Participants.

Author GM and nine naïve observers, (6 female, mean ±sd age: 24±6) participated in the study. All participants had normal or corrected to normal vision and normal stereo vision. Prior to testing, participants were screened using the Titmus stereopsis test and only participants with stereoacuity of 40 arcseconds or better were included in the study.

Apparatus.

The experiment was programmed with the Psychophysics Toolbox Version 3 [56, 57] in Matlab (MathWorks). Stimuli were presented on an BenQ XL2720Z LCD monitor with a resolution of 1920 × 1080 pixels (display dot pitch 0.311 mm) at 120 Hz. The monitor was run from an NVidia Quadro K 420 graphics processing unit. Observers were seated in a dimly lit room, 45 cm in front of the monitor with their heads stabilized in a chin and forehead rest and wore active stereoscopic shutter-glasses (NVIDIA 3DVision) to control dichoptic stimulus presentation. The cross talk of the dichoptic system was 1% measured with a Spectrascan 6500 photometer.

Stimuli.

Stimuli were 1/f pink noise stereograms presented on a uniformly gray background; examples for each experimental condition are shown in Fig 1a. The stimuli contained oblique (45 or 135 degrees) sinusoidal disparity corrugations of varying amplitude and spatial frequency, generated as in [38] (see also [58]). The stimuli were presented as disks or rings with 1 degree cosinusoidal edges. The central fixation target was a 0.25 degree black disk with 0.125 degree cosinusoidal edge. In pilot testing, we verified that it was not possible to perform the experiment without dichoptic stimulus presentation (i.e. the oblique sinusoidal corrugation did not generate visible compression and expansion artifacts in the pink noise patterns).

Procedure.

Each trial, observers were presented with a black fixation dot on a uniformly gray background. As soon as the response from the previous trial had been recorded, the stimulus for the current trial was shown for 250 milliseconds. This was too brief a time for observers to benefit from changes in fixation, since stimulus-driven saccade latencies are on average greater than 200 ms [59], saccade durations range from 20 to 200 ms [60], and visual sensitivity is reduced during and after a saccade [61, 62]. Once the stimulus had been extinguished, observers were required to indicate, via button press, whether the disparity corrugation was top-tilted leftwards or rightwards. Observers were given unlimited time to respond. The following trial commenced as soon as observers provided a response. Each trial, the amount of peak-to-trough disparity was under the control of a three-down, one-up staircase [63] that adjusted the disparity magnitude to a level that produced 79% correct responses.

Design.

We measured how observer’s disparity sensitivity (1/disparity threshold) varied, as a function of the spatial frequency of the sinusoidal disparity corrugation, throughout different portions of the visual field. We tested four visual field conditions. In the central visual field condition, stimuli were presented within a disk with a 3 degree radius centered at fixation. In the near and far peripheral visual field conditions, stimuli were presented within rings spanning 3-9 and 9-21 degrees into the visual periphery, respectively. Lastly, in the full visual field condition, stimuli were presented within a disk with a 21 degree radius, and thus spanned the full extent of the visual field tested in this study. In each condition, we measured disparity thresholds at six spatial frequencies: 0.04 0.09, 0.18, 0.35, 0.71, 1.41 cycles/degree. Thresholds were measured via 24 randomly interleaved staircases [63]. The raw data from 75 trials from each staircase were combined and fitted with a cumulative normal function by weighted least-squares regression (in which the data are weighted by their binomial standard deviation). Disparity discrimination thresholds were estimated from the 75% correct point of the psychometric function.

It is well known that disparity sensitivity varies lawfully as a function of spatial frequency following a bell-shaped function [64, 65]. This function is well described by a log-parabola model [38]. Therefore, we first converted disparity threshold estimates into disparity sensitivity (sensitivity = 1/threshold). Then, for each visual field condition, we fit the sensitivity data to a three-parameter log parabola Disparity Sensitivity Function (DSF) [38, 66] defined as: (1) where γ_max represents the peak gain (i.e. peak sensitivity), f_max is the peak frequency (i.e. the spatial frequency at which the peak gain occurs), and β is the bandwidth at half height (in octaves) of the function. The sensitivity data were fit to this equation, via least-squares regression, to obtain parameter estimates that could then be compared across experimental conditions.

Retino-cortical mapping.

