We define textures in terms of statistical correlations between luminance levels at nearby locations, generalizing the methods developed for binary images (Victor and Conte, 2012; Hermundstad et al., 2014) to three luminance levels. If we consider four ‘checks’ arranged in a $2\times 2$ square, the three luminance levels lead to $3^4=81$ possible patterns, and their frequency of occurrence in an image is equivalently parameterized by intensity correlations within the square. Thus there is an 81-dimensional space of ternary textures defined by correlations within square arrangements of checks. However, translation invariance constrains these 81 probabilities, reducing the number of independent statistics and thus the dimension of the texture space.

We can quantify the statistics of such textures in an image patch by gliding a $2\times 2$ block (a ‘glider’) over the patch and analyzing the luminance levels at different locations within this block (Figure 1A). At the most basic level, we can measure the luminance histogram at each of the four check locations in the glider. Check intensities can take three values (0, 1, or 2, for black, gray, or white), and the corresponding frequencies of occurrence must add to one, leaving two free parameters. If the histograms at each of the four locations within the glider were independent, this would lead to $4\times 2=8$ texture dimensions. However, because of translation invariance in natural images, the luminance histograms at each location must be the same, leaving only two independent dimensions of texture space from the single-check statistics.

Figure 1

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Next, we can analyze the statistics of luminance levels at pairs of locations within the glider. Taking into account translation invariance, there are four ways to position these pairs (Figure 1B), each with a different orientation. For each orientation, we can calculate either the sum $A+B$ or the difference $A-B$ of the luminance values ${\displaystyle A}$ and ${\displaystyle B}$ at the two locations. This yields eight possible texture groups (Figure 1B). Within each group, we build texture coordinates by counting the fraction of occurrences in which $A\pm B$ is equal to 0, 1, or 2, up to a multiple of 3; that is, we are building a histogram of $A\pm B$ modulo 3, here denoted ${\displaystyle \mathrm{m}\mathrm{o}\mathrm{d}(A\pm B,3)}$. The appearance of the modulo function here is a consequence of using a Fourier transform in the space of luminance levels, which is convenient for incorporating translation invariance into our coordinate system (see Materials and methods and Appendix 1). The fractions of values of ${\displaystyle \mathrm{m}\mathrm{o}\mathrm{d}(A+B,3)}$ equal to 0, 1, or 2 must add up to 1, so that each texture group is characterized by two independent coordinates, that is, a plane in texture space. These 8 planes constitute 16 independent dimensions of the texture space, in addition to the two dimensions needed to capture the histogram statistics at individual locations.

We can similarly analyze the joint statistics of luminance levels at three checks within the glider, or all four together. There are four ways to position 3-check gliders within the $2\times 2$ square, and, for each of these positions, eight parameters are needed to describe their occurrence frequencies, once the first- and second-order statistics have been fixed. This leads to $4\times 8=32$ third-order parameters. For configurations of all four checks, 16 parameters are required, once first-, second-, and third-order parameters are fixed. These $32+16=48$ parameters, in addition to the 18 parameters described above, lead to a 66-dimensional texture space. This provides a complete parameterization of the $2\times 2$ configurations with three gray levels. See Methods and Appendix 1 for a detailed derivation and a generalization to higher numbers of gray levels.

In order to probe human psychophysical sensitivity to visual textures, we need an algorithm for generating texture patches at different locations in texture space. To do so, we use an approach that generalizes the methods from Victor and Conte, 2012. Briefly, we randomly populate entries of the first row and/or column of the texture patch, and we then sequentially fill the rest of the entries in the patch using a Markov process that is designed to sample from a maximum-entropy distribution in which some of the texture coordinates are fixed (see Methods and Appendix 2 for details). Examples of texture patches obtained by co-varying coordinates within a single texture group are shown in Figure 1C and D. Examples of textures obtained by co-varying coordinates in two texture groups are shown in Figure 2. We refer to the first case as ‘simple’ planes, and the second as ‘mixed’ planes.

Figure 2

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When applying these methods to the analysis of natural images, we bin luminance values to produce equal numbers of black, white, and gray checks (details below), thus equalizing the previously studied brightness statistics in scenes (e.g., Zemon et al., 1988; Chubb et al., 2004; Jin et al., 2008; Ratliff et al., 2010; Tkacik et al., 2010; Tkačik et al., 2011; Komban et al., 2014; Kremkow et al., 2014). This procedure allowed us to focus on higher-order correlations. In previous work, we found that median-binarized natural images show the highest variability in pairwise correlations, and that observers are correspondingly most sensitive to variations in these statistics (Tkačik et al., 2011; Hermundstad et al., 2014). In view of this, we focused our attention on pairwise statistics. For three gray levels, these comprise a 16-dimensional ‘salient’ subspace of the overall texture space.

The ‘variance is salience’ hypothesis can be further tested by asking whether symmetries of the natural distribution of textures are reflected in invariances of psychophysical thresholds. Binary texture coordinates (Hermundstad et al., 2014) are not affected by many of these symmetry transformations, and so a test requires textures containing at least three gray levels. For instance, reflecting a texture around the vertical axis has no effect on second-order statistics in the binary case, but it leads to a flip around the $p_0$ direction in the ${\displaystyle {\beta }_{+-}}$ simple plane in ternary texture space (dashed green line in Figure 4C). To see this, recall that the coordinates in the ${\displaystyle {\beta }_{+-}}$ plane are given by the probabilities that ${\displaystyle \mathrm{m}\mathrm{o}\mathrm{d}(A_1-A_2,3)=h}$ for the three possible values of ${\displaystyle h}$ (Figure 1). Under a left-right flip, the values $A_1$ and $A_2$ are exchanged, leading to ${\displaystyle \mathrm{m}\mathrm{o}\mathrm{d}(A_1-A_2,3)=\mathrm{m}\mathrm{o}\mathrm{d}(-h,3)}$. This means that the $h=1$ direction gets mapped to $h=2$, the $h=2$ direction gets mapped to $h=1$, and the $h=0$ direction remains unaffected. More details, and a generalization to additional symmetry transformations, can be found in Appendix 3. We find that the distribution of natural images is symmetric about the $p_0$ direction and is thus unaffected by this transformation, predicting that psychophysical thresholds should also be unaffected when textures are flipped about the vertical axis. This is indeed the case (Figure 6B). Similarly, the natural image distribution is symmetric under flips about the horizontal axis, and also under rotations by 90, 180, and 270 degrees, predicting perceptual invariances that are borne out in the psychophysical data (Figure 6B).

