a having a scale and a central frequency ⇠0 . Since the rotation and scale of the texton is handl the (✓, z) parameters, we can impose without loss of generality the normalization ⇠0 = (1, 0 the special case where ! 0 , g is a grating of frequency ⇠0 , and the image I is a dense mi of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositi Its proof can be found in the supplementary materials. We call this Gaussian ﬁeld a Motion C (MC), and it is parameterized by the envelopes ( P Z, P ⇥, P V ) and has central frequency and s (⇠0, v0) . Note that it is possible to consider any arbitrary textons g , which would give rise to complicated parameterizations for the power spectrum ˆ g , but we decided here to stick to the si case of gratings. Proposition 2. When g(x) = ei h x, ⇠0 i, the image I deﬁned in Proposition 1 is a stationary Gau ﬁeld of covariance having the power-spectrum 8 (⇠, ⌧) 2 R2 ⇥ R , ˆ(⇠, ⌧) = P Z ( || ⇠ || ) || ⇠ ||2 P ⇥ ( \ ⇠) L ( P || V v0 ||) ✓ ⌧ || ⇠ || || v0 || cos( \ v0 \ ⇠) ◆ R ⇡ Rotations + zooms: Independency: Radial speed variations:
80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 We detail here this model where the warpings are rotations and scalings (see Figure 1). This al to account for the characteristic orientations and sizes (or spatial scales) in a scene with respe the observer 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, 'a(x) def. = zR ✓(x), where R✓ is the planar rotation of angle ✓ . We now give some physical and biological motiva underlying our particular choice for the distributions of the parameters. We assume that the dist tions P Z and P ⇥ of spatial scales z and orientations ✓ , respectively (see Figure 1), are indepen and have densities, thus considering 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, P A(a) = P Z(z) P ⇥(✓). The speed vector ⌫ are assumed to be randomly ﬂuctuating around a central speed v0 , so that 8 ⌫ 2 R2 , P V (⌫) = P || V v0 ||( || ⌫ v0 || ). In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen deﬁne the texton g to be equal to a standard receptive ﬁeld of V1 , i.e. an oriented Gabor-like a having a scale and a central frequency ⇠0 . Since the rotation and scale of the texton is handle the (✓, z) parameters, we can impose without loss of generality the normalization ⇠0 = (1, 0 the special case where ! 0 , g is a grating of frequency ⇠0 , and the image I is a dense mix of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositio Its proof can be found in the supplementary materials. We call this Gaussian ﬁeld a Motion C (MC), and it is parameterized by the envelopes ( P Z, P ⇥, P V ) and has central frequency and s (⇠0, v0) . Note that it is possible to consider any arbitrary textons g , which would give rise to m complicated parameterizations for the power spectrum ˆ g , but we decided here to stick to the sim case of gratings. Proposition 2. When g(x) = ei h x, ⇠0 i, the image I deﬁned in Proposition 1 is a stationary Gaus + a ✓ where R✓ is the planar rotation of angle ✓ . We now give some physical and biological motivat underlying our particular choice for the distributions of the parameters. We assume that the distri tions P Z and P ⇥ of spatial scales z and orientations ✓ , respectively (see Figure 1), are independ and have densities, thus considering 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, P A(a) = P Z(z) P ⇥(✓). The speed vector ⌫ are assumed to be randomly ﬂuctuating around a central speed v0 , so that 8 ⌫ 2 R2 , P V (⌫) = P || V v0 ||( || ⌫ v0 || ). In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sense deﬁne the texton g to be equal to a standard receptive ﬁeld of V1 , i.e. an oriented Gabor-like ato having a scale and a central frequency ⇠0 . Since the rotation and scale of the texton is handled the (✓, z) parameters, we can impose without loss of generality the normalization ⇠0 = (1, 0) . the special case where ! 0 , g is a grating of frequency ⇠0 , and the image I is a dense mixt of drifting gratings, whose power-spectrum has a closed form expression detailed in Proposition Its proof can be found in the supplementary materials. We call this Gaussian ﬁeld a Motion Clo (MC), and it is parameterized by the envelopes ( P Z, P ⇥, P V ) and has central frequency and sp (⇠0, v0) . Note that it is possible to consider any arbitrary textons g , which would give rise to m complicated parameterizations for the power spectrum ˆ g , but we decided here to stick to the sim case of gratings. Proposition 2. When g(x) = ei h x, ⇠0 i, the image I deﬁned in Proposition 1 is a stationary Gauss ﬁeld of covariance having the power-spectrum 8 (⇠, ⌧) 2 R2 ⇥ R , ˆ(⇠, ⌧) = P Z ( || ⇠ || ) || ⇠ ||2 P ⇥ ( \ ⇠) L ( P || V v0 ||) ✓ ⌧ || ⇠ || || v0 || cos( \ v0 \ ⇠) ◆ 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 where R✓ is the planar rotation of angle ✓ . We now give some physical and biological motiv underlying our particular choice for the distributions of the parameters. We assume that the dis tions P Z and P ⇥ of spatial scales z and orientations ✓ , respectively (see Figure 1), are indepe and have densities, thus considering 8 a = (✓, z) 2 [ ⇡, ⇡) ⇥ R⇤ +, P A(a) = P Z(z) P ⇥(✓). The speed vector ⌫ are assumed to be randomly ﬂuctuating around a central speed v0 , so that 8 ⌫ 2 R2 , P V (⌫) = P || V v0 ||( || ⌫ v0 || ). In order to obtain “optimal” responses to the stimulation (as advocated by [18]), it makes sen deﬁne the texton g to be equal to a standard receptive ﬁeld of V1 , i.e. an oriented Gabor-like a having a scale and a central frequency ⇠0 . Since the rotation and scale of the texton is handl the (✓, z) parameters, we can impose without loss of generality the normalization ⇠0 = (1, 0 the special case where ! 0 , g is a grating of frequency ⇠0 , and the image I is a dense mi of drifting gratings, whose power-spectrum has a closed form expression detailed in Propositi Its proof can be found in the supplementary materials. We call this Gaussian ﬁeld a Motion C (MC), and it is parameterized by the envelopes ( P Z, P ⇥, P V ) and has central frequency and s (⇠0, v0) . Note that it is possible to consider any arbitrary textons g , which would give rise to complicated parameterizations for the power spectrum ˆ g , but we decided here to stick to the si case of gratings. Proposition 2. When g(x) = ei h x, ⇠0 i, the image I deﬁned in Proposition 1 is a stationary Gau ﬁeld of covariance having the power-spectrum 8 (⇠, ⌧) 2 R2 ⇥ R , ˆ(⇠, ⌧) = P Z ( || ⇠ || ) || ⇠ ||2 P ⇥ ( \ ⇠) L ( P || V v0 ||) ✓ ⌧ || ⇠ || || v0 || cos( \ v0 \ ⇠) ◆ R ⇡ Rotations + zooms: Independency: Radial speed variations: Proposition: for g(x) = e i x1 , I1 satisﬁes ⇠1 ⇠2 ⌧ ⇠2 ⌧ + hv0, ⇠i = 0 Speed variations. Spatial angular variations. Spatial radial variations. 8 (⇠, ⌧) 2 R2 ⇥ R, ˆ(⇠, ⌧) = P Z (||⇠||) ||⇠||2 P ✓ (\⇠) L(P ||V v0 || ) ✓ ⌧ + hv0, ⇠i ||⇠|| ◆ where L ( f )( u ) def. = R ⇡/2 ⇡/2 f ( u/ cos( ' ))d '
expression) should be used to obtain an ct equivalence with the sPDE discretization mentionned in Section 2.4. e distributions of the parameters are thus chosen as P Z(z) / z0 z e ln( z z 0 ) 2 2 ln(1+ 2 Z ) , P ⇥(✓) / e cos(2( ✓ ✓ 0)) 2 ⇥ and P || V v0 ||(r) / e r2 2 2 V . (6) z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z ope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z ure 2: Graphical representation of the covariance (left) —note the cone-like shape of the elopes– and an example of synthesized dynamics for narrow-band and broad-band Motion uds (right). gging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- ing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude a slightly different bell-function (with a more complicated expression) should be used to obtain exact equivalence with the sPDE discretization mentionned in Section 2.4. The distributions of the parameters are thus chosen as P Z(z) / z0 z e ln( z z 0 ) 2 2 ln(1+ 2 Z ) , P ⇥(✓) / e cos(2( ✓ ✓ 0)) 2 ⇥ and P || V v0 ||(r) / e r2 2 2 V . ( z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencie Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of t envelopes– and an example of synthesized dynamics for narrow-band and broad-band Moti Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion clou one obtains a parameterization which is very similar to the one originally introduced in [9]. The fo lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitu and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows exact equivalence with the sPDE discretization mentionned in Section 2.4. The distributions of the parameters are thus chosen as P Z(z) / z0 z e ln( z z 0 ) 2 2 ln(1+ 2 Z ) , P ⇥(✓) / e cos(2( ✓ ✓ 0)) 2 ⇥ and P || V v0 ||(r) / e r2 2 2 V . z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequenci Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of envelopes– and an example of synthesized dynamics for narrow-band and broad-band Mot Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion clo one obtains a parameterization which is very similar to the one originally introduced in [9]. The f lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitu and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 show 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 The distributions of the parameters are thus chosen as P Z(z) / z0 z e ln( z z 0 ) 2 2 ln(1+ 2 Z ) , P ⇥(✓) / e cos(2( ✓ ✓ 0)) 2 ⇥ and z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space MC of Figure 2: Graphical representation of the covariance (left) — envelopes– and an example of synthesized dynamics for narro Clouds (right). 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 The distributions of the parameters are thus chosen as P Z(z) / z0 z e ln( z z 0 ) 2 2 ln(1+ 2 Z ) , P ⇥(✓) / e cos(2( ✓ ✓ 0)) 2 ⇥ z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space MC Figure 2: Graphical representation of the covariance (lef envelopes– and an example of synthesized dynamics for n Clouds (right). z0 Z ⇠1 ⇠1 ✓0 z0 Z lope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z gure 2: Graphical representation of the covariance (left) —note the cone-like shape of the velopes– and an example of synthesized dynamics for narrow-band and broad-band Motion ouds (right). ugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, e obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- wing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude d orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a aphical display of the inﬂuence of these parameters. Speed Freq. orient. Freq. amplitude (mean, dispersion) (v0, V ) (✓0, ⇥) (z0, Z) mark 2. Note that the ﬁnal envelope of ˆ is in agreement with the formulation that is used in [8]. owever, this latter derivation was based on a heuristic following a trial-and-error strategy between odelers and psychophysicists. Herein, we justiﬁed these different points in a constructive manner. 4 sPDE Formulation and Numerical Synthesis Algorithm
I with speed v0 can be obtained from a MC I0 with zero speed by the constant speed time rping I(x, t) def. = I0(x v0t, t). (2) now restrict our attention to I0 . consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation (sPDE) of form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) (3) is equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are stationary utions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. corresponding to temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in space and ? he spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at enforcing additional correlation in time of the model. Section 2.2 explains how to choose (↵, , ⌃W ) so t the stationary solutions of (3) have the power spectrum given in (1) (in the case that v0 = 0 ), are motion clouds. is sPDE formulation is important since we aim to deal with dynamic stimulation, which should described by a causal equation which is local in time. This is crucial for numerical simulation explained in Section 2.4) but also to simplify the application of MC inside a bayesian model of ychophysical experiments (see Section 3). hile it is beyond the scope of this paper to study theoretically this equation, one can shows exis- ce and uniqueness results of stationary solutions for this class of sPDE under stability conditions the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁed in our I(x, t) def. = I0(x v0t, t). We now restrict our attention to I0 . We consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation (sPDE the form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) This equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are station solutions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. correspondin the temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in space an is the spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at enforc an additional correlation in time of the model. Section 2.2 explains how to choose (↵, , ⌃W ) that the stationary solutions of (3) have the power spectrum given in (1) (in the case that v0 = i.e. are motion clouds. This sPDE formulation is important since we aim to deal with dynamic stimulation, which sho be described by a causal equation which is local in time. This is crucial for numerical simulat (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian mode psychophysical experiments (see Section 3). While it is beyond the scope of this paper to study theoretically this equation, one can shows e tence and uniqueness results of stationary solutions for this class of sPDE under stability conditi on the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁed in simulations. Note also that one can show that in fact the stationary solutions to (3) all share same law. These solutions can be obtained by solving the sODE (4) forward for time t > t0 w arbitrary boundary conditions at time t = t0 , and letting t0 ! 1. This is consistent with numerical scheme detailed in Section 2.4. 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 I(x, t) def. = I0(x v0t, t). We now restrict our attention to I0 . We consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation ( the form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) This equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are s solutions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. corresp the temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in sp is the spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at an additional correlation in time of the model. Section 2.2 explains how to choose (↵, that the stationary solutions of (3) have the power spectrum given in (1) (in the case that i.e. are motion clouds. This sPDE formulation is important since we aim to deal with dynamic stimulation, whic be described by a causal equation which is local in time. This is crucial for numerical s (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian psychophysical experiments (see Section 3). While it is beyond the scope of this paper to study theoretically this equation, one can sh tence and uniqueness results of stationary solutions for this class of sPDE under stability c on the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁ simulations. Note also that one can show that in fact the stationary solutions to (3) all same law. These solutions can be obtained by solving the sODE (4) forward for time t > arbitrary boundary conditions at time t = t0 , and letting t0 ! 1. This is consistent Constant speed motion: sPDE: where