To mimic the retino-cortical mapping of the primate visual system that provides a space-variant representation of the visual scene, we employ the central blind-spot model: each Cartesian image is transformed into its cortical representation through a log-polar transformation [28, 32, 71–73]. We chose this specific model with respect to other models in the literature (e.g. [74]) for several reasons: it captures the essential aspects of the retino-cortical mapping, it can be implemented efficiently, it provides a good preservation of image information [75, 76], and it allows us to provide an analytic description of cortical processing.

In the central blind-spot model, the mapping from the Cartesian domain (x, y) to the cortical domain of coordinates (ξ, η) is described by the following equations: (3) where a parameterizes the non-linearity of the mapping, q is related to the angular resolution, ρ₀ is the radius of the central blind spot, and are the polar coordinates derived from the Cartesian ones. All points with ρ < ρ₀ are ignored (hence the central blind spot).

Discrete log-polar mapping.

Our aim was to test the model using the same experimental stimuli and procedures employed with human observers. Therefore, the log-polar transformation must be applied to digital images. Given a Cartesian image of N_c × N_r pixels, and defined ρ_max = 0.5min(N_c, N_r), we obtain an R × S (rings × sectors) discrete cortical image of coordinates (u, v) by taking: (4) where ⌊⋅⌋ denotes the integer part, q = S/(2π), and the non-linearity of the mapping is a = (ρ_max/ρ₀)^1/R.

Fig 8 shows the log-polar pixels, which can be thought of as the log-polar receptive fields, in the Cartesian domain (Fig 8b) and in the cortical domain (Fig 8c): the Cartesian area (i.e. the log-polar pixel) that refers to a given cortical pixel defines the cortical pixel’s receptive field. The non-linearity of the log-polar transformation can be described as follows: by referring to Fig 8b and 8c, a uniform (green) row of cortical units is mapped to a (green) sector of space variant receptive fields, and a vertical (cyan) column of cortical units is mapped to a (cyan) circular set of uniform receptive fields. By inverting Eq 3 the centers of the receptive fields can be computed, and these points present a non-uniform distribution throughout the retinal plane (see the yellow circles overlying the Cartesian images in Fig 8a). The magenta circular curve in Fig 8b, with radius S/2π, represents the locus where the size of log-polar pixels is equal to the size of Cartesian pixels. In particular, in the area inside the magenta circular curve (the fovea) a single Cartesian pixel contributes to many log-polar pixels (oversampling), whereas outside this region multiple Cartesian pixels will contribute to a single log-polar pixel. To avoid spatial aliasing due to the undersampling, we employ overlapping receptive fields. Specifically, we use overlapping circular Gaussian receptive fields [77, 78], which are the most biologically plausible and optimally preserve image information [75]. An example of a transformation from Cartesian to cortical domain is shown in Fig 8a and 8d. The cortical image (Fig 8d) clearly demonstrates the non-linear effects of the log-polar mapping.

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Fig 8. Log-polar mapping and cortical processing.

(top) Log-polar mapping scheme for the central blind-spot model (Eq 3). (a) A standard Cartesian image with overlying log-polar pixels, the receptive fields (yellow circles). (b) Cartesian domain with the superposition of the circular overlapping log-polar receptive fields and (c) the corresponding cortical domain, where the squares denote the neural units. The green sector of receptive fields map to the horizontal row of (green) neural units and the cyan circle of receptive fields to a column of (cyan) neural units. The magenta circle delimits the oversampling (fovea) and undersampling areas (periphery). (d) The cortical representation of the standard Cartesian image. The cortical image is zoomed to improve the visualization. (bottom) A uniform processing in the cortical domain maps to a space-variant processing in the retinal domain. (a) The retinal space variant filtered image that is the backward mapping of the cortical uniform filtered image of subfigure (h). (f) The retinal filters that correspond to the filters in the cortical domain (g): a uniform filtering in the cortical domain results in a space-variant filtering operation in the retinal domain, where both the scale (red circle) and the orientation (green circle) of the filters vary. (h) The cortical filtered image obtained by applying the filter depicted in subfigure (g) on the cortical image shown in subfigure (d). The specific values of the log-polar parameters are: R = 130, S = 203, ρ_o = 3, CR = 3.9, W_max = 4.8. The spatial support of the filter is 31 × 31 cortical pixels.

https://doi.org/10.1371/journal.pcbi.1007699.g008

This discrete log-polar mapping provides a significant data reduction while preserving a large field of view and high resolution at the fovea [31, 79, 80]. To characterize the amount of data reduction provided by this transformation, we can can define the compression ratio (CR) of the cortical image with respect to the Cartesian one as: (5)

This compression ratio CR thus describes the data reduction occurring in the human visual system (that our computational model mimics), and will also affect the execution time of the simulated model.