Reflecting a texture about the vertical axis also has an interesting effect on the ${\displaystyle {\beta }_{\genfrac{}{}{0ex}{}{+\phantom{-}}{\phantom{+}-}}}$ plane: it not only flips the texture about the $p_0$ direction, but it also maps the texture onto the plane corresponding to the opposite diagonal orientation, ${\displaystyle {\beta }_{\genfrac{}{}{0ex}{}{\phantom{-}+}{-\phantom{+}}}}$. The fact that a flip about the $p_0$ direction is a symmetry of natural images is thus related to the fact that the diagonal pairwise correlations are the same regardless of the orientation of the diagonal. This fact was already observed in the binary analysis (Hermundstad et al., 2014), and is related to the invariance under 90-degree rotations observed here (Figure 6B).

It is important to note that these symmetries were not guaranteed to exist for either natural images or human psychophysics. Most of the textures that we are using are not themselves invariant under rotations (see the examples from Figure 1C,D). This means that invariances of predicted thresholds arise from symmetries in the overall shape of the distribution of natural textures. Similarly, had observed thresholds been unrelated to natural texture statistics, we could have found a discrepancy between the symmetries observed in natural images and those observed in human perception. As an example, the up and down directions differ in meaning, as do vertical and horizontal directions. A system that preserves these semantic differences would not be invariant under flips and rotations. The fact that the psychophysical thresholds are, in fact, invariant under precisely those transformations that leave the natural image distribution unchanged supports the idea that this is an adaptation to symmetries present in the natural visual world.

Natural images also have a well-known asymmetry between bright and dark contrasts (Ratliff et al., 2010; Tkacik et al., 2010) that is reflected in the anatomy and physiology of visual circuits, and in visual behavior (Ratliff et al., 2010; Tkacik et al., 2010; Zemon et al., 1988; Chubb et al., 2004; Jin et al., 2008; Komban et al., 2014; Kremkow et al., 2014). Our psychophysical data also show a bright/dark asymmetry. For instance, in Figure 4C, the threshold contour is not symmetric under the exchange of black and white checks, which has the effect of reflecting thresholds about the upper-left axis in the ${\displaystyle {\beta }_{++}}$ plane (dashed orange line in the figure). Such bright-dark asymmetries lead to a wide distribution of relative changes in detection threshold upon the exchange of black and white (red violin plot in Figure 6B for the exch(B,W) transformation). Our natural image analysis does not show this asymmetry (blue violin plot in Figure 6B for the exch(B,W) transformation) because of our preprocessing, which follows previous work (Hermundstad et al., 2014). As observed in Ratliff et al., 2010, the bright-dark asymmetry rests on two main characteristics of natural images: a skewed distribution of light intensities such that the mean intensity is larger than the median, and a power-law spectrum of spatial correlations. Both of these are reduced or removed by our preprocessing pipeline, which starts by taking the logarithm of intensity values and thereby reduces the skewness in the intensity distribution, and continues with a whitening stage that removes the overall $1/f$ spatial correlation spectrum seen in natural images. The final ternarization step additionally reduces any remaining dark-bright asymmetry, since we ensure that each of the three gray levels occurs in equal proportions in the preprocessed patches. This explains why we do not see this asymmetry in our natural-image analysis.

The code and data used to generate all of the results in the paper can be found on GitHub (RRID:SCR_002630), at https://github.com/ttesileanu/TextureAnalysis (Tesileanu et al., 2020; copy archived at https://github.com/elifesciences-publications/TextureAnalysis).

A texture is defined here by the statistical properties of $2\times 2$ blocks of checks, each of which takes the value 0, 1, or 2, corresponding to the three luminance levels (black, gray, or white; see Appendix 1 for a generalization to more gray levels). The $3^4=81$ probabilities for all the possible configurations of such blocks form an overcomplete coordinate system because the statistical properties of textures are independent of position. To build a non-redundant parametrization of texture space, we use a construction based on a discrete Fourier transform (see Appendix 1). Starting with the luminance values ${\displaystyle A_i,i=1,\dots ,4}$, of checks in a $2\times 2$ texture block (arranged as in Figure 1A), we define the coordinates ${\displaystyle {\displaystyle {\sigma }_{\genfrac{}{}{0ex}{}{s_1{s}_2}{s_3{s}_4}}(h)}}$ which are equal to the fraction of locations where the linear combination $s_1{A}_1+s_2{A}_2+s_3{A}_3+s_4{A}_4$ has remainder equal to ${\displaystyle h}$ after division by three (the number of gray levels). In the case of three gray levels, the coefficients $s_i$ can be +1, −1, or 0.

Each set of coefficients $s_i$ identifies a texture group, and within each texture group we have three probability values, one for each value of ${\displaystyle h}$. Since the probabilities sum up to 1, each texture group can be represented as a plane, and more specifically, as a triangle in a plane, since the probabilities are also non-negative. This is the representation shown in Figure 1C,D and used in subsequent figures. For compactness of notation, when referring to the coefficients $s_i$, we write + and − instead of +1 and −1, and omit coefficients that are 0, e.g., ${\displaystyle {\sigma }_{\genfrac{}{}{0ex}{}{+-}{\phantom{+}-}}}$ instead of ${\displaystyle {\displaystyle {\sigma }_{\genfrac{}{}{0ex}{}{+1-1}{\phantom{+}0-1}}}}$. We also use $\gamma$ (rather than the generic symbol ${\displaystyle \sigma }$) for 1-point correlations, $\beta$ for 2-point correlations, $\theta$ for 3-point correlations, and $\alpha$ for 4-point correlation, matching the notation used in the binary case (Victor and Conte, 2012; Hermundstad et al., 2014). For instance, ${\displaystyle {\beta }_{\genfrac{}{}{0ex}{}{+}{-}}}$ is the plane identified by the linear combination ${\displaystyle A_1-A_3(\mathrm{m}\mathrm{o}\mathrm{d}3)}$.

Finding the location in texture space that matches the statistics of a given image patch is straightforward given the definition above: we simply glide a $2\times 2$ block over the image patch and count the fraction of locations where the combination $s_1{A}_1+s_2{A}_2+s_3{A}_3+s_4{A}_4$ takes each of its possible values modulo three, for each texture group identified by the coefficients $s_i$. (As a technical detail, we glide the smallest shape that contains non-zero coefficients. For example, for ${\displaystyle {\beta }_{+-}}$, we glide a $1\times 2$ region instead of a $2\times 2$ one. This differs from gliding the $2\times 2$ block for all orders only through edge effects, and thus the difference decreases as the patch size ${\displaystyle R}$ increases.)