I with speed v0 can be obtained from a MC I0 with zero speed by the constant speed time rping I(x, t) def. = I0(x v0t, t). (2) now restrict our attention to I0 . consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation (sPDE) of form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) (3) is equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are stationary utions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. corresponding to temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in space and ? he spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at enforcing additional correlation in time of the model. Section 2.2 explains how to choose (↵, , ⌃W ) so t the stationary solutions of (3) have the power spectrum given in (1) (in the case that v0 = 0 ), are motion clouds. is sPDE formulation is important since we aim to deal with dynamic stimulation, which should described by a causal equation which is local in time. This is crucial for numerical simulation explained in Section 2.4) but also to simplify the application of MC inside a bayesian model of ychophysical experiments (see Section 3). hile it is beyond the scope of this paper to study theoretically this equation, one can shows exis- ce and uniqueness results of stationary solutions for this class of sPDE under stability conditions the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁed in our I(x, t) def. = I0(x v0t, t). We now restrict our attention to I0 . We consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation (sPDE the form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) This equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are station solutions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. correspondin the temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in space an is the spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at enforc an additional correlation in time of the model. Section 2.2 explains how to choose (↵, , ⌃W ) that the stationary solutions of (3) have the power spectrum given in (1) (in the case that v0 = i.e. are motion clouds. This sPDE formulation is important since we aim to deal with dynamic stimulation, which sho be described by a causal equation which is local in time. This is crucial for numerical simulat (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian mode psychophysical experiments (see Section 3). While it is beyond the scope of this paper to study theoretically this equation, one can shows e tence and uniqueness results of stationary solutions for this class of sPDE under stability conditi on the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁed in simulations. Note also that one can show that in fact the stationary solutions to (3) all share same law. These solutions can be obtained by solving the sODE (4) forward for time t > t0 w arbitrary boundary conditions at time t = t0 , and letting t0 ! 1. This is consistent with numerical scheme detailed in Section 2.4. 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 I(x, t) def. = I0(x v0t, t). We now restrict our attention to I0 . We consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation ( the form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) This equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are s solutions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. corresp the temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in sp is the spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at an additional correlation in time of the model. Section 2.2 explains how to choose (↵, that the stationary solutions of (3) have the power spectrum given in (1) (in the case that i.e. are motion clouds. This sPDE formulation is important since we aim to deal with dynamic stimulation, whic be described by a causal equation which is local in time. This is crucial for numerical s (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian psychophysical experiments (see Section 3). While it is beyond the scope of this paper to study theoretically this equation, one can sh tence and uniqueness results of stationary solutions for this class of sPDE under stability c on the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁ simulations. Note also that one can show that in fact the stationary solutions to (3) all same law. These solutions can be obtained by solving the sODE (4) forward for time t > arbitrary boundary conditions at time t = t0 , and letting t0 ! 1. This is consistent imulations. Note also that one can show that in fact the stationary solutions to (3) all share the ame law. These solutions can be obtained by solving the sODE (4) forward for time t > t0 with rbitrary boundary conditions at time t = t0 , and letting t0 ! 1. This is consistent with the umerical scheme detailed in Section 2.4. .2 Equivalence Between Spectral and sPDE MC Formulations The sPDE equation (3) corresponds to a set of independent stochastic ODEs over the spatial Fourier omain, which reads, for each frequency ⇠ , 8 t 2 R , @ 2 ˆ I0(⇠, t) @t 2 + ˆ ↵(⇠) @ ˆ I0(⇠, t) @t + ˆ(⇠)ˆ I0(⇠, t) = ˆW (⇠) ˆ w(⇠, t) (4) where ˆ I0(⇠, t) denotes the Fourier transform with respect to the space variable x only. Here, ˆW (⇠) 2 s the spatial power spectrum of @W @t , which means that ⌃W (x, y) = c(x y) where ˆ c(⇠) = ˆ 2 W (⇠). (5) Here ˆ w(⇠, t) ⇠ N (0, 1) and w is a white noise in space and time. This formulation makes explicit hat (ˆ ↵(⇠), ˆ(⇠)) should be chosen in order to make the temporal covariance of the resulting process qual (or at least approximate) the temporal covariance appearing in (1) in the motion-less setting since we deal here with I0 ), i.e. when v0 = 0 . This covariance should be localized around 0 and on-oscillating. It thus make sense to constrain (ˆ ↵(⇠), 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 motion-less setting since we deal here with I0 ), i.e. when v0 = 0 . This covariance should be localized around 0 and on-oscillating. It thus make sense to constrain (ˆ ↵(⇠), 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:
I with speed v0 can be obtained from a MC I0 with zero speed by the constant speed time rping I(x, t) def. = I0(x v0t, t). (2) now restrict our attention to I0 . consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation (sPDE) of form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) (3) is equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are stationary utions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. corresponding to temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in space and ? he spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at enforcing additional correlation in time of the model. Section 2.2 explains how to choose (↵, , ⌃W ) so t the stationary solutions of (3) have the power spectrum given in (1) (in the case that v0 = 0 ), are motion clouds. is sPDE formulation is important since we aim to deal with dynamic stimulation, which should described by a causal equation which is local in time. This is crucial for numerical simulation explained in Section 2.4) but also to simplify the application of MC inside a bayesian model of ychophysical experiments (see Section 3). hile it is beyond the scope of this paper to study theoretically this equation, one can shows exis- ce and uniqueness results of stationary solutions for this class of sPDE under stability conditions the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁed in our I(x, t) def. = I0(x v0t, t). We now restrict our attention to I0 . We consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation (sPDE the form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) This equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are station solutions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. correspondin the temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in space an is the spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at enforc an additional correlation in time of the model. Section 2.2 explains how to choose (↵, , ⌃W ) that the stationary solutions of (3) have the power spectrum given in (1) (in the case that v0 = i.e. are motion clouds. This sPDE formulation is important since we aim to deal with dynamic stimulation, which sho be described by a causal equation which is local in time. This is crucial for numerical simulat (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian mode psychophysical experiments (see Section 3). While it is beyond the scope of this paper to study theoretically this equation, one can shows e tence and uniqueness results of stationary solutions for this class of sPDE under stability conditi on the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁed in simulations. Note also that one can show that in fact the stationary solutions to (3) all share same law. These solutions can be obtained by solving the sODE (4) forward for time t > t0 w arbitrary boundary conditions at time t = t0 , and letting t0 ! 1. This is consistent with numerical scheme detailed in Section 2.4. 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 I(x, t) def. = I0(x v0t, t). We now restrict our attention to I0 . We consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation ( the form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) This equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are s solutions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. corresp the temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in sp is the spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at an additional correlation in time of the model. Section 2.2 explains how to choose (↵, that the stationary solutions of (3) have the power spectrum given in (1) (in the case that i.e. are motion clouds. This sPDE formulation is important since we aim to deal with dynamic stimulation, whic be described by a causal equation which is local in time. This is crucial for numerical s (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian psychophysical experiments (see Section 3). While it is beyond the scope of this paper to study theoretically this equation, one can sh tence and uniqueness results of stationary solutions for this class of sPDE under stability c on the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁ simulations. Note also that one can show that in fact the stationary solutions to (3) all same law. These solutions can be obtained by solving the sODE (4) forward for time t > arbitrary boundary conditions at time t = t0 , and letting t0 ! 1. This is consistent imulations. Note also that one can show that in fact the stationary solutions to (3) all share the ame law. These solutions can be obtained by solving the sODE (4) forward for time t > t0 with rbitrary boundary conditions at time t = t0 , and letting t0 ! 1. This is consistent with the umerical scheme detailed in Section 2.4. .2 Equivalence Between Spectral and sPDE MC Formulations The sPDE equation (3) corresponds to a set of independent stochastic ODEs over the spatial Fourier omain, which reads, for each frequency ⇠ , 8 t 2 R , @ 2 ˆ I0(⇠, t) @t 2 + ˆ ↵(⇠) @ ˆ I0(⇠, t) @t + ˆ(⇠)ˆ I0(⇠, t) = ˆW (⇠) ˆ w(⇠, t) (4) where ˆ I0(⇠, t) denotes the Fourier transform with respect to the space variable x only. Here, ˆW (⇠) 2 s the spatial power spectrum of @W @t , which means that ⌃W (x, y) = c(x y) where ˆ c(⇠) = ˆ 2 W (⇠). (5) Here ˆ w(⇠, t) ⇠ N (0, 1) and w is a white noise in space and time. This formulation makes explicit hat (ˆ ↵(⇠), ˆ(⇠)) should be chosen in order to make the temporal covariance of the resulting process qual (or at least approximate) the temporal covariance appearing in (1) in the motion-less setting since we deal here with I0 ), i.e. when v0 = 0 . This covariance should be localized around 0 and on-oscillating. It thus make sense to constrain (ˆ ↵(⇠), 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 motion-less setting since we deal here with I0 ), i.e. when v0 = 0 . This covariance should be localized around 0 and on-oscillating. It thus make sense to constrain (ˆ ↵(⇠), 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 non-oscillating. 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 ⇠ .