The log-polar transformation models the space variant image resolution: the size of the receptive fields increases as a function of the eccentricity (the distance between the center of the receptive field and the fovea). We can define the relationship between the receptive field size (in particular, the maximum receptive field size W_max) and the parameters of the mapping as follows: (6)

Eq 6 provides a measure of the scale at which the periphery of the Cartesian image is processed. Moreover, the parameters of the log-polar mapping also influence the proportion of cortical units used to over-represent the fovea: we can define the percentage of the cortical area used to represent the fovea (χ). This can be derived from Eq 6 by setting the receptive field size to 1 and inverting the equation to find the corresponding u (see Eq 4), and by then dividing by the overall size of the modeled cortex R: (7)

By exploiting Eqs 6 and 7 we can control the growth of the size of the receptive fields and the over-representation of the fovea in order to reproduce data from the literature on the size-to-eccentricity relationship [73, 81, 82].

Cortical processing.

In the human visual system, visual processing is performed by networks of units (cells) described by their receptive fields. This neural network can be approximated by sets of filter banks whose responses to visual stimuli mimic those of neurons throughout the human visual system. The proposed model for disparity estimation could therefore embed the processing of V1 binocular simple units directly into the log-polar receptive fields. Specifically, the log-polar transform could be modified by using, as receptive fields, filters that perform V1-like feature extraction. However, to minimize the model’s computational load, we can consider that filter banks embedded in the log-polar transform can be “implemented” as a filtering process applied directly to the cortical image [31, 83]. We can demonstrate that the extraction of visual features can be carried out directly in the cortical domain by using solutions developed for the Cartesian domain without any modifications. To do so, in the following we analyze the relationships between the different parameters of a discrete log-polar mapping and of a bank of multi-scale and multi-orientation band-pass filters [84].

To maintain equivalence between Cartesian and cortical visual processing, the discreet log polar mapping should provide an isotropic sampling of Cartesian coordinates. To avoid anisotropy, circular sampling must be (locally) equal to radial sampling, since the cortical space consists of a uniform network of neural units. Sampling points can be derived by considering the inverse of the cortical mapping (Eq 3). Specifically, the circular sampling interval is (2π/S)ρ₀ a^u−1 and the radial sampling interval is ρ₀ a^u−1(a − 1). To maintain isotropic sampling these sampling intervals must be equal, therefore the relationship between rings and sectors of the log-polar mapping must follow the rule: (8)

From a geometric point of view, the optimal relationship between R and S, expressed by Eq 8, is the one that optimizes the log-polar pixel aspect ratio making it as close as possible to 1.

The receptive fields of V1 simple cells are classically modeled as band-pass filters [85], thus we define the following complex-valued Gabor filter [86]: (9) where σ defines the spatial scale, f_s the peak spatial frequency, and ψ is the phase of the sinusoidal modulation. By considering filters that are normalized by their energy, we have .

In order to process the cortically-transformed images, it is necessary to characterize the filters, defined in the Cartesian domain, with respect to the cortical domain, i.e. to map the filters into the cortical domain, thus obtaining g(x(ξ, η), y(ξ, η), θ, σ, ψ). As a consequence of the non-linearity of the log-polar mapping, the mapped filters are distorted [87, 88], thus a filtering operation directly in the cortical domain could introduce undesired distortions in the outputs. Here, we show that under specific conditions these distortions can be kept to a minimum: under these assumptions, it is possible to directly work in the cortical domain, by considering spatial filters sampled in log-polar coordinates g(ξ, η, θ, σ, ψ).

At a global level (e.g. see Fig 8d) log-polar transformed images exhibit large distortions. However, we can consider what occurs at a more local level, at the scale of the receptive field of a single Gabor filter. First, we consider that the log-polar mapping can be expressed in terms of general coordinates transformation [89], thus the Jacobian matrix of the coordinates transformation allows us to describe how the receptive field locally changes. Specifically, the scalar coefficient ρ₀ a^ξln(a) represents the scale factor of the log-polar vector, and the matrix describes the rotation η due to the mapping. Fig 8g shows a set of cortical filters and Fig 8f their retinal counterpart (i.e. the inverse log-polar transform): the red circle highlights the scale factor (i.e. the spatial support) of the filter and the green one its rotation. It is worth to note that the column of equally oriented filters in the cortical domain maps on a circle of filters in the retinal domain and each retinal filter is also at a different orientation. Specifically, vertically-oriented filters on the cortex correspond to azimuthally/tangentially-oriented filters on the retina; horizontally-oriented filters on the cortex correspond to radially-oriented filters on the retina.