In order to generate a patch that corresponds to a given location in texture space, we use a maximum entropy construction that is an extension of the methods from Victor and Conte, 2012. There are efficient algorithms based on 2d Markov models that can generate all single-group textures, namely, textures in which only the probabilities within a single texture group deviate from (1/3, 1/3, 1/3). For textures involving several groups, the construction is more involved, and in fact some combinations of texture coordinates cannot be achieved in a real texture. This restriction applies as textures become progressively more ‘saturated’, that is, near the boundary of the space. In contrast, near the origin of the space all combinations can be achieved via the ‘donut’ construction for mixing textures, described in Victor and Conte, 2012. Details are provided in Appendix 2.

The psychophysical task is adapted from Victor et al., 2005, and requires that the subject identify the location of a $16\times 64$-check target within a $64\times 64$-check array. The target is positioned near one of the four sides of the square array (chosen at random), with an 8-check margin. Target and background are distinguished by the texture used to color the checks: one is always the i.i.d. (unbiased) texture with three gray levels; the other is a texture specified by one or two of the coordinates defined in the text. In half of the trials, the target is structured and the background is i.i.d.; in the other half of the trials, the target is i.i.d. and the background is structured. To determine psychophysical sensitivity in a specific direction in the space of image statistics, we proceed as follows Hermundstad et al., 2014; Victor and Conte, 2012; Victor et al., 2013; Victor et al., 2015. We measure subject performance in this 4-alternative forced-choice task across a range of ‘texture contrasts’, that is, distances from the origin in the direction of interest. Fraction correct, as a function of texture contrast, is fit to a Weibull function, and threshold is taken as the texture contrast corresponding to a fraction correct of 0.625, that is, halfway between chance (0.25) and ceiling (1.0). Typically, 12 different directions in one plane of stimulus space are studied in a randomly interleaved fashion. Each of these 12 directions is sampled at 3 values of texture contrast, chosen in pilot experiments to yield performance between chance and ceiling. These trials are organized into 15 blocks of 288 trials each (a total of 4320 trials), so that each direction is sampled 360 times.

Stimuli, as described above, were presented on a mean-gray background for 120 ms, followed by a mask consisting of an array of i.i.d. checks, each half the size of the stimulus checks. The display size was $15\times 15$ deg; viewing distance was 103 cm. Each of the $64\times 64$ array stimulus checks consisted of $10\times 10$ hardware pixels, and measured $14\times 14$ min. The display device was an LCD monitor with a refresh rate of 100 Hz, driven by a Cambridge Research ViSaGe system. The monitor was calibrated with a photometer prior to each day of data collection to ensure that the luminance of the gray checks was halfway between that of the black checks (<0.1) and white checks (23 cd/m²).

Subjects were normal volunteers (three male, three female), ages 20 to 57, with visual acuities, corrected if necessary, of 20/20 or better. Of the six subjects, MC is an experienced psychophysical observer with thousands of hours of experience; the other subjects (SR, NM, WC, ZA, JWB) had approximately 10 (JWB), 40 (NM, WC, ZA) or 100 (SR) hours of experience at the start of the study, as subjects in related experiments. MC is an author. NM, WC, and ZA were naïve to the purposes of the experiment.

This work was carried out with the subjects’ informed consent, and in accordance with the Code of Ethics of the World Medical Association (Declaration of Helsinki) and the approval of the Institutional Review Board of Weill Cornell.

The average thresholds used in the main text were calculated by using the geometric mean of the subject thresholds, after applying a per-subject scaling factor chosen to best align the overall sensitivities of all the subjects; these multipliers ranged from 0.855 to 1.15. Rescaling a single consensus set of thresholds in this fashion accounted for 98.8% of the variance of individual thresholds across subjects. The average error bars were calculated by taking the root-mean-squared of the per-subject error bars in log space (determined from a bootstrap resampling of the Weibull-function fits, as in Victor et al., 2005), and then exponentiating.

Images were taken from the UPenn Natural Image Database (Tkačik et al., 2011) and preprocessed as shown in Figure 3A (see Materials and methods for details). Starting with a logarithmic encoding of luminance, we downsampled each image by averaging over $N\times N$ blocks of pixels to reduce potential camera sampling artifacts. We then split the images into non-overlapping patches of size ${\displaystyle R}$, filtered the patches to remove the average pairwise correlation expected in natural images (van Hateren, 1992), and finally ternarized patches to produce equal numbers of black, gray, and white checks (Figures 3A, C and D). For most figures shown in the main text, we used $N=2$ and $R=32$. Each patch was then analyzed in terms of its texture content and mapped to a point in ternary texture space following the procedure described in the main text. Finally, to avoid biases due to blurring artifacts, we fit a two-Gaussian mixture model to the texture distribution and used this to separate in-focus from blurred patches (Figure 3B; Hermundstad et al., 2014; details in Materials and methods).

To generate the whitening filter that we used to remove average pairwise correlations, we started with the same preprocessing steps as for the texture analysis, up to and including the splitting into non-overlapping patches (first three steps in Figure 3A). We then took the average over all the patches of the power spectrum, which was obtained by taking the magnitude-squared of the 2d Fourier transform. Taking the reciprocal square root of each value in the resulting matrix yielded the Fourier transform of the filtering matrix.

Following the procedure from Hermundstad et al., 2014, we fit a Gaussian mixture model with non-shared covariance matrices to the distribution of natural images in order to identify patches that are out of focus or motion blurred. This assigned each image patch to one of two multivariate Gaussian distributions. To identify which mixture component contained the sharper patches, we chose the component that had the higher median value of a measure of sharpness based on a Laplacian filter. Specifically, each patch was normalized so that its median luminance was set to 1, then convolved with the matrix ${\displaystyle \left(\begin{array}{ccc}0& 1& 0\\ 1& -4& 1\\ 0& 1& 0\end{array}\right)}$. The sharpness of the patch was calculated as the median absolute value over the pixels of the convolution result. This analysis was performed before any of the preprocessing steps, including the transformation to log intensities, and was restricted to the pixels that did not border image edges. There was thus no need to make assumptions regarding pixel values outside images.