I with speed v0 can be obtained from a MC I0 with zero speed by the constant speed time rping I(x, t) def. = I0(x v0t, t). (2) now restrict our attention to I0 . consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation (sPDE) of form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) (3) is equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are stationary utions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. corresponding to temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in space and ? he spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at enforcing additional correlation in time of the model. Section 2.2 explains how to choose (↵, , ⌃W ) so t the stationary solutions of (3) have the power spectrum given in (1) (in the case that v0 = 0 ), are motion clouds. is sPDE formulation is important since we aim to deal with dynamic stimulation, which should described by a causal equation which is local in time. This is crucial for numerical simulation explained in Section 2.4) but also to simplify the application of MC inside a bayesian model of ychophysical experiments (see Section 3). hile it is beyond the scope of this paper to study theoretically this equation, one can shows exis- ce and uniqueness results of stationary solutions for this class of sPDE under stability conditions the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁed in our I(x, t) def. = I0(x v0t, t). We now restrict our attention to I0 . We consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation (sPDE the form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) This equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are station solutions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. correspondin the temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in space an is the spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at enforc an additional correlation in time of the model. Section 2.2 explains how to choose (↵, , ⌃W ) that the stationary solutions of (3) have the power spectrum given in (1) (in the case that v0 = i.e. are motion clouds. This sPDE formulation is important since we aim to deal with dynamic stimulation, which sho be described by a causal equation which is local in time. This is crucial for numerical simulat (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian mode psychophysical experiments (see Section 3). While it is beyond the scope of this paper to study theoretically this equation, one can shows e tence and uniqueness results of stationary solutions for this class of sPDE under stability conditi on the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁed in simulations. Note also that one can show that in fact the stationary solutions to (3) all share same law. These solutions can be obtained by solving the sODE (4) forward for time t > t0 w arbitrary boundary conditions at time t = t0 , and letting t0 ! 1. This is consistent with numerical scheme detailed in Section 2.4. 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 I(x, t) def. = I0(x v0t, t). We now restrict our attention to I0 . We consider Gaussian random ﬁelds deﬁned by a stochastic partial differential equation ( the form D (I0) = @W @t (x) where D (I0) def. = @ 2 I0 @t 2 (x) + ↵ ? @I0 @t (x) + ? I0(x) This equation should be satisﬁed for all (x, t) , and we look for Gaussian ﬁelds that are s solutions of this equation. In this sPDE, the driving noise @W @t is white in time (i.e. corresp the temporal derivative of a Brownian motion in time) and has a 2-D covariance ⌃W in sp is the spatial convolution operator. The parameters (↵, ) are 2-D spatial ﬁlters that aim at an additional correlation in time of the model. Section 2.2 explains how to choose (↵, that the stationary solutions of (3) have the power spectrum given in (1) (in the case that i.e. are motion clouds. This sPDE formulation is important since we aim to deal with dynamic stimulation, whic be described by a causal equation which is local in time. This is crucial for numerical s (as explained in Section 2.4) but also to simplify the application of MC inside a bayesian psychophysical experiments (see Section 3). While it is beyond the scope of this paper to study theoretically this equation, one can sh tence and uniqueness results of stationary solutions for this class of sPDE under stability c on the ﬁlers (↵, ) (see for instance [8]) that we found numerically to be always satisﬁ simulations. Note also that one can show that in fact the stationary solutions to (3) all same law. These solutions can be obtained by solving the sODE (4) forward for time t > arbitrary boundary conditions at time t = t0 , and letting t0 ! 1. This is consistent imulations. Note also that one can show that in fact the stationary solutions to (3) all share the ame law. These solutions can be obtained by solving the sODE (4) forward for time t > t0 with rbitrary boundary conditions at time t = t0 , and letting t0 ! 1. This is consistent with the umerical scheme detailed in Section 2.4. .2 Equivalence Between Spectral and sPDE MC Formulations The sPDE equation (3) corresponds to a set of independent stochastic ODEs over the spatial Fourier omain, which reads, for each frequency ⇠ , 8 t 2 R , @ 2 ˆ I0(⇠, t) @t 2 + ˆ ↵(⇠) @ ˆ I0(⇠, t) @t + ˆ(⇠)ˆ I0(⇠, t) = ˆW (⇠) ˆ w(⇠, t) (4) where ˆ I0(⇠, t) denotes the Fourier transform with respect to the space variable x only. Here, ˆW (⇠) 2 s the spatial power spectrum of @W @t , which means that ⌃W (x, y) = c(x y) where ˆ c(⇠) = ˆ 2 W (⇠). (5) Here ˆ w(⇠, t) ⇠ N (0, 1) and w is a white noise in space and time. This formulation makes explicit hat (ˆ ↵(⇠), ˆ(⇠)) should be chosen in order to make the temporal covariance of the resulting process qual (or at least approximate) the temporal covariance appearing in (1) in the motion-less setting since we deal here with I0 ), i.e. when v0 = 0 . This covariance should be localized around 0 and on-oscillating. It thus make sense to constrain (ˆ ↵(⇠), 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 motion-less setting since we deal here with I0 ), i.e. when v0 = 0 . This covariance should be localized around 0 and on-oscillating. It thus make sense to constrain (ˆ ↵(⇠), 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 non-oscillating. 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 ⌫ ( ⇠ ◆
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
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 '
168 169 170 171 172 173 174 175 176 177 denoted u = ˆ ⌫(⇠) . This impulse response is easily shown to be r(t) = te t/u 1R+ (t) . iance of ˆ I0(⇠, · ) is thus, after some computation, equal to ˆ 2 W (⇠)r ? ¯ r = ˆ 2 W (⇠)h( · /u) ) / (1 + | t | )e | t |. Taking the Fourier transform of this equality, the power spectrum ˆ0 of ads ˆ0(⇠, ⌧) = ˆ 2 W (⇠)ˆ ⌫(⇠)h(ˆ ⌫(⇠)⌧) where h(u) = 1 (1 + u 2 ) 2 e it should be noted that this h function is the same as the one introduced in (6). The e MC of I MC and MC 0 of I MC 0 are related by the relation ˆ MC 0 (⇠, ⌧) = ˆ MC (⇠, ⌧ h ⇠, v0 i ) = 1 || ⇠ ||2 P Z( || ⇠ || ) P ⇥ ( \ ⇠) h ✓ ⌧ V h ⇠, v0 i ◆ . used the expression (1) for ˆ MC and the value of L ( P || V v0 ||) given by (6). Condition (7) s that expression (2.2) and (2.2) coincide, and thus ˆ0 = ˆ MC 0 . ression for P || V v0 || (6) states that in order to obtain a perfect equivalence between the MC deﬁned by (1) and e function has L 1 (h) to be well-deﬁned. It means we need to compute the inverse of the of the linear operator L 8 u 2 R , L (f)(u) = Z ⇡ ⇡ f( u/ cos('))d'. ction h . Manipulation of the integral deﬁning L shows that one can actually write in closed (h) . The variable substitution x = cos(') allows to rewrite L as a Mellin convolution ld be then inverted using Mellin transform, see Figure 2. One obtains L 1 (h)(u) = 2 u 2 ⇡(1 + u 2 ) 2 u 2 (u 2 + 4)(log(u) log( p u 2 + 1 + 1)) ⇡(u 2 + 1) 5 / 2 . 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 || V v0 || V where L is deﬁned in (1), equation (4) admits a solution I which is a stationary Gaussian power spectrum (1) when setting ˆ 2 W (⇠) = 1 ˆ ⌫(⇠) || ⇠ ||2 P Z( || ⇠ || ) P ⇥( \ ⇠), and ˆ ⌫(⇠) = 1 V || ⇠ ||. Proof. For this proof, we denote I MC the motion cloud deﬁned by (1), and I a stationa of the sPDE deﬁned by (3). We aim at showing that under the speciﬁcation (7), they hav covariance. This is equivalent to showing that I MC 0 (x, t) = I MC (x+ct, t) has the same cov I0 . One shows that for any ﬁxed ⇠ , equation (4) admits a unique (in law) stationary soluti which is a stationary Gaussian process of zero mean and with a covariance which is ˆ where r is the impulse response (i.e. taking formally a = ) of the ODE r 00 + 2r 0 /u + r where we denoted u = ˆ ⌫(⇠) . This impulse response is easily shown to be r(t) = te The covariance of ˆ I0(⇠, · ) is thus, after some computation, equal to ˆ 2 W (⇠)r ? ¯ r = ˆ 2 W where h(t) / (1 + | t | )e | t |. Taking the Fourier transform of this equality, the power spec I0 thus reads ˆ0(⇠, ⌧) = ˆ 2 W (⇠)ˆ ⌫(⇠)h(ˆ ⌫(⇠)⌧) where h(u) = 1 (1 + u 2 ) 2 and where it should be noted that this h function is the same as the one introduced in covariance MC of I MC and MC 0 of I MC 0 are related by the relation ˆ MC 0 (⇠, ⌧) = ˆ MC (⇠, ⌧ h ⇠, v0 i ) = 1 || ⇠ ||2 P Z( || ⇠ || ) P ⇥ ( \ ⇠) h ✓ ⌧ V h ⇠, v0 i ◆ . where we used the expression (1) for ˆ MC and the value of L ( P || V v0 ||) given by (6). Co MC Theorem: selecting stationary solutions of the sPDE have covariance 8 (⇠, ⌧) 2 R2 ⇥ R, ˆ(⇠, ⌧) = P Z (||⇠||) ||⇠||2 P ✓ (\⇠) L(P ||V v0 || ) ✓ ⌧ + hv0, ⇠i ||⇠|| ◆ Motion cloud covariance: 8 (⇠, ⌧) 2 R2 ⇥ R, ˆsPDE (⇠, ⌧) = P Z (||⇠||) ||⇠||2 P ✓ (\⇠) h ✓ ⌧ + hv0, ⇠i V ||⇠|| ◆ where h(u) = (1 + u2) 2 Proposition: perfect equivalence for P ||V v0 || = L 1(h)(·/ V ) L ( f )( u ) def. = R ⇡/2 ⇡/2 f ( u/ cos( ' ))d '