Next, we want to analyze how the distortion affects the receptive field shape as a function of the distance from its center p₀ = (ξ₀, η₀): we can consider that the ratio g(x(ξ, η), y(ξ, η), θ, σ, ψ)/g(ξ, η, θ, σ, ψ) around a given point should be equal to 1. Since the filter g(⋅) is an exponential function, we can evaluate the difference h(⋅) between their arguments. We can approximate such a difference by using a Taylor expansion of a multi-variable function: (10) where (⋅)^T denotes the transpose, and H(⋅) the Hessian matrix. In the following we only focus on the terms that are relevant to describe how the distortion affects the receptive field shape: essentially, this depends on the partial derivatives of (x(ξ, η), y(ξ, η)) that constitute the gradient and the Hessian of h(⋅). The first order term takes into account how the mapping depends on the spatial position of the receptive field center. Indeed, the gradient has terms that are in common with the Jacobian matrix of the coordinates transformation, thus it describes the scale factor and the rotation of the receptive field as a function of the position p₀. The approximation error can be expressed by the second order term of the Taylor expansion: thus, there is an error that increases as a quadratic function of the distance p − p₀ (i.e. from the receptive field center), and an error that depends on the Hessian matrix that is related to the log-polar parameters. For instance, the mixed partial derivative of x(ξ, η) is ρ₀ln(a)a^ξsin(η), thus we can consider that the error related to the log-polar parameters is proportional to ρ₀ln(a) = (ρ₀/R)ln(ρ_max/ρ₀). It increases as a function of ρ₀ (given a fixed ρ_max) and decreases as R increases, which in turn decreases the compression ratio (Eq 5). Fig 8f and 8g shows that such distortions can be negligible, though the spatial support of the displayed filters is large for sake of visualization. Fig 8h shows the cortical image (Fig 8d) filtered by the filter that is drawn in different cortical positions in Fig 8g. In Fig 8e the retinal (i.e. space variant) processing is shown, which is obtained through the inverse log-polar mapping of Fig 8h.

Cortical computational model of disparity estimation.

We consider a pair of (grayscale) cortical images I^L(p) and I^R(p), for all positions p = (ξ, η) that are the cortical representations of an input stereo pair of Cartesian images. Our goal is to define a computational model that is able to encode in its cortical activity the information related to the disparity present in the Cartesian images. The cortical images are a warped version of the Cartesian images. The representation of disparity is a vector quantity. We thus define the disparity map δ(p) = (d_ξ, d_η)(p) as the difference between the pair of cortical images at each position p. To compute this cortical disparity map, the proposed model is composed of several processing stages.

V1 binocular energy computation and normalization.

In the proposed model we consider two sub-populations of neurons at the V1 level: binocular simple cells and complex cells. V1 simple cells are characterized by a preferred spatial orientation θ and a preferred phase difference Δψ between the left- and right-eye components of a cell’s receptive field. We model the receptive fields of V1 simple cells as Gabor filters (see Eq 9). The spatial support of the filters is defined as a function of their spatial radial peak frequency f_s and bandwidth B: . We consider one standard deviation of the amplitude spectrum as the cut-off frequency.

Following the phase-shift model [33, 34], we define the receptive fields of the binocular simple cell as S^L(p, θ, σ, ψ^L) = ℜ[g^L(p, θ, σ, ψ^L)] and S^R(p, θ, σ, ψ^R) = ℜ[g^R(p, θ, σ, ψ^R)]. These receptive fields are centered at the same position in the left- and right-eye images, and have a binocular phase difference Δψ = ψ^L − ψ^R. For each spatial orientation, a set of K binocular phase differences are chosen to obtain tuning to different disparities: d = Δψ/f_s.

We define the response of binocular simple cells as (11)

We can compute the response R_q(p, θ, σ, Δψ) of a quadrature binocular simple cell by using the imaginary part of the Gabor filters.

The response of a complex cell is described by the binocular energy (the sum of the squared responses of a quadrature pair of binocular simple cells) [33, 35, 36]: (12) by considering that d = Δψ/f_s. By taking into account the extensions of the binocular energy model proposed in [90, 91], we apply a static non-linearity to the complex cell response described in Eq 12.