Threshold predictions from natural image statistics were obtained as in Hermundstad et al., 2014. We fit a multivariate Gaussian to the distribution of texture patches after removing the blurry component, and used the inverse of the standard deviation in the texture direction of interest as a prediction for the psychophysical threshold in that direction. The overall scale of the predictions is not fixed by this procedure. We chose the scaling so as to minimize the error between the ${\displaystyle n}$ measurements $x_i$ and the predictions $y_i$, $\min \frac{1}{n}\sum _{i=1}^n{\left(\frac{\log y_i-\log x_i}{{ϵ}_i}\right)}^2$, where ${ϵ}_i$ are the measurement uncertainties in log space. This scaling factor was our single fitting parameter.

In Figure 6 we show several comparisons of two sets of thresholds $x_i$ and $y_i$. These are either the set of measured thresholds and the set of natural image predictions for specific preprocessing options; or sets of either measured or predicted thresholds before and after the action of a symmetry transformation. We measure mismatch by the difference between the natural logarithms of the two quantities, $\log y_i-\log x_i$, which is approximately equal to the relative error when the mismatches are not too large ($\log y_i-\log x_i=\log y_i/x_i=\log \left(1+\frac{y_i-x_i}{x_i}\right)\approx \frac{y_i-x_i}{x_i}$). In panel A of the figure all 311 measured values and the corresponding predictions were used. For panel B, this set was restricted in two ways. First, for we ignored the measurements for which we did not have psychophysics data for the transformed direction. And second, we ignored directions on which the transformation acted trivially (see Appendix 3).

This Supplement provides the details for the present parameterization of local image statistics. It generalizes the approach developed in Victor and Conte, 2012 for binary images along the lines indicated in that manuscript’s Appendix A, so that it is applicable to an arbitrary number ${\displaystyle G}$ of gray levels. In the present study, ${\displaystyle G=3}$, but the analysis is equally applicable to any prime value ${\displaystyle G}$, including the $G=2$ case that has been the focus of previous work (Briguglio et al., 2013; Victor and Conte, 2012; Victor et al., 2017; Victor et al., 2013; Victor et al., 2015). When ${\displaystyle G}$ is composite, the basic approach remains valid but, as mentioned below, there are some additional considerations.

We consider the statistics of a $2\times 2$ neighborhood of checks in images with ${\displaystyle G}$ gray levels. The starting point is an enumeration of the probabilities of each kind of $2\times 2$ block, $p\left(\begin{array}{cc}\hfill A_1\hfill & \hfill A_2\hfill \\ \hfill A_3\hfill & \hfill A_4\hfill \end{array}\right)$, where each $A_k$ denotes the gray level of a check, which we denote by an integer from 0 to ${\displaystyle G-1}$. There are ${\displaystyle G^4}$ such configurations, but the probabilities are not independent: they must sum to 1, and they must be stationary in space. For example, the probability of $1\times 2$ blocks computed by marginalizing over the lower two checks must equal to the probability of $1\times 2$ blocks computed by marginalizing over the upper two checks.

Our main goal is to obtain a coordinate system that removes these linear dependencies. The first step is to construct new coordinates $\phi \left(\begin{array}{cc}\hfill s_1\hfill & \hfill s_2\hfill \\ \hfill s_3\hfill & \hfill s_4\hfill \end{array}\right)$ by discrete Fourier transformation with respect to the gray level value in each check. As this is a discrete transform, the arguments $s_k$ are also integers from 0 to ${\displaystyle G-1}$. Equivalently, we can use other sets of integers that lead to unique values of the complex exponential; for instance, when ${\displaystyle G=3}$, it is equivalent to use the set $\{0,+1,-1\}$ or the set $\{0,1,2\}$. We have

The original block probabilities $p\left(\begin{array}{cc}\hfill A_1\hfill & \hfill A_2\hfill \\ \hfill A_3\hfill & \hfill A_4\hfill \end{array}\right)$ can be obtained from the Fourier transform coordinates (Equation (1)) by standard inversion: 

Fourier transform coordinates can be similarly described for any configuration of checks, including subsets of the $2\times 2$ neighborhood.

A basic property of the Fourier transform coordinates is that setting an argument to zero corresponds to marginalizing over the corresponding check. For example, consider the probabilities of the configurations of the upper $1\times 2$ block of checks, $p\left(\begin{array}{cc}\hfill A_1\hfill & \hfill A_2\hfill \end{array}\right)$, which are determined by marginalizing $p\left(\begin{array}{cc}\hfill A_1\hfill & \hfill A_2\hfill \\ \hfill A_3\hfill & \hfill A_4\hfill \end{array}\right)$ over $A_3$ and $A_4$ . Their Fourier transform coordinates are given by

It follows from Equation (1) that

Thus, the stationarity condition

is equivalent to

Similarly, the stationarity condition for ${\displaystyle 2\times 1}$ blocks is equivalent to

the condition that the single-check probabilities are equal in all four positions is equivalent to

and the condition that the sum of all block probabilities is one is equivalent to