195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 ﬁelds typically decays too fast in time. The detailed derivation of the AR(2) implementation can be found in the supplementary materials. The discretization computes a (possibly inﬁnite) discrete set of 2-D frames (I ( ` ) 0 )` > `0 separa a time step , and we approach at time t = ` the derivatives as @I0( · , t) @t ⇡ 1 (I ( ` ) 0 I ( ` 1) 0 ) and @ 2 I0( · , t) @t 2 ⇡ 2 (I ( ` +1) 0 + I ( ` 1) 0 2I ( ` ) 0 ), which leads to the following explicit recursion 8 ` > `0, I ( ` +1) 0 = (2 ↵ 2 ) ? I ( ` ) 0 + ( + ↵) ? I ( ` 1) 0 + 2 W ( ` ) , where is the 2-D Dirac distribution and where (W ( ` ) )` are i.i.d. 2-D Gaussian ﬁeld with di tion N (0, ⌃W ) , and (I ( `0 1) 0 , I ( `0 1) 0 ) can be arbitrary initialized. One can show that when `0 ! 1 (to allow for a long enough “warmup” phase to reach a imate time-stationarity) and ! 0 , then I0 deﬁned by interpolating I0 ( · , `) = I ( ` ) con (in the sense of ﬁnite dimensional distributions) toward a solution I0 of the sPDE (3). W to [9] for a similar result in the 1-D case (stochastic ODE). We implemented the recursion computing the 2-D convolutions with FFT’s on a GPU, which allows us to generate high res videos in real time, without the need to explicitly store the synthesized video. 3 Experimental Likelihood vs. the MC Model In our paper, we propose to directly ﬁt the likelihood P M | V,Z(m | v, z) from the experiment chophysical curve. While this makes sense from a data-analysis point of view, this required modeling hypothesis, in particular, that the likelihood is Gaussian with a variance 2 z indep of the parameter v to be estimated by the observer. s formulation (3). Indeed, numerical simulations show that AR(1) mporal artifacts: in particular, the time correlation of AR(1) random time. The detailed derivation of the AR(2) implementation of MC y materials. ossibly inﬁnite) discrete set of 2-D frames (I ( ` ) 0 )` > `0 separated by at time t = ` the derivatives as ( ` 1) 0 ) and @ 2 I0( · , t) @t 2 ⇡ 2 (I ( ` +1) 0 + I ( ` 1) 0 2I ( ` ) 0 ), licit recursion ↵ 2 ) ? I ( ` ) 0 + ( + ↵) ? I ( ` 1) 0 + 2 W ( ` ) , (8) tion and where (W ( ` ) )` are i.i.d. 2-D Gaussian ﬁeld with distribu- 0 1) ) can be arbitrary initialized. 1 (to allow for a long enough “warmup” phase to reach approx- ! 0 , then I0 deﬁned by interpolating I0 ( · , `) = I ( ` ) converges nal distributions) toward a solution I0 of the sPDE (3). We refer 1-D case (stochastic ODE). We implemented the recursion (8) by with FFT’s on a GPU, which allows us to generate high resolution eed to explicitly store the synthesized video. ood vs. the MC Model ctly ﬁt the likelihood P M | V,Z(m | v, z) from the experimental psy- Discretization: ﬁnite di↵erence in time
195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 ﬁelds typically decays too fast in time. The detailed derivation of the AR(2) implementation can be found in the supplementary materials. The discretization computes a (possibly inﬁnite) discrete set of 2-D frames (I ( ` ) 0 )` > `0 separa a time step , and we approach at time t = ` the derivatives as @I0( · , t) @t ⇡ 1 (I ( ` ) 0 I ( ` 1) 0 ) and @ 2 I0( · , t) @t 2 ⇡ 2 (I ( ` +1) 0 + I ( ` 1) 0 2I ( ` ) 0 ), which leads to the following explicit recursion 8 ` > `0, I ( ` +1) 0 = (2 ↵ 2 ) ? I ( ` ) 0 + ( + ↵) ? I ( ` 1) 0 + 2 W ( ` ) , where is the 2-D Dirac distribution and where (W ( ` ) )` are i.i.d. 2-D Gaussian ﬁeld with di tion N (0, ⌃W ) , and (I ( `0 1) 0 , I ( `0 1) 0 ) can be arbitrary initialized. One can show that when `0 ! 1 (to allow for a long enough “warmup” phase to reach a imate time-stationarity) and ! 0 , then I0 deﬁned by interpolating I0 ( · , `) = I ( ` ) con (in the sense of ﬁnite dimensional distributions) toward a solution I0 of the sPDE (3). W to [9] for a similar result in the 1-D case (stochastic ODE). We implemented the recursion computing the 2-D convolutions with FFT’s on a GPU, which allows us to generate high res videos in real time, without the need to explicitly store the synthesized video. 3 Experimental Likelihood vs. the MC Model In our paper, we propose to directly ﬁt the likelihood P M | V,Z(m | v, z) from the experiment chophysical curve. While this makes sense from a data-analysis point of view, this required modeling hypothesis, in particular, that the likelihood is Gaussian with a variance 2 z indep of the parameter v to be estimated by the observer. s formulation (3). Indeed, numerical simulations show that AR(1) mporal artifacts: in particular, the time correlation of AR(1) random time. The detailed derivation of the AR(2) implementation of MC y materials. ossibly inﬁnite) discrete set of 2-D frames (I ( ` ) 0 )` > `0 separated by at time t = ` the derivatives as ( ` 1) 0 ) and @ 2 I0( · , t) @t 2 ⇡ 2 (I ( ` +1) 0 + I ( ` 1) 0 2I ( ` ) 0 ), licit recursion ↵ 2 ) ? I ( ` ) 0 + ( + ↵) ? I ( ` 1) 0 + 2 W ( ` ) , (8) tion and where (W ( ` ) )` are i.i.d. 2-D Gaussian ﬁeld with distribu- 0 1) ) can be arbitrary initialized. 1 (to allow for a long enough “warmup” phase to reach approx- ! 0 , then I0 deﬁned by interpolating I0 ( · , `) = I ( ` ) converges nal distributions) toward a solution I0 of the sPDE (3). We refer 1-D case (stochastic ODE). We implemented the recursion (8) by with FFT’s on a GPU, which allows us to generate high resolution eed to explicitly store the synthesized video. ood vs. the MC Model ctly ﬁt the likelihood P M | V,Z(m | v, z) from the experimental psy- AR(2) regression (in place of a ﬁrst order AR(1) model). Using higher order recursions is crucia be consistent with the continuous formulation (3). Indeed, numerical simulations show that AR iterations lead to unacceptable temporal artifacts: in particular, the time correlation of AR(1) rand ﬁelds typically decays too fast in time. The detailed derivation of the AR(2) implementation of M can be found in the supplementary materials. The discretization computes a (possibly inﬁnite) discrete set of 2-D frames (I ( ` ) 0 )` > `0 separated a time step , and we approach at time t = ` the derivatives as @I0( · , t) @t ⇡ 1 (I ( ` ) 0 I ( ` 1) 0 ) and @ 2 I0( · , t) @t 2 ⇡ 2 (I ( ` +1) 0 + I ( ` 1) 0 2I ( ` ) 0 ), which leads to the following explicit recursion 8 ` > `0, I ( ` +1) 0 = (2 ↵ 2 ) ? I ( ` ) 0 + ( + ↵) ? I ( ` 1) 0 + 2 W ( ` ) , where is the 2-D Dirac distribution and where (W ( ` ) )` are i.i.d. 2-D Gaussian ﬁeld with distri tion N (0, ⌃W ) , and (I ( `0 1) 0 , I ( `0 1) 0 ) can be arbitrary initialized. One can show that when `0 ! 1 (to allow for a long enough “warmup” phase to reach appr imate time-stationarity) and ! 0 , then I0 deﬁned by interpolating I0 ( · , `) = I ( ` ) conver (in the sense of ﬁnite dimensional distributions) toward a solution I0 of the sPDE (3). We re to [9] for a similar result in the 1-D case (stochastic ODE). We implemented the recursion (8) computing the 2-D convolutions with FFT’s on a GPU, which allows us to generate high resolut videos in real time, without the need to explicitly store the synthesized video. 3 Experimental Likelihood vs. the MC Model Discretization: ﬁnite di↵erence in time Auto-regressive model: AR(2)
195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 ﬁelds typically decays too fast in time. The detailed derivation of the AR(2) implementation can be found in the supplementary materials. The discretization computes a (possibly inﬁnite) discrete set of 2-D frames (I ( ` ) 0 )` > `0 separa a time step , and we approach at time t = ` the derivatives as @I0( · , t) @t ⇡ 1 (I ( ` ) 0 I ( ` 1) 0 ) and @ 2 I0( · , t) @t 2 ⇡ 2 (I ( ` +1) 0 + I ( ` 1) 0 2I ( ` ) 0 ), which leads to the following explicit recursion 8 ` > `0, I ( ` +1) 0 = (2 ↵ 2 ) ? I ( ` ) 0 + ( + ↵) ? I ( ` 1) 0 + 2 W ( ` ) , where is the 2-D Dirac distribution and where (W ( ` ) )` are i.i.d. 2-D Gaussian ﬁeld with di tion N (0, ⌃W ) , and (I ( `0 1) 0 , I ( `0 1) 0 ) can be arbitrary initialized. One can show that when `0 ! 1 (to allow for a long enough “warmup” phase to reach a imate time-stationarity) and ! 0 , then I0 deﬁned by interpolating I0 ( · , `) = I ( ` ) con (in the sense of ﬁnite dimensional distributions) toward a solution I0 of the sPDE (3). W to [9] for a similar result in the 1-D case (stochastic ODE). We implemented the recursion computing the 2-D convolutions with FFT’s on a GPU, which allows us to generate high res videos in real time, without the need to explicitly store the synthesized video. 3 Experimental Likelihood vs. the MC Model In our paper, we propose to directly ﬁt the likelihood P M | V,Z(m | v, z) from the experiment chophysical curve. While this makes sense from a data-analysis point of view, this required modeling hypothesis, in particular, that the likelihood is Gaussian with a variance 2 z indep of the parameter v to be estimated by the observer. s formulation (3). Indeed, numerical simulations show that AR(1) mporal artifacts: in particular, the time correlation of AR(1) random time. The detailed derivation of the AR(2) implementation of MC y materials. ossibly inﬁnite) discrete set of 2-D frames (I ( ` ) 0 )` > `0 separated by at time t = ` the derivatives as ( ` 1) 0 ) and @ 2 I0( · , t) @t 2 ⇡ 2 (I ( ` +1) 0 + I ( ` 1) 0 2I ( ` ) 0 ), licit recursion ↵ 2 ) ? I ( ` ) 0 + ( + ↵) ? I ( ` 1) 0 + 2 W ( ` ) , (8) tion and where (W ( ` ) )` are i.i.d. 2-D Gaussian ﬁeld with distribu- 0 1) ) can be arbitrary initialized. 