The response of the V1 layer of our model, when considering a finite set of orientations θ = θ₁…θ_N, can be defined, through a divisive normalization to remove confounds due to variations in the local amount of contrast [92, 93], as (13) where 0 < ε ≪ 1 is a small constant to avoid dividing by zero in regions where no binocular energy is computed (i.e. no texture is present). For simplicity we omit from the notation the spatial scale σ. At this level, V1 responses are tuned to the spatial orientation and magnitude of the stimulus. The model neurons are tuned to disparity orthogonal to their orientation on the cortex; e.g. a horizontally-oriented cortical RF is tuned to the radial component of retinal disparity. It’s important to recognise that the tuning is to 1D disparity—a cell will respond strongly if the component of stimulus disparity along its preferred direction matches the magnitude of disparity that the cell is tuned to, regardless of stimulus disparity in the orthogonal direction.

In order to mimic natural neural activity, we consider that neural noise is present [90]. We model this neural noise as: E^V1(p, θ, d) = E^V1(p, θ, d)+ n_V1(p). The noise is uniformly distributed and its value is a fraction of the local average neural activity.

MT cells response.

Orientation-independent disparity tuning is obtained at the MT level of the model by pooling afferent V1 responses in the spatial and orientation domains, followed by a non-linearity [39, 94].

The responses of an MT cell, tuned to the magnitude d and direction ϕ of the vector disparity δ, can be expressed as follows: (14) where denotes a Gaussian kernel (standard deviation σ_pool) for the spatial pooling, F(s) = exp(s) is a static non-linearity, specifically an exponential function [39, 92], λ is the gain of the non-linearity, and w_ϕ represents the MT linear weights that give origin to the MT tuning. Spatial pooling accounts for the fact that MT receptive fields are larger than V1 receptive fields, and has the effect of improving the accuracy of disparity estimation [39]. The static non-linearity is employed since linear models fail to account for the response patterns of MT cells, whereas an exponential nonlinearity provides a good description of the MT firing patterns [92] and improves the accuracy of disparity estimation [39].

Similarly to what occurs at the V1 layer, we model neural noise at the MT level as: E^MT(p, ϕ, d) = E^MT(p, ϕ, d)+ n_MT(p).

Experimental evidence suggests that w_ϕ is a smooth function with central excitation and lateral inhibition. Therefore, by considering the MT linear weights shown in [92], we define w_ϕ(θ) as (15)

Vector disparity is thus encoded as a distributed representation through a population of MT neurons that span over the 2-D disparity space with a preferred set of tuning directions (ϕ = ϕ₁…ϕ_P) in [0, 2π] and tuning magnitudes (d = d₁…d_K). Thus, this processing stage contributes to represent the disparity stimulus in terms of its parameters, i.e. directions and magnitude, with respect to the V1 representation of the stimulus that is described in terms of the cells’ parameters.

Such a representation mimics the neural distributed representation of information. However, from a computational point of view, cosine functions shifted over various orientations (see Eq 15) are described by the linear combination of an orthonormal basis (i.e., sine and cosine functions). Thus, all the V1 afferent information can be encoded by a population of MT neurons tuned to the directions ϕ = 0 and ϕ = π/2, only, with varying tuning magnitudes (see Eq 14).

This observation may help account for the larger selectivity for horizontal disparity reported in the literature [95–97]. Since a neural population tuned to two directions (at an angular difference of ϕ = π/2) can encode the full vector disparity, a neural population of MT units tuned to a retinal disparity range slightly larger than [−π/4, π/4] is able to recover the full vector disparity, i.e. a population of MT cells tuned around the horizontal axis might account also for the selectivity to vertical disparity [39].

Our model implementation however does not incorporate this anisotropy, nor does it account for the fact that the anisotropy between horizontal and vertical disparity tuning has been found already at the V1 level [98]. Indeed, our model is not meant to incorporate all known properties of V1 (such as the differences in crossed/uncrossed disparity tuning across upper and lower visual field [43]). However, we highlight how the vertical/horizontal anisotropy may arise at the MT layer, since this is where we have orientation-independent disparity tuning and is therefore where we can first explicitly estimate vector disparity.

Multi-scale analysis.

A standard approach to handle multi-scale analysis is to adopt the following steps [39]: (i) a pyramidal decomposition with L levels [40] and (ii) a coarse-to-fine refinement [41]. This is a computationally efficient way to take into account the presence of different spatial frequency channels in the visual cortex and of large range of disparities and spatial frequencies in the real visual signal.