In sum, the stationarity conditions can be stated in terms of the Fourier transform coordinates as follows: if any of the arguments of $\phi \left(\begin{array}{cc}\hfill s_1\hfill & \hfill s_2\hfill \\ \hfill s_3\hfill & \hfill s_4\hfill \end{array}\right)$ are zero, then they can be replaced by empty spaces, and the value of $\phi \left(\begin{array}{cc}\hfill s_1\hfill & \hfill s_2\hfill \\ \hfill s_3\hfill & \hfill s_4\hfill \end{array}\right)$ must be unchanged by translating the nonzero values within the $2\times 2$ neighborhood. It follows that the Fourier transform coordinates of a stationary distribution are specified by: $\phi \left(\begin{array}{c}\hfill s\hfill \end{array}\right)$, equal to the common value of the four expressions in Equation (8); $\phi \left(\begin{array}{cc}\hfill s_1\hfill & \hfill s_2\hfill \end{array}\right)$, equal to the common value of the two expressions in Equation (6); $\phi \left(\begin{array}{c}\hfill s_1\hfill \\ \hfill s_3\hfill \end{array}\right)$, equal to the common value of the two expressions in Equation (7); $\phi \left(\begin{array}{cc}\hfill s_1\hfill & \hfill \hfill \\ \hfill \hfill & \hfill s_4\hfill \end{array}\right)$ and $\phi \left(\begin{array}{cc}\hfill \hfill & \hfill s_2\hfill \\ \hfill s_3\hfill & \hfill \hfill \end{array}\right)$, defining pairwise correlations of two-check configurations that cannot be translated within the $2\times 2$ neighborhood; $\phi \left(\begin{array}{cc}\hfill \hfill & \hfill s_2\hfill \\ \hfill s_3\hfill & \hfill s_4\hfill \end{array}\right)$, $\phi \left(\begin{array}{cc}\hfill s_1\hfill & \hfill \hfill \\ \hfill s_3\hfill & \hfill s_4\hfill \end{array}\right)$, $\phi \left(\begin{array}{cc}\hfill s_1\hfill & \hfill s_2\hfill \\ \hfill \hfill & \hfill s_4\hfill \end{array}\right)$, and $\phi \left(\begin{array}{cc}\hfill s_1\hfill & \hfill s_2\hfill \\ \hfill s_3\hfill & \hfill \hfill \end{array}\right)$, defining three-check correlations, and $\phi \left(\begin{array}{cc}\hfill s_1\hfill & \hfill s_2\hfill \\ \hfill s_3\hfill & \hfill s_4\hfill \end{array}\right)$. In all of these cases, the arguments $s_k$ are nonzero. Thus, the total number of parameters, obtained by allowing each of the $s_k$ to range from 1 to $G-1$, is ${\displaystyle (G-1)+4(G-1{)}^2+4(G-1{)}^3+(G-1{)}^4=G(G-1)(G^2+G-1)}$; this is 10 for $G=2$ and 66 for $G=3$.

The Fourier transform coordinates incorporate the stationarity constraints, and they also have the convenient property that the origin of the space, that is, the image whose coordinates are all zero, is an image with identically-distributed gray levels in which every block probability $p\left(\begin{array}{cc}\hfill A_1\hfill & \hfill A_2\hfill \\ \hfill A_3\hfill & \hfill A_4\hfill \end{array}\right)$ is equal to $1/G^4$. The Fourier transform coordinates have another convenient property (using the approach of Appendix B of Victor and Conte, 2012): near the origin of the space, entropy is asymptotically proportional to the square of the Euclidean distance from the origin. This means that the Fourier transform coordinates are ‘calibrated’: for an ideal observer, small deviations in any direction from the origin are equally discriminable.

However, since for $G>2$, the Fourier transform coordinates are complex numbers, an arbitrary choice of them will typically not correspond to a realizable set of block probabilities, since the block probabilities must be all real and in the range [0, 1].

To address this problem, we note a relationship among the Fourier transform coordinates, which, when ${\displaystyle G}$ is prime, partitions the set of independent coordinates into disjoint subsets of size ${\displaystyle G-1}$. A further linear transformation within this set yields real-valued coordinates, which correspond to the coordinate system used here (for ${\displaystyle G=3}$) and in previous studies (for $G=2$).

To derive these real-valued coordinates, we use modular arithmetic. When ${\displaystyle G}$ is prime, for every integer ${\displaystyle q\in \{1,\dots ,G-1\}}$ there is a unique integer ${\displaystyle r\in \{1,\dots ,G-1\}}$ for which ${\displaystyle qr=1(\mathrm{m}\mathrm{o}\mathrm{d}G)}$, which we denote $q^{-1}$. Thus, if $S^{\prime }=\left(\begin{array}{cc}\hfill s_1^{\prime }\hfill & \hfill s_2^{\prime }\hfill \\ \hfill s_3^{\prime }\hfill & \hfill s_4^{\prime }\hfill \end{array}\right)$ can be obtained from $S=\left(\begin{array}{cc}\hfill s_1\hfill & \hfill s_2\hfill \\ \hfill s_3\hfill & \hfill s_4\hfill \end{array}\right)$ by $S^{\prime }=qS$, then it follows that ${\displaystyle S}$ can be obtained from $S^{\prime }$ by $S=q^{-1}{S}^{\prime }$, that is, the relationship is reciprocal. It also follows that every Fourier transform coordinate $S^{\prime }$ is some scalar multiple of a ‘monic’ Fourier transform coordinate, that is, one whose first nonzero element is 1. The reason is that we can take the multiplier ${\displaystyle q}$ to be the first nonzero coordinate of $S^{\prime }$ and write $S=q^{-1}{S}^{\prime }$.

Next, we note that all of the Fourier transform coordinates whose arguments are of the form $qS$ have an exponential in Equation (1) that depends only on the value of ${\displaystyle \sum _{k=1}^4{s}_k{A}_k(\mathrm{m}\mathrm{o}\mathrm{d}G)}$. This motivates grouping the terms of Equation (1) according to this value. We therefore define

that is, ${\displaystyle {\displaystyle {\sigma }_{\left(\begin{array}{cc}\phantom{b}{s}_1& s_2\\ \phantom{b}{s}_3& s_4\end{array}\right)}(h)}}$ is the sum of the probabilities of all blocks for which ${\displaystyle \sum _{k=1}^4{s}_k{A}_k=h(\mathrm{m}\mathrm{o}\mathrm{d}G)}$. The monic ${\displaystyle \sigma }$’s, that is, the ${\displaystyle \sigma }$’s whose first nonzero $s_k$ is equal to 1, are our desired real-valued coordinates. For each such ${\displaystyle \sigma }$, the coordinates are a vector of the ${\displaystyle G}$ numbers ${\displaystyle {\displaystyle {\sigma }_{\left(\begin{array}{cc}\phantom{b}{s}_1& s_2\\ \phantom{b}{s}_3& s_4\end{array}\right)}(h)}}$, for ${\displaystyle h=\{0,\dots ,G-1\}}$. As Equation (10) shows, ${\displaystyle {\displaystyle {\sigma }_{\left(\begin{array}{cc}\phantom{b}{s}_1& s_2\\ \phantom{b}{s}_3& s_4\end{array}\right)}(h)}}$ is the probability that a linear combination of gray levels whose coefficients are specified by the $s_k$ will result in a value of ${\displaystyle h}$. They are thus all in the range [0, 1], and their sum is 1. For $G=2$, the pair $\sigma (0)$ and $\sigma (1)$ subject to $\sigma (0)+\sigma (1)=1$ is a one-dimensional domain, parametrized by $\sigma (1)-\sigma (0)$; other than a change in sign, these are the coordinates used in Briguglio et al., 2013; Victor and Conte, 2012; Victor et al., 2017; Victor et al., 2013; Victor et al., 2015. For ${\displaystyle G=3}$, as is the case here, the triplet $\sigma (0)$, $\sigma (1)$, and $\sigma (2)$ is subject to $\sigma (0)+\sigma (1)+\sigma (2)=1$ and is naturally plotted in a triangular ‘alloy plot’ with centroid at (1/3, 1/3, 1/3), the image with independent, identically-distributed gray levels. Note that ${\displaystyle 2=-1(\mathrm{m}\mathrm{o}\mathrm{d}3)}$, yielding the notation in the main text where the coefficients $s_i$ took values 0, +1, and −1 instead of 0, 1, and 2.