1 (to allow for a long enough “warmup” phase to reach approx- ! 0 , then I0 deﬁned by interpolating I0 ( · , `) = I ( ` ) converges nal distributions) toward a solution I0 of the sPDE (3). We refer 1-D case (stochastic ODE). We implemented the recursion (8) by with FFT’s on a GPU, which allows us to generate high resolution eed to explicitly store the synthesized video. ood vs. the MC Model ctly ﬁt the likelihood P M | V,Z(m | v, z) from the experimental psy- AR(2) regression (in place of a ﬁrst order AR(1) model). Using higher order recursions is crucia be consistent with the continuous formulation (3). Indeed, numerical simulations show that AR iterations lead to unacceptable temporal artifacts: in particular, the time correlation of AR(1) rand ﬁelds typically decays too fast in time. The detailed derivation of the AR(2) implementation of M can be found in the supplementary materials. The discretization computes a (possibly inﬁnite) discrete set of 2-D frames (I ( ` ) 0 )` > `0 separated a time step , and we approach at time t = ` the derivatives as @I0( · , t) @t ⇡ 1 (I ( ` ) 0 I ( ` 1) 0 ) and @ 2 I0( · , t) @t 2 ⇡ 2 (I ( ` +1) 0 + I ( ` 1) 0 2I ( ` ) 0 ), which leads to the following explicit recursion 8 ` > `0, I ( ` +1) 0 = (2 ↵ 2 ) ? I ( ` ) 0 + ( + ↵) ? I ( ` 1) 0 + 2 W ( ` ) , where is the 2-D Dirac distribution and where (W ( ` ) )` are i.i.d. 2-D Gaussian ﬁeld with distri tion N (0, ⌃W ) , and (I ( `0 1) 0 , I ( `0 1) 0 ) can be arbitrary initialized. One can show that when `0 ! 1 (to allow for a long enough “warmup” phase to reach appr imate time-stationarity) and ! 0 , then I0 deﬁned by interpolating I0 ( · , `) = I ( ` ) conver (in the sense of ﬁnite dimensional distributions) toward a solution I0 of the sPDE (3). We re to [9] for a similar result in the 1-D case (stochastic ODE). We implemented the recursion (8) computing the 2-D convolutions with FFT’s on a GPU, which allows us to generate high resolut videos in real time, without the need to explicitly store the synthesized video. 3 Experimental Likelihood vs. the MC Model Discretization: ﬁnite di↵erence in time Auto-regressive model: AR(2)
noise with orthogonal direction Non-stationarity in time: change ( ↵, , W ) with ` , rotations/zooms, etc. Translating anisotropic noise morphs into isotropic noise
noise with orthogonal direction Non-stationarity in time: change ( ↵, , W ) with ` , rotations/zooms, etc. Translating anisotropic noise morphs into isotropic noise
( v, 0). Two alternative forced choice (2AFC) Nuisance: spatial frequency z def. = z0. 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 The distributions of the parameters are thus chosen as P Z(z) / z0 z e ln( z z 0 ) 2 2 ln(1+ 2 Z ) , P ⇥(✓) / e cos(2( ✓ ✓ 0)) 2 ⇥ and P || V v0 ||(r) / e r2 2 2 V . (6) z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a graphical display of the inﬂuence of these parameters. Speed Freq. orient. Freq. amplitude (mean, dispersion) (v0, V ) (✓0, ⇥) (z0, Z) Remark 2. Note that the ﬁnal envelope of ˆ is in agreement with the formulation that is used in [8]. 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 The distributions of the parameters are thus chosen as P Z(z) / z0 z e ln( z z 0 ) 2 2 ln(1+ 2 Z ) , P ⇥(✓) / e cos(2( ✓ ✓ 0)) 2 ⇥ and P || V v0 ||(r) / e r2 2 2 V . (6) z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a graphical display of the inﬂuence of these parameters. Speed Freq. orient. Freq. amplitude (mean, dispersion) (v0, V ) (✓0, ⇥) (z0, Z) Remark 2. Note that the ﬁnal envelope of ˆ is in agreement with the formulation that is used in [8]. (v2, z2) (v1, z1) (v2, z2)
( v, 0). Two alternative forced choice (2AFC) Nuisance: spatial frequency z def. = z0. 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 The distributions of the parameters are thus chosen as P Z(z) / z0 z e ln( z z 0 ) 2 2 ln(1+ 2 Z ) , P ⇥(✓) / e cos(2( ✓ ✓ 0)) 2 ⇥ and P || V v0 ||(r) / e r2 2 2 V . (6) z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a graphical display of the inﬂuence of these parameters. Speed Freq. orient. Freq. amplitude (mean, dispersion) (v0, V ) (✓0, ⇥) (z0, Z) Remark 2. Note that the ﬁnal envelope of ˆ is in agreement with the formulation that is used in [8]. 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 The distributions of the parameters are thus chosen as P Z(z) / z0 z e ln( z z 0 ) 2 2 ln(1+ 2 Z ) , P ⇥(✓) / e cos(2( ✓ ✓ 0)) 2 ⇥ and P || V v0 ||(r) / e r2 2 2 V . (6) z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a graphical display of the inﬂuence of these parameters. Speed Freq. orient. Freq. amplitude (mean, dispersion) (v0, V ) (✓0, ⇥) (z0, Z) Remark 2. Note that the ﬁnal envelope of ˆ is in agreement with the formulation that is used in [8]. (v2, z2) (v1, z1) (v2, z2)
end of the trial, a grey screen appears asking the participant to report which one of the two intervals was perceived as moving faster by pressing one of two buttons, that is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected so that ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16, 0.47 } , where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across trials), and where z values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitions of each of the 25 possible combinations of these parameters are made per block of 250 trials and at least four such blocks were collected per condition tested. The outcome of these experiments are summarized by psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the value ˆ 'v ? ,z ? (v, z) is the empirical probability (averaged over the 40 trials) that a stimulus generated with parameters (v? , z) is moving faster than a stimulus with parameters (v, z? ) . To asses the validity of our model, we tested four different scenarios by considering all possible choices among z? = 0.78 c/ , v? 2 { 5 /s , 10 /s} , ⌧? 2 { 0.1s, 0.2s } , which corresponds to combinations of low/high speeds and temporal frequency bandwidths. Stimuli were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic p227f monitor with resolution 1024 ⇥ 768 at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9 controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three observers with normal or corrected to normal vision were used. They gave their informed consent and the experiments received ethical approval from the Aix-Marseille Ethics Committee in accordance with the declaration of Helsinki. a small dark ﬁxation spot was followed by two stimulus intervals of 250 ms each, separated by a grey 250 ms inter-stimulus interval. The ﬁrst stimulus has parameters (v1, z1) and the second one has parameters (v2, z2) . At the end of the trial, a grey screen appears asking the participant to report which one of the two intervals was perceived as moving faster by pressing one of two buttons, that is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected so that ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16, 0.47 } , where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across trials), and where z values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitions of each of the 25 possible combinations of these parameters are made per block of 250 trials and at least four such blocks were collected per condition tested. The outcome of these experiments are summarized by psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the value ˆ 'v ? ,z ? (v, z) is the empirical probability (averaged over the 40 trials) that a stimulus generated with parameters (v? , z) is moving faster than a stimulus with parameters (v, z? ) . To asses the validity of our model, we tested four different scenarios by considering all possible choices among z? = 0.78 c/ , v? 2 { 5 /s , 10 /s} , ⌧? 2 { 0.1s, 0.2s } , which corresponds to combinations of low/high speeds and temporal frequency bandwidths. Stimuli were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic p227f monitor with resolution 1024 ⇥ 768 at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9 controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three observers with normal or corrected to normal vision were used. They gave their informed consent and the experiments received ethical approval from the Aix-Marseille Ethics Committee in accordance with the declaration of Helsinki. 0 V z0 frequency bandwidth, as illustrated on the left of Figure 2. oice (2AFC) paradigm. In each trial a grey ﬁxation screen with wed by two stimulus intervals of 250 ms each, separated by a The ﬁrst stimulus has parameters (v1, z1) and the second one of the trial, a grey screen appears asking the participant to report perceived as moving faster by pressing one of two buttons, that r each trial, (v1, z1) and (v2, z2) are selected so that where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16, 0.47 } , , 1) (i.e. the ordering is randomized across trials), and where z egree (c/ ) and v values in /s. Ten repetitions of each of the 25 ameters are made per block of 250 trials and at least four such tested. The outcome of these experiments are summarized by or all (v v? , z z? ) 2 V ⇥ Z , the value ˆ 'v ? ,z ? (v, z) is the r the 40 trials) that a stimulus generated with parameters (v? , z) h parameters (v, z? ) . we tested four different scenarios by considering all possible v? 2 { 5 /s , 10 /s} , ⌧? 2 { 0.1s, 0.2s } , of low/high speeds and temporal frequency bandwidths. Stimuli S 10.6.8 and displayed on a 20” Viewsonic p227f monitor with utines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9 ervers sat 57 cm from the screen in a dark room. Three observers v z (v?, z?)