However, our model implements a log-polar mapping, thus its space variance, i.e. the linear increase of the filter size with respect to the eccentricity, can be exploited to efficiently implement a multi-scale analysis. Specifically, a pyramidal approach can be considered as a “vertical” multi-scale (the variation of the filter size at a single location), whereas the log-polar spatial sampling acts as an “horizontal” multi-scale (the variation of the filter size across different location [42]). The “vertical” multi-scale is also addressed in the literature as “cortical pyramids”.

Cue combination across the visual field.

Human observers and model were tested with annular stimuli spanning sub-portions of the visual field, as well as with full field stimuli spanning the whole region of the visual field visual within a 21 degree radius. When considering the responses of the model to the foveal, mid-peripheral, and far-peripheral stimuli, only the neural units corresponding to the stimulated field regions exhibited any neural activity (as described by Eq 14) and contributed to the model output. When analyzing the responses of the model to the full-filed stimuli, we pooled the neural activities of the distinct MT populations across the three considered annular regions.

Decoding.

To assess whether the proposed computational model is able to effectively encode information about the features of the visual signal, and whether the model DSF is similar to the DSF of human observers, we decode the population responses of the MT neurons [90], which encode the disparity stimulus parameters in their distributed representation. The population responses of the MT neurons essentially highlight the most probable disparity values. We adopt a linear combination approach to decode the MT population response as in [39, 99, 100]: (16)

Note that when considering P tuning directions (ϕ₁…ϕ_P), Eq 16 would normally contain a 2/P normalization term (see [39] for how this term is derived). Here we consider only 2 tuning directions, thus P = 2 and the normalization term is 1.

Next, we backwards transform into the retinal domain the disparity map described by Eq 16. To easily detect whether the disparity corrugation is top-tilted leftwards or rightwards, we apply the Fourier transform to the retinal disparity map and check the position of the peak of its magnitude.

Simulation parameters.

The simulation parameters selected to obtain the results presented in Fig 1 were adapted from the simulation parameters reported in [39], which were originally tuned to perform on computer vision benchmarks [101–104]. Since the proposed algorithm is meant to model human stereo vision, not compete on computer vision benchmarks, we modified the simulation parameters to reflect the known properties of the human visual system. Most parameter choices were derived from the literature, and the rest were selected based on pilot work [37] where we compared model performance to the normative data from Reynaud et al. [38]. The most notable differences between the current model and the one presented in [39] are:

• The foveated architecture and the related cortical processing that were not present in [39]: the log-polar paradigm, employed in the proposed computational model, is crucial for replicating the patterns of human data
• The algorithm presented in [39] did not contain neural noise, which is instead present in the human visual system [32] and was thus incorporated into the current model
• In [39] a multi-scale approach was adopted with 11 sub-octave scales in order to recover a large range of disparities (common in computer vision) by using Gabor filters with peak frequency of 0.26 cycles/pixel. However, in the current model, only 1 scale was employed, since as we’ve noted, the log-polar spatial sampling acts as a “sliding” multi-scale

The specific model parameters employed here were:

• pixels, the cortical disparity range to which the neural units are sensitive (this range is constrained by the spatial peak frequency f_s of the filters). Note that the retinal disparity range increases linearly (with receptive field size) across the model’s visual field, from ±0.43 arcmin at the fovea to ±25 arcmin in the model’s periphery.
• K = 5, the sampling of the disparity range, i.e. the number of neural units for a given spatial orientation θ.
• the V1 static non-linearity is a power function with exponent 0.5.
• σ_pool = 3.66 pixels, the spatial pooling of V1 responses (its standard deviation).
• λ = 0.65, the gain of the exponential static non-linearity at the MT level.
• N = 12, the number of spatial orientations, i.e. the number of neural units that sample the spatial orientation θ.
• the neural noise is set to 34% and 18% of the local average neural activity at the V1 and MT levels, respectively.
• f_s = 0.13 cycles/pixel, the radial peak frequency of the Gabor filters.
• σ = 5.12 pixels, the standard deviation of the Gabor filters.
• the Gabor filters are zero-mean.
• R = 318, the number of rings of the log-polar mapping.
• ρ₀ = 9 pixels, the radius of the central blind spot.
• CR = 6.4, the compression ratio of the cortical image compared to the Cartesian image.