It remains to show that all of the degrees of freedom in the Fourier Transform coordinates $\phi$ are captured by the ${\displaystyle \sigma }$’s. Since the Fourier transform coordinates are partitioned into disjoint subsets, it suffices to examine the transformation between each of these subsets (i.e., between the Fourier transform coordinates that are scalar multiples of a particular monic ${\displaystyle S}$, and the corresponding ${\displaystyle \sigma }$). These subsets correspond to the simple planes introduced in the main text. It follows from Equations (1) and (10) that, for ${\displaystyle q\ne 0(\mathrm{m}\mathrm{o}\mathrm{d}G)}$,

Via discrete Fourier inversion, this equation implies

Note that the $q=0$ term of this equation is given by Equation (9), the normalization condition. Thus,

Equations (11) and (13) display the bidirectional transformation between the ${\displaystyle G}$-vector ${\displaystyle \sigma }$ and the ${\displaystyle G-1}$ Fourier transform coordinates $qS$. Since this transformation is a discrete Fourier transform (and hence, a multiple of a unitary transformation), it preserves the property that entropy is a proportional to the square of the Euclidean distance from the origin.

When ${\displaystyle G}$ is not prime, the decomposition of Fourier transform coordinates into disjoint sets indexed by monic coordinates is no longer possible. For example, with ${\displaystyle G=4}$, both $S^{\prime }=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$ and $S^{\prime \prime }=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 0\hfill \end{array}\right)$ are monic but the sets $q{S}^{\prime }$ and $q{S}^{\prime \prime }$ have a non-empty intersection: ${\displaystyle 2S^{\mathrm{\prime }}=2S^{\mathrm{\prime }\mathrm{\prime }}=\left(\begin{array}{cc}2& 0\\ 0& 0\end{array}\right)\left(\mathrm{m}\mathrm{o}\mathrm{d}4\right)}$. This necessitates a more elaborate version of the above approach, with separate strata for each factor of ${\displaystyle G}$. The disjoint sets of Fourier transform coordinates are no longer all of the same size, and the transformations (Equation 11) and (Equation 13) have a more elaborate form, although the overall parameter count is the same. As this case is not relevant to the present experiments, we do not discuss it further.

Creation of the stimuli used here requires synthesis of images whose statistics are specified by one or two of the coordinates described above. We adopted the strategy of Victor and Conte, 2012, Table 2, for this purpose. (i) Coordinates that are of lower order than the specified coordinates are set to zero. The rationale is that sensitivity to lower-order coordinates is generally greater than sensitivity to higher-order coordinates, so this allows detection to be driven by the higher-order coordinates. (ii) Coordinates that are of higher order than the specified coordinates are chosen to achieve an image ensemble whose entropy is maximized. The rationale is that this sets the higher-order coordinates at the value that is implied by the lower-order coordinates, without any further spatial structure. (iii) Coordinates that are of the same order as the specified coordinates are chosen to satisfy the ‘Pickard rules’ (Pickard, 1980), as this allows for stimulus generation via a Markov process in the plane, and hence, maximizes entropy given the constraints of the specified textures. (iv) In a few cases (all involving third-order statistics), the specified parameters are inconsistent with the Pickard rules, and in these cases, specific choices of texture parameters are made to allow for convenient texture synthesis. In all cases in which the unspecified texture parameters are assigned a nonzero value, this nonzero value decreases to zero rapidly (at least quadratically) near the origin. Thus, the construction guarantees that the surface corresponding to any two specified texture coordinates is tangent to their coordinate plane at the origin.

In general, the above strategy also applies for $G\ge 3$. We focus on specifying pairs of second-order coordinates, the stimuli used in this study. When the specified coordinates share the same linear combination (e.g., ${\displaystyle {\beta }_{12}[0]}$ and ${\displaystyle {\beta }_{12}[1]}$) or when the two specified coordinates involve different linear combinations but the same checks (e.g., ${\displaystyle {\beta }_{\genfrac{}{}{0ex}{}{1\phantom{2}}{\phantom{2}1}}[0]}$ and ${\displaystyle {\beta }_{\genfrac{}{}{0ex}{}{1\phantom{2}}{\phantom{1}2}}[0]}$), images may be synthesized by a Markov process that operates in the direction of the coupled checks. For example, if ${\displaystyle {\beta }_{1s_2}}$ is specified, a Markov process is used to create each row. The Markov generation procedure guarantees that the image is a sample from a maximum-entropy ensemble consistent with the specified coordinates. It implicitly defines the other texture coordinates in a manner consistent with the above policies established for $G=2$.

For combinations of coordinates involving non-identical pairs of checks (e.g., ${\displaystyle {\beta }_{1s_2}}$ and ${\displaystyle {\beta }_{\genfrac{}{}{0ex}{}{1}{s_3}}}$), a new construction is needed in some cases. The basic issue is that the Markov process that generates the correlations along rows may be inconsistent with the Markov process that generates the columns. To see how this can happen, consider ${\displaystyle {\beta }_{11}[0]=1}$ and ${\displaystyle {\beta }_{\genfrac{}{}{0ex}{}{1}{2}}[1]=1}$. Taking into account stationarity, ${\displaystyle {\beta }_{11}[0]=1}$ means that ${\displaystyle A_1+A_2=A_3+A_4=0}$; ${\displaystyle {\beta }_{\genfrac{}{}{0ex}{}{1}{2}}[1]=1}$ means that $A_1+2A_3=A_2+2A_4=1$. The inconsistency arises because the first set of equalities implies that $A_2=-A_1$ and $A_4=-A_3$, so $A_1+2A_3=-(A_2+2A_4)$, which is inconsistent with the second set of equalities. The inconsistency can also be seen from an algebraic viewpoint. The horizontal correlation ${\displaystyle {\beta }_{11}}$ corresponds to a left-to-right Markov process that biases check pairs whose gray levels satisfy a multiplicative recurrence, $x_{n+1}=-x_n=2x_n$. The vertical correlation ${\displaystyle {\beta }_{\genfrac{}{}{0ex}{}{1}{2}}}$ corresponds to a top-to-bottom Markov process that biases check pairs whose gray levels satisfy an additive recurrence, $x_{n+1}=x_n-1$. The inconsistency arises because these transformations do not commute.