end of the trial, a grey screen appears asking the participant to report which one of the two intervals was perceived as moving faster by pressing one of two buttons, that is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected so that ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16, 0.47 } , where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across trials), and where z values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitions of each of the 25 possible combinations of these parameters are made per block of 250 trials and at least four such blocks were collected per condition tested. The outcome of these experiments are summarized by psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the value ˆ 'v ? ,z ? (v, z) is the empirical probability (averaged over the 40 trials) that a stimulus generated with parameters (v? , z) is moving faster than a stimulus with parameters (v, z? ) . To asses the validity of our model, we tested four different scenarios by considering all possible choices among z? = 0.78 c/ , v? 2 { 5 /s , 10 /s} , ⌧? 2 { 0.1s, 0.2s } , which corresponds to combinations of low/high speeds and temporal frequency bandwidths. Stimuli were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic p227f monitor with resolution 1024 ⇥ 768 at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9 controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three observers with normal or corrected to normal vision were used. They gave their informed consent and the experiments received ethical approval from the Aix-Marseille Ethics Committee in accordance with the declaration of Helsinki. a small dark ﬁxation spot was followed by two stimulus intervals of 250 ms each, separated by a grey 250 ms inter-stimulus interval. The ﬁrst stimulus has parameters (v1, z1) and the second one has parameters (v2, z2) . At the end of the trial, a grey screen appears asking the participant to report which one of the two intervals was perceived as moving faster by pressing one of two buttons, that is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected so that ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16, 0.47 } , where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across trials), and where z values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitions of each of the 25 possible combinations of these parameters are made per block of 250 trials and at least four such blocks were collected per condition tested. The outcome of these experiments are summarized by psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the value ˆ 'v ? ,z ? (v, z) is the empirical probability (averaged over the 40 trials) that a stimulus generated with parameters (v? , z) is moving faster than a stimulus with parameters (v, z? ) . To asses the validity of our model, we tested four different scenarios by considering all possible choices among z? = 0.78 c/ , v? 2 { 5 /s , 10 /s} , ⌧? 2 { 0.1s, 0.2s } , which corresponds to combinations of low/high speeds and temporal frequency bandwidths. Stimuli were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic p227f monitor with resolution 1024 ⇥ 768 at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9 controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three observers with normal or corrected to normal vision were used. They gave their informed consent and the experiments received ethical approval from the Aix-Marseille Ethics Committee in accordance with the declaration of Helsinki. Output: psychometric curves ˆ 'v?,z? (·, z?) 0 1 v v? ˆ 'v?,z? (·, z 6= z?) 0 1 v v? Bias 0 V z0 frequency bandwidth, as illustrated on the left of Figure 2. oice (2AFC) paradigm. In each trial a grey ﬁxation screen with wed by two stimulus intervals of 250 ms each, separated by a The ﬁrst stimulus has parameters (v1, z1) and the second one of the trial, a grey screen appears asking the participant to report perceived as moving faster by pressing one of two buttons, that r each trial, (v1, z1) and (v2, z2) are selected so that where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16, 0.47 } , , 1) (i.e. the ordering is randomized across trials), and where z egree (c/ ) and v values in /s. Ten repetitions of each of the 25 ameters are made per block of 250 trials and at least four such tested. The outcome of these experiments are summarized by or all (v v? , z z? ) 2 V ⇥ Z , the value ˆ 'v ? ,z ? (v, z) is the r the 40 trials) that a stimulus generated with parameters (v? , z) h parameters (v, z? ) . we tested four different scenarios by considering all possible v? 2 { 5 /s , 10 /s} , ⌧? 2 { 0.1s, 0.2s } , of low/high speeds and temporal frequency bandwidths. Stimuli S 10.6.8 and displayed on a 20” Viewsonic p227f monitor with utines were written using Matlab 7.10.0 and Psychtoolbox 3.0.9 ervers sat 57 cm from the screen in a dark room. Three observers v z (v?, z?) ˆ 'v?,z? (v, z) def. = “empirical probability that MC with parameter ( v?, z ) is perceived faster than MC with parameter (v, z?)”
320 321 322 323 resolution 1024 ⇥ 768 at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolb controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three o with normal or corrected to normal vision were used. They gave their informed consent experiments received ethical approval from the Aix-Marseille Ethics Committee in accorda the declaration of Helsinki. 3.2 Bayesian modeling To make full use of our MC paradigm in analyzing the obtained results, we follow the meth of the Bayesian observer used for instance in [12]. We assume the observer makes its decisi a Maximum A Posteriori (MAP) estimator ˆ vz(m) = argmin v [ log( P M | V,Z(m | v, z)) log( P V | Z(v | z))] 6 v z P M|V,Z P V |Z ˆ v = ˆ vz(m) random noise stimuli deterministic prior likelihood nuisance parameter MAP estimator: m internal representation
320 321 322 323 resolution 1024 ⇥ 768 at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolb controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three o with normal or corrected to normal vision were used. They gave their informed consent experiments received ethical approval from the Aix-Marseille Ethics Committee in accorda the declaration of Helsinki. 3.2 Bayesian modeling To make full use of our MC paradigm in analyzing the obtained results, we follow the meth of the Bayesian observer used for instance in [12]. We assume the observer makes its decisi a Maximum A Posteriori (MAP) estimator ˆ vz(m) = argmin v [ log( P M | V,Z(m | v, z)) log( P V | Z(v | z))] 6 v z P M|V,Z P V |Z ˆ v = ˆ vz(m) random noise stimuli deterministic prior likelihood nuisance parameter MAP estimator: Mv,z ⇠ P M|V,Z (·, v, z) ˆ Vv,z def. = ˆ vz(Mv,z) (z, v) Response model: m internal representation
320 321 322 323 resolution 1024 ⇥ 768 at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolb controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three o with normal or corrected to normal vision were used. They gave their informed consent experiments received ethical approval from the Aix-Marseille Ethics Committee in accorda the declaration of Helsinki. 3.2 Bayesian modeling To make full use of our MC paradigm in analyzing the obtained results, we follow the meth of the Bayesian observer used for instance in [12]. We assume the observer makes its decisi a Maximum A Posteriori (MAP) estimator ˆ vz(m) = argmin v [ log( P M | V,Z(m | v, z)) log( P V | Z(v | z))] 6 v z P M|V,Z P V |Z ˆ v = ˆ vz(m) random noise stimuli deterministic prior likelihood nuisance parameter MAP estimator: Theoretical psychophysical curve: Mv,z ⇠ P M|V,Z (·, v, z) ˆ Vv,z def. = ˆ vz(Mv,z) (z, v) Response model: 'v?,z? def. = E( ˆ Vv?,z > ˆ Vv,z? ) m internal representation 0 1 v v? ˆ 'v?,z? (·, z) 'v?,z? (·, z)
320 321 322 323 resolution 1024 ⇥ 768 at 100 Hz. Routines were written using Matlab 7.10.0 and Psychtoolb controlled the stimulus display. Observers sat 57 cm from the screen in a dark room. Three o with normal or corrected to normal vision were used. They gave their informed consent experiments received ethical approval from the Aix-Marseille Ethics Committee in accorda the declaration of Helsinki. 3.2 Bayesian modeling To make full use of our MC paradigm in analyzing the obtained results, we follow the meth of the Bayesian observer used for instance in [12]. We assume the observer makes its decisi a Maximum A Posteriori (MAP) estimator ˆ vz(m) = argmin v [ log( P M | V,Z(m | v, z)) log( P V | Z(v | z))] 6 v z P M|V,Z P V |Z ˆ v = ˆ vz(m) random noise stimuli deterministic prior likelihood nuisance parameter MAP estimator: Theoretical psychophysical curve: Mv,z ⇠ P M|V,Z (·, v, z) ˆ Vv,z def. = ˆ vz(Mv,z) (z, v) Response model: 'v?,z? def. = E( ˆ Vv?,z > ˆ Vv,z? ) m internal representation 0 1 v v? ˆ 'v?,z? (·, z) 'v?,z? (·, z) Inverse Bayesian estimation Compute ( P M|V,Z , P V |Z) by comparing ˆ ' and ' .
2 R of the observed stimulus. For simplicity, we ssume that the observer estimates z from m without bias. o simplify both the exposition and the numerical analysis, we assume that the likelihood is Gaus- an, with a variance independent of v . Furthermore, we assume that the prior is Laplacian as this ives a good description of the a priori statistics of speeds in natural images [2]: P M | V,Z(m | v, z) = 1 p 2⇡ z e |m v|2 2 2 z and P V | Z(v | z) / eazv 1[0,vmax](v). (8) here vmax > 0 is a cutoff speed ensuring that P V | Z is a well deﬁned density even if az > 0 . oth az and z are unknown parameters of the model, and are obtained from the outcome of the xperiments by a ﬁtting process we now explain. .3 Likelihood and Prior Estimation ollowing for instance [12], the theoretical psychophysical curve obtained by a Bayesian decision model is 'v ? ,z ? (v, z) def. = E (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 v|2 2 2 z and P V | Z(v | z) / eazv 1[0,vmax](v). (8) where vmax > 0 is a cutoff speed ensuring that P V | Z is a well deﬁned density even if az > 0 . Both az and z are unknown parameters of the model, and are obtained from the outcome of the experiments by a ﬁtting process we now explain. 3.3 Likelihood and Prior Estimation Following for instance [12], the theoretical psychophysical curve obtained by a Bayesian decision model is 'v ? ,z ? (v, z) def. = E (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 ⇠ PM|V PV is ambiguous too many things to estimate low likelihood variance hypothesis (low noise) low dimensional parameterization:
2 R of the observed stimulus. For simplicity, we ssume that the observer estimates z from m without bias. o simplify both the exposition and the numerical analysis, we assume that the likelihood is Gaus- an, with a variance independent of v . Furthermore, we assume that the prior is Laplacian as this ives a good description of the a priori statistics of speeds in natural images [2]: P M | V,Z(m | v, z) = 1 p 2⇡ z e |m v|2 2 2 z and P V | Z(v | z) / eazv 1[0,vmax](v). (8) here vmax > 0 is a cutoff speed ensuring that P V | Z is a well deﬁned density even if az > 0 . oth az and z are unknown parameters of the model, and are obtained from the outcome of the xperiments by a ﬁtting process we now explain. .3 Likelihood and Prior Estimation ollowing for instance [12], the theoretical psychophysical curve obtained by a Bayesian decision model is 'v ? ,z ? (v, z) def. = E (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 v|2 2 2 z and P V | Z(v | z) / eazv 1[0,vmax](v). (8) where vmax > 0 is a cutoff speed ensuring that P V | Z is a well deﬁned density even if az > 0 . Both az and z are unknown parameters of the model, and are obtained from the outcome of the experiments by a ﬁtting process we now explain. 3.3 Likelihood and Prior Estimation Following for instance [12], the theoretical psychophysical curve obtained by a Bayesian decision model is 'v ? ,z ? (v, z) def. = E (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 ⇠ PM|V PV is ambiguous too many things to estimate low likelihood variance hypothesis (low noise) low dimensional parameterization: Proposition: Mvz ⇠ N(v, 2 z ) ˆ Vvz ⇠ N(v az 2 z , 2 z ) Mv,z ⇠ P M|V,Z (·, v, z) ˆ Vv,z def. = ˆ vz(Mv,z) (z, v) Response model: v P Mv,z Pˆ Vv,z v az 2 z Bias
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
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?