To handle these cases, we use the following construction. It produces samples from an image ensemble that match the specified second-order statistics, and for which all other second-order statistics, as well as first- and third-order statistics, are zero. As a first step, two textures are created: a Markov process along rows, providing for correlations of the form ${\displaystyle {\beta }_{1s_2}}$, and a Markov process along columns, providing for correlations of the form ${\displaystyle {\beta }_{\genfrac{}{}{0ex}{}{1}{s_3}}}$. Then, taking inspiration from the ‘dead leaves’ generative model of images (Zylberberg et al., 2012), we randomly choose rows from the first texture and columns from the second texture and sequentially place them on the plane, occluding whatever has been placed earlier. Eventually, this ‘falling sticks’ construction covers the plane, and this completes the next stage of the construction.

The resulting texture has the same kinds of correlations as the two starting textures, but the correlations are diluted, because a ‘stick’ in one direction may be overlaid by an orthogonal stick placed at a later time. To calculate this dilution, we consider the three ‘sticks’ that could contribute to a given $1\times 2$ block of the final texture: one horizontal stick and two vertical sticks. Since they are dropped in a random order, there is a 1/3 chance that any of them is the one that is placed last. If the last stick is horizontal (which happens with probability 1/3), then the two checks of the $1\times 2$ block reflect the correlation structure of the underlying texture. If the last stick is vertical (which happens with probability 2/3), then these two checks are uncorrelated, since they are derived from independent Markov processes. That is,

Thus, to obtain a falling-sticks texture with a given set of correlations, one must choose

Expressed in terms of distance from randomness, this demonstrates the threefold dilution:

This limits the maximum correlation strength of the final texture, but it suffices for the present purposes since the achievable correlation strengths are generally far above perceptual threshold. A similar analysis applies to the vertical pairwise correlations.

Note that first-order correlations are zero since the two starting textures had an equal number of checks of all gray levels, and third-order correlations within a $2\times 2$ block are zero because they always involve checks that originated in independent Markov processes. Some fourth-order correlations are not zero, but their deviations from zero are small: this is because they arise from $2\times 2$ neighborhoods in which the two last ‘sticks’ are both horizontal (1/6 of the time) or both vertical (1/6 of the time), and these numerical factors multiply a product of two terms involving ${\sigma }^{\text{component}}-1/G$.

Finally, a Metropolis mixing step (Metropolis et al., 1953) is applied, to maximize entropy without changing the $2\times 2$ block probabilities (see Victor and Conte, 2012 for details). This eliminates any spurious long-range correlations that may have arisen from the ‘sticks’ of the underlying textures.

In order to test to what extent the natural image predictions or the psychophysical measurements are invariant under symmetry transformations, we need to understand the effect of such transformations on texture coordinates ${\displaystyle {\displaystyle {\sigma }_{\genfrac{}{}{0ex}{}{s_1{s}_2}{s_3{s}_4}}}}$. Two observations are key. First, every geometric transformation we are interested in—reflections and rotations by multiples of 90°—correspond to permutations of the check locations $A_1$, $A_2$, $A_3$, $A_4$. For instance, a horizontal flip exchanges $A_1$ with $A_2$ and $A_3$ with $A_4$. Second, for the particular case of ternary textures, all the permutations of the three gray levels correspond to affine transformations modulo 3. Specifically, given a check value ${\displaystyle A}$, consider the transformation ${\displaystyle A\to xA+y(\mathrm{m}\mathrm{o}\mathrm{d}3)}$. ${\displaystyle (x,y)=(1,0)}$ is the identity; $(x,y)=(1,1)$ and $(x,y)=(1,2)$ are the non-trivial cyclic permutations; and $x=2$, $y\in \{0,1,2\}$ yields the three pairwise exchanges. We exclude ${\displaystyle x=0}$ since this would correspond to removing all luminance variations in the image patches.

As we now show, the net result of a geometric transformation and a color permutation always corresponds to a permutation of the texture coordinates. Consider a general permutation ${\rho }^{-1}$ on the four check locations, $A_k\to A_{{\rho }^{-1}(k)}$ (it will become clear below why we use the inverse here), and a general affine transformation on the gray levels, ${\displaystyle A\to xA+y(\mathrm{m}\mathrm{o}\mathrm{d}G)}$, with $x\ne 0$. The equation appearing in the definition of the ${\displaystyle {\displaystyle {\sigma }_{\genfrac{}{}{0ex}{}{s_1{s}_2}{s_3{s}_4}}(h)}}$ direction (Equation 10) becomes:

Note that here, ${\displaystyle h}$ is a label identifying the coordinate direction, and therefore is not transformed by the affine transformation applied to the luminance values $A_k$. Since $x\ne 0$, we find

Thus, the effect of the transformation was to convert the original texture direction into a different one. To properly identify the transformed direction, we need to put it in monic form; that is, we need to ensure that the first non-zero coefficient is set to 1. Suppose that this coefficient appears at position $k_0$, and is equal to $s_{\rho (k_0)}$. We can write:

where 

and therefore ${\tilde{s}}_{k_0}=1$. Thus the direction ${\displaystyle {\displaystyle {\sigma }_{\genfrac{}{}{0ex}{}{s_1{s}_2}{s_3{s}_4}}(h)}}$ gets mapped to ${\displaystyle {\displaystyle {\sigma }_{\genfrac{}{}{0ex}{}{{\tilde{s}}_1{\tilde{s}}_2}{{\tilde{s}}_3{\tilde{s}}_4}}(ah-b(\mathrm{m}\mathrm{o}\mathrm{d}G))}}$ where $a=s_{\rho (k_0)}^{-1}{x}^{-1}$ and $b=ay\sum s_k$. Thus, all transformations correspond to relabeling of the texture coordinates. This approach holds for any prime ${\displaystyle G}$, but for ${\displaystyle G>3}$ the affine transformations will no longer be sufficient to model all gray level permutations.