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:
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:
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?
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?
315 316 317 318 319 320 321 322 323 is moving faster than a stimulus with parameters (v, z? ) . To asses the validity of our model, we tested four different scenarios by cons choices among z? = 0.78 c/ , v? 2 { 5 /s , 10 /s} , ⌧? 2 { 0.1s, 0.2s } , which corresponds to combinations of low/high speeds and temporal frequency b were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic resolution 1024 ⇥ 768 at 100 Hz. Routines were written using Matlab 7.10.0 and controlled the stimulus display. Observers sat 57 cm from the screen in a dark roo with normal or corrected to normal vision were used. They gave their inform experiments received ethical approval from the Aix-Marseille Ethics Committee the declaration of Helsinki. 3.2 Bayesian modeling To make full use of our MC paradigm in analyzing the obtained results, we follo of the Bayesian observer used for instance in [12]. We assume the observer mak a Maximum A Posteriori (MAP) estimator ˆ vz(m) = argmin v [ log( P M | V,Z(m | v, z)) log( P V | Z(v | z)) 6 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 3.1 Methods The task is to discriminate the speed v 2 R of MC stimuli moving w v0 = (v, 0) . We assign as independent experimental variable the aver we denote in the following z (we drop the index 0 to ease readabilit parameters are set to the following values V = 1 ⌧?z0 , ✓0 = ⇡ 2 , ⇥ = ⇡ 6 , Z = 0. Note that V is thus dependent of the value of z = z0 to ensure that ⌧? parameter ⌧? controls the temporal frequency bandwidth, as illustrated We used a two alternative forced choice (2AFC) paradigm. In each tri a small dark ﬁxation spot was followed by two stimulus intervals of grey 250 ms inter-stimulus interval. The ﬁrst stimulus has parameter has parameters (v2, z2) . At the end of the trial, a grey screen appears a which one of the two intervals was perceived as moving faster by pres is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) ar ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0 Z = { 0.31, where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomize values are expressed in cycles per degree (c/ ) and v values in /s. Ten possible combinations of these parameters are made per block of 250 blocks were collected per condition tested. The outcome of these exp psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 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 inter-stimulus interval. The ﬁrst stimulus has parameters (v1, z1) and t has parameters (v2, z2) . At the end of the trial, a grey screen appears asking the partic which one of the two intervals was perceived as moving faster by pressing one of tw is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected so th ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16, 0. where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across trials) values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitions of possible combinations of these parameters are made per block of 250 trials and at l blocks were collected per condition tested. The outcome of these experiments are s psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the value ˆ 'v ? empirical probability (averaged over the 40 trials) that a stimulus generated with para is moving faster than a stimulus with parameters (v, z? ) . 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 The task is to discriminate the speed v 2 R of MC stimuli moving with a horizo v0 = (v, 0) . We assign as independent experimental variable the average spatial f we denote in the following z (we drop the index 0 to ease readability in this se parameters are set to the following values V = 1 ⌧?z0 , ✓0 = ⇡ 2 , ⇥ = ⇡ 6 , Z = 0.78 c/ . Note that V is thus dependent of the value of z = z0 to ensure that ⌧? = 1 V z0 sta parameter ⌧? controls the temporal frequency bandwidth, as illustrated on the left We used a two alternative forced choice (2AFC) paradigm. In each trial a grey ﬁx a small dark ﬁxation spot was followed by two stimulus intervals of 250 ms eac grey 250 ms inter-stimulus interval. The ﬁrst stimulus has parameters (v1, z1) an has parameters (v2, z2) . At the end of the trial, a grey screen appears asking the pa which one of the two intervals was perceived as moving faster by pressing one of is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected s ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16 where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across tri values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitions possible combinations of these parameters are made per block of 250 trials and blocks were collected per condition tested. The outcome of these experiments ar psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the value ' empirical probability (averaged over the 40 trials) that a stimulus generated with p is moving faster than a stimulus with parameters (v, z? ) . 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 The task is to discriminate the speed v 2 R of MC stimuli moving with a hori v0 = (v, 0) . We assign as independent experimental variable the average spatia we denote in the following z (we drop the index 0 to ease readability in this parameters are set to the following values V = 1 ⌧?z0 , ✓0 = ⇡ 2 , ⇥ = ⇡ 6 , Z = 0.78 c/ . Note that V is thus dependent of the value of z = z0 to ensure that ⌧? = 1 V z0 parameter ⌧? controls the temporal frequency bandwidth, as illustrated on the le We used a two alternative forced choice (2AFC) paradigm. In each trial a grey a small dark ﬁxation spot was followed by two stimulus intervals of 250 ms e grey 250 ms inter-stimulus interval. The ﬁrst stimulus has parameters (v1, z1) has parameters (v2, z2) . At the end of the trial, a grey screen appears asking the which one of the two intervals was perceived as moving faster by pressing one is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0 where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitio possible combinations of these parameters are made per block of 250 trials an blocks were collected per condition tested. The outcome of these experiments psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the valu empirical probability (averaged over the 40 trials) that a stimulus generated wit is moving faster than a stimulus with parameters (v, z? ) . P Z(z) / z e , P ⇥(✓) / e and P || V v0 ||(r) / e . (6) z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a graphical display of the inﬂuence of these parameters. Speed Freq. orient. Freq. amplitude (mean, dispersion) (v0, V ) (✓0, ⇥) (z0, Z) Remark 2. Note that the ﬁnal envelope of ˆ is in agreement with the formulation that is used in [8]. However, this latter derivation was based on a heuristic following a trial-and-error strategy between modelers and psychophysicists. Herein, we justiﬁed these different points in a constructive manner. P Z(z) / z e , P ⇥(✓) / e and P || V v0 ||(r) / e . (6) z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a graphical display of the inﬂuence of these parameters. Speed Freq. orient. Freq. amplitude (mean, dispersion) (v0, V ) (✓0, ⇥) (z0, Z) Remark 2. Note that the ﬁnal envelope of ˆ is in agreement with the formulation that is used in [8]. However, this latter derivation was based on a heuristic following a trial-and-error strategy between modelers and psychophysicists. Herein, we justiﬁed these different points in a constructive manner.
315 316 317 318 319 320 321 322 323 is moving faster than a stimulus with parameters (v, z? ) . To asses the validity of our model, we tested four different scenarios by cons choices among z? = 0.78 c/ , v? 2 { 5 /s , 10 /s} , ⌧? 2 { 0.1s, 0.2s } , which corresponds to combinations of low/high speeds and temporal frequency b were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic resolution 1024 ⇥ 768 at 100 Hz. Routines were written using Matlab 7.10.0 and controlled the stimulus display. Observers sat 57 cm from the screen in a dark roo with normal or corrected to normal vision were used. They gave their inform experiments received ethical approval from the Aix-Marseille Ethics Committee the declaration of Helsinki. 3.2 Bayesian modeling To make full use of our MC paradigm in analyzing the obtained results, we follo of the Bayesian observer used for instance in [12]. We assume the observer mak a Maximum A Posteriori (MAP) estimator ˆ vz(m) = argmin v [ log( P M | V,Z(m | v, z)) log( P V | Z(v | z)) 6 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 3.1 Methods The task is to discriminate the speed v 2 R of MC stimuli moving w v0 = (v, 0) . We assign as independent experimental variable the aver we denote in the following z (we drop the index 0 to ease readabilit parameters are set to the following values V = 1 ⌧?z0 , ✓0 = ⇡ 2 , ⇥ = ⇡ 6 , Z = 0. Note that V is thus dependent of the value of z = z0 to ensure that ⌧? parameter ⌧? controls the temporal frequency bandwidth, as illustrated We used a two alternative forced choice (2AFC) paradigm. In each tri a small dark ﬁxation spot was followed by two stimulus intervals of grey 250 ms inter-stimulus interval. The ﬁrst stimulus has parameter has parameters (v2, z2) . At the end of the trial, a grey screen appears a which one of the two intervals was perceived as moving faster by pres is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) ar ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0 Z = { 0.31, where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomize values are expressed in cycles per degree (c/ ) and v values in /s. Ten possible combinations of these parameters are made per block of 250 blocks were collected per condition tested. The outcome of these exp psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 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 inter-stimulus interval. The ﬁrst stimulus has parameters (v1, z1) and t has parameters (v2, z2) . At the end of the trial, a grey screen appears asking the partic which one of the two intervals was perceived as moving faster by pressing one of tw is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected so th ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16, 0. where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across trials) values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitions of possible combinations of these parameters are made per block of 250 trials and at l blocks were collected per condition tested. The outcome of these experiments are s psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the value ˆ 'v ? empirical probability (averaged over the 40 trials) that a stimulus generated with para is moving faster than a stimulus with parameters (v, z? ) . 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 The task is to discriminate the speed v 2 R of MC stimuli moving with a horizo v0 = (v, 0) . We assign as independent experimental variable the average spatial f we denote in the following z (we drop the index 0 to ease readability in this se parameters are set to the following values V = 1 ⌧?z0 , ✓0 = ⇡ 2 , ⇥ = ⇡ 6 , Z = 0.78 c/ . Note that V is thus dependent of the value of z = z0 to ensure that ⌧? = 1 V z0 sta parameter ⌧? controls the temporal frequency bandwidth, as illustrated on the left We used a two alternative forced choice (2AFC) paradigm. In each trial a grey ﬁx a small dark ﬁxation spot was followed by two stimulus intervals of 250 ms eac grey 250 ms inter-stimulus interval. The ﬁrst stimulus has parameters (v1, z1) an has parameters (v2, z2) . At the end of the trial, a grey screen appears asking the pa which one of the two intervals was perceived as moving faster by pressing one of is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected s ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16 where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across tri values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitions possible combinations of these parameters are made per block of 250 trials and blocks were collected per condition tested. The outcome of these experiments ar psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the value ' empirical probability (averaged over the 40 trials) that a stimulus generated with p is moving faster than a stimulus with parameters (v, z? ) . 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 The task is to discriminate the speed v 2 R of MC stimuli moving with a hori v0 = (v, 0) . We assign as independent experimental variable the average spatia we denote in the following z (we drop the index 0 to ease readability in this parameters are set to the following values V = 1 ⌧?z0 , ✓0 = ⇡ 2 , ⇥ = ⇡ 6 , Z = 0.78 c/ . Note that V is thus dependent of the value of z = z0 to ensure that ⌧? = 1 V z0 parameter ⌧? controls the temporal frequency bandwidth, as illustrated on the le We used a two alternative forced choice (2AFC) paradigm. In each trial a grey a small dark ﬁxation spot was followed by two stimulus intervals of 250 ms e grey 250 ms inter-stimulus interval. The ﬁrst stimulus has parameters (v1, z1) has parameters (v2, z2) . At the end of the trial, a grey screen appears asking the which one of the two intervals was perceived as moving faster by pressing one is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0 where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitio possible combinations of these parameters are made per block of 250 trials an blocks were collected per condition tested. The outcome of these experiments psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the valu empirical probability (averaged over the 40 trials) that a stimulus generated wit is moving faster than a stimulus with parameters (v, z? ) . P Z(z) / z e , P ⇥(✓) / e and P || V v0 ||(r) / e . (6) z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a graphical display of the inﬂuence of these parameters. Speed Freq. orient. Freq. amplitude (mean, dispersion) (v0, V ) (✓0, ⇥) (z0, Z) Remark 2. Note that the ﬁnal envelope of ˆ is in agreement with the formulation that is used in [8]. However, this latter derivation was based on a heuristic following a trial-and-error strategy between modelers and psychophysicists. Herein, we justiﬁed these different points in a constructive manner. P Z(z) / z e , P ⇥(✓) / e and P || V v0 ||(r) / e . (6) z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a graphical display of the inﬂuence of these parameters. Speed Freq. orient. Freq. amplitude (mean, dispersion) (v0, V ) (✓0, ⇥) (z0, Z) Remark 2. Note that the ﬁnal envelope of ˆ is in agreement with the formulation that is used in [8]. However, this latter derivation was based on a heuristic following a trial-and-error strategy between modelers and psychophysicists. Herein, we justiﬁed these different points in a constructive manner.
0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 PSE bias (µz,z⇤ ) Subject 1 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 0.15 0.10 0.05 0.00 0.05 0.10 0.15 Subject 2 v⇤ = 5, ⌧ = 100 v⇤ = 5, ⌧ = 200 v⇤ = 10, ⌧ = 100 v⇤ = 10, ⌧ = 200 1.0 1.1 1.2 1.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 0.15 0.10 0.05 0.00 0.05 0.10 0.15 Subject 2 v⇤ = 5, ⌧ = 100 v⇤ = 5, ⌧ = 200 v⇤ = 10, ⌧ = 100 v⇤ = 10, ⌧ = 200 Experimental Findings 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 is moving faster than a stimulus with parameters (v, z? ) . To asses the validity of our model, we tested four different scenarios by cons choices among z? = 0.78 c/ , v? 2 { 5 /s , 10 /s} , ⌧? 2 { 0.1s, 0.2s } , which corresponds to combinations of low/high speeds and temporal frequency b were generated on a Mac running OS 10.6.8 and displayed on a 20” Viewsonic resolution 1024 ⇥ 768 at 100 Hz. Routines were written using Matlab 7.10.0 and controlled the stimulus display. Observers sat 57 cm from the screen in a dark roo with normal or corrected to normal vision were used. They gave their inform experiments received ethical approval from the Aix-Marseille Ethics Committee the declaration of Helsinki. 3.2 Bayesian modeling To make full use of our MC paradigm in analyzing the obtained results, we follo of the Bayesian observer used for instance in [12]. We assume the observer mak a Maximum A Posteriori (MAP) estimator ˆ vz(m) = argmin v [ log( P M | V,Z(m | v, z)) log( P V | Z(v | z)) 6 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 3.1 Methods The task is to discriminate the speed v 2 R of MC stimuli moving w v0 = (v, 0) . We assign as independent experimental variable the aver we denote in the following z (we drop the index 0 to ease readabilit parameters are set to the following values V = 1 ⌧?z0 , ✓0 = ⇡ 2 , ⇥ = ⇡ 6 , Z = 0. Note that V is thus dependent of the value of z = z0 to ensure that ⌧? parameter ⌧? controls the temporal frequency bandwidth, as illustrated We used a two alternative forced choice (2AFC) paradigm. In each tri a small dark ﬁxation spot was followed by two stimulus intervals of grey 250 ms inter-stimulus interval. The ﬁrst stimulus has parameter has parameters (v2, z2) . At the end of the trial, a grey screen appears a which one of the two intervals was perceived as moving faster by pres is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) ar ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0 Z = { 0.31, where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomize values are expressed in cycles per degree (c/ ) and v values in /s. Ten possible combinations of these parameters are made per block of 250 blocks were collected per condition tested. The outcome of these exp psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 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 inter-stimulus interval. The ﬁrst stimulus has parameters (v1, z1) and t has parameters (v2, z2) . At the end of the trial, a grey screen appears asking the partic which one of the two intervals was perceived as moving faster by pressing one of tw is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected so th ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16, 0. where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across trials) values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitions of possible combinations of these parameters are made per block of 250 trials and at l blocks were collected per condition tested. The outcome of these experiments are s psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the value ˆ 'v ? empirical probability (averaged over the 40 trials) that a stimulus generated with para is moving faster than a stimulus with parameters (v, z? ) . 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 The task is to discriminate the speed v 2 R of MC stimuli moving with a horizo v0 = (v, 0) . We assign as independent experimental variable the average spatial f we denote in the following z (we drop the index 0 to ease readability in this se parameters are set to the following values V = 1 ⌧?z0 , ✓0 = ⇡ 2 , ⇥ = ⇡ 6 , Z = 0.78 c/ . Note that V is thus dependent of the value of z = z0 to ensure that ⌧? = 1 V z0 sta parameter ⌧? controls the temporal frequency bandwidth, as illustrated on the left We used a two alternative forced choice (2AFC) paradigm. In each trial a grey ﬁx a small dark ﬁxation spot was followed by two stimulus intervals of 250 ms eac grey 250 ms inter-stimulus interval. The ﬁrst stimulus has parameters (v1, z1) an has parameters (v2, z2) . At the end of the trial, a grey screen appears asking the pa which one of the two intervals was perceived as moving faster by pressing one of is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected s ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0.16 where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across tri values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitions possible combinations of these parameters are made per block of 250 trials and blocks were collected per condition tested. The outcome of these experiments ar psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the value ' empirical probability (averaged over the 40 trials) that a stimulus generated with p is moving faster than a stimulus with parameters (v, z? ) . 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 The task is to discriminate the speed v 2 R of MC stimuli moving with a hori v0 = (v, 0) . We assign as independent experimental variable the average spatia we denote in the following z (we drop the index 0 to ease readability in this parameters are set to the following values V = 1 ⌧?z0 , ✓0 = ⇡ 2 , ⇥ = ⇡ 6 , Z = 0.78 c/ . Note that V is thus dependent of the value of z = z0 to ensure that ⌧? = 1 V z0 parameter ⌧? controls the temporal frequency bandwidth, as illustrated on the le We used a two alternative forced choice (2AFC) paradigm. In each trial a grey a small dark ﬁxation spot was followed by two stimulus intervals of 250 ms e grey 250 ms inter-stimulus interval. The ﬁrst stimulus has parameters (v1, z1) has parameters (v2, z2) . At the end of the trial, a grey screen appears asking the which one of the two intervals was perceived as moving faster by pressing one is whether v1 > v2 or v2 > v1 . Given references values (v? , z? ) , for each trial, (v1, z1) and (v2, z2) are selected ⇢ vi = v? , zi 2 z? + Z vj 2 v? + V , zj = z? where ⇢ V = { 2, 1, 0, 1, 2 } , Z = { 0.31, 0.15, 0, 0 where (i, j) = (1, 2) or (i, j) = (2, 1) (i.e. the ordering is randomized across values are expressed in cycles per degree (c/ ) and v values in /s. Ten repetitio possible combinations of these parameters are made per block of 250 trials an blocks were collected per condition tested. The outcome of these experiments psychometric curves ˆ 'v ? ,z ? , where for all (v v? , z z? ) 2 V ⇥ Z , the valu empirical probability (averaged over the 40 trials) that a stimulus generated wit is moving faster than a stimulus with parameters (v, z? ) . P Z(z) / z e , P ⇥(✓) / e and P || V v0 ||(r) / e . (6) z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a graphical display of the inﬂuence of these parameters. Speed Freq. orient. Freq. amplitude (mean, dispersion) (v0, V ) (✓0, ⇥) (z0, Z) Remark 2. Note that the ﬁnal envelope of ˆ is in agreement with the formulation that is used in [8]. However, this latter derivation was based on a heuristic following a trial-and-error strategy between modelers and psychophysicists. Herein, we justiﬁed these different points in a constructive manner. P Z(z) / z e , P ⇥(✓) / e and P || V v0 ||(r) / e . (6) z0 Z V ⇠1 ⌧ ⇠2 ⇠1 ✓0 z0 ⇥ Z Slope: \ v0 Two different projections of ˆ in Fourier space t MC of two different spatial frequencies z Figure 2: Graphical representation of the covariance (left) —note the cone-like shape of the envelopes– and an example of synthesized dynamics for narrow-band and broad-band Motion Clouds (right). Plugging these expressions (6) into the deﬁnition (5) of the power spectrum of the motion cloud, one obtains a parameterization which is very similar to the one originally introduced in [9]. The fol- lowing table articulates the speed v0 and frequency (✓0, z0) central parameters in term of amplitude and orientation, each one being coupled with the relevant dispersion parameters. Figure 2 shows a graphical display of the inﬂuence of these parameters. Speed Freq. orient. Freq. amplitude (mean, dispersion) (v0, V ) (✓0, ⇥) (z0, Z) Remark 2. Note that the ﬁnal envelope of ˆ is in agreement with the formulation that is used in [8]. However, this latter derivation was based on a heuristic following a trial-and-error strategy between modelers and psychophysicists. Herein, we justiﬁed these different points in a constructive manner. µz,z? z z µz,z?
to estimate/manipulate. ! equivalent to spot noise. ! sub-class of Gaussian spot-noises. ! meaningful texton-based generative model. ! equivalent to a dynamic s-PDE model. Motion clouds: Psychophysics: ! Interpretable through inverse Bayesian estimation. ! Making use of MC parameters (not yet fully).
to estimate/manipulate. ! equivalent to spot noise. ! sub-class of Gaussian spot-noises. ! meaningful texton-based generative model. ! equivalent to a dynamic s-PDE model. Motion clouds: Psychophysics: ! Interpretable through inverse Bayesian estimation. ! Making use of MC parameters (not yet fully). Future work: application to VSD in-vivo imaging.