To find the effect of the geometric and gray-level transformations on the theoretical predictions, we apply this reshuffling to the columns of the $N_{\text{patches}}\times 99$ matrix giving the distribution of natural image patches in texture space (99 here corresponds to the three probability values in each of the 33 simple texture planes). We then recalculate the threshold predictions and check how much these changed. This is equivalent to first performing the geometric and gray level transformations directly on each image and then rerunning the whole analysis, but is substantially more efficient.

For the psychophysics, we apply the transformation to each direction in texture space for which we have thresholds. In some cases, the transformed direction is not contained in the experimental dataset; we cannot check for symmetry when this happens. We thus only compare the original and transformed thresholds in cases where the transformed direction maps onto one of the directions in the original dataset.

Note that some symmetry transformations leave certain directions in texture space invariant. For instance, a left-right flip leaves the entire ${\displaystyle {\beta }_{11}}$ plane invariant since it only flips the order of the terms in the sum ${\displaystyle A_1+A_2}$. When this happens, the thresholds obtained in those directions are unaffected by the transformation and thus the fact that they do not change cannot be used as evidence that the symmetry is obeyed. For this reason, all invariant directions are ignored when looking at the effect of a transformation. This explains the variability in the number of points in the different plots from Figure 6B.

We performed several statistical tests to assess the quality of the match that we found between measurements and natural image predictions. For the permutation tests below, we used a scalar measure of the discrepancy that is given by median absolute error calculated in log space, after accounting for an overall scaling factor between psychophysical measurements and natural image predictions. More precisely, suppose we have ${\displaystyle n}$ measurements $x_i$ and ${\displaystyle n}$ predictions $y_i$. We define the mean-centered log thresholds

where $⟨.⟩$ represents the mean over all measurement directions. Then the mismatch between ${\displaystyle x}$ and ${\displaystyle y}$ is given by

Note that the difference between the natural logarithms of two quantities that are relatively close to each other, $a_1\approx a_2\approx \overline{a}\equiv (a_1+a_2)/2$, is approximately equal to the relative error between the two quantities,

where $\mathrm{\Delta }a=a_1-a_2$, and the big-$𝒪$ notation shows that the approximation error scales as the square of the absolute prediction error. In fact, the log error is even better approximated by a symmetric form of the relative error, in which the denominator is replaced by the mean between measurement and prediction: 

We see that in this case the approximation error scales with the cubed absolute prediction error. In our case, prediction errors are low enough that the ratio between the log error and the symmetric relative error $\mathrm{\Delta }a/\overline{a}$ is between 1 and 1.034 for all 311 thresholds.

1. Permutation test over all thresholds. This test estimates how likely it is that the observed value of the difference ${\displaystyle D}$ could have been obtained by chance in a model in which all the measured thresholds $x_i$ are drawn independently from a common random distribution. We generated 10,000 random permutations of the dataset $x_i$, $x_i^{\mu }$ (each of which contain 311 data points in 4 single and 22 mixed planes), and for each of these calculated the difference $D^{\mu }$ between the observed values and the predictions $y_i$. We then calculated the fraction of samples ${\displaystyle \mu }$ for which the difference was smaller than the observed one, $D^{\mu }\le D$, which is an estimate of the ${\displaystyle p}$-value.

2. Permutation test preserving ellipticity. The naïve permutation test above generates independent thresholds even for directions that are nearby within a given texture plane. To build a more stringent test where the threshold contours are kept close to elliptical, we used a different sampling procedure where permutations were applied only within texture groups, and were forced to be cyclic. In this way, thresholds obtained for adjacent texture directions were kept adjacent to each other, preserving the correlations implied by the elliptical contours. More specifically, assume that we index the $n_{\text{groups}}$ texture groups for which we have data by ${\displaystyle \sigma }$, and let $x_{\sigma }$ be the subset of elements of ${\displaystyle x}$ corresponding to group ${\displaystyle \sigma }$. Also let $ℛ$ be the shift operator that circularly permutes the elements in a vector to the right, such that the first element becomes the second, the second becomes the third and so on, with the last element being moved to the first position. For each resampling of the measurements ${\displaystyle x}$, we sampled $n_{\text{groups}}$ non-negative integers $k_{\sigma }$ and performed the transformations $x_{\sigma }\to {ℛ}^{k_{\sigma }}{x}_{\sigma }$. The largest value for $k_{\sigma }$ was chosen to be equal to the number of elements in $x_{\sigma }$. For each of the 10,000 samples obtained in this way we calculated the ${\displaystyle D}$ statistic and proceeded as above to get a ${\displaystyle p}$-value.

3. Exponent estimation. The predictions from natural image statistics were obtained by taking the inverse of the standard deviation of the natural texture distribution in each direction. This differs by a square root from the efficient-coding prediction for Gaussian inputs and linear gain (Hermundstad et al., 2014). This is not unreasonable since natural scene statistics are not exactly Gaussian, and we do not expect brain processing to be precisely linear. However, to more fully investigate possible non-linearities, we considered general power-law transformations of the predicted thresholds, and used the data to estimate the exponent ${\displaystyle \eta }$. Specifically, given the mean-centered log predictions ${\widehat{y}}_i$, we asked whether $\eta {\widehat{y}}_i$ is a better approximation to the measurements ${\widehat{x}}_i$ (this implies that $y_i^{\eta }$ provides a better prediction for the $x_i$ values, which is why we refer to ${\displaystyle \eta }$ as an exponent). We can write down the model

• where ${ϵ}_i$ are errors drawn from a standard normal distribution, ${ϵ}_i\sim 𝒩(0,1)$. This model interpolates between the null model above that assumes all thresholds are drawn from the same distribution (when the exponent $\eta =0$), and a model in which they are given by the values predicted from natural images, plus noise (with exponent $\eta =1$). To find ${\displaystyle \eta }$ and ${\displaystyle \sigma }$, we used a Bayesian approach. The log posterior distribution for the parameters ${\displaystyle \eta }$ and ${\displaystyle \sigma }$ given the measured data ${\widehat{x}}_i$ is:

• where ${\displaystyle n}$ is the number of measurements, and we assumed that ${\displaystyle \eta }$ and ${\displaystyle \sigma }$ are independent in our prior distribution. We then used slice sampling (taking advantage of the slicesample function in Matlab, RRID:SCR_001622) to draw from the posterior distribution. We used 10,000 samples in total, and we discarded the first 5000 as a burn-in period. We used a flat prior for ${\displaystyle \eta }$ and $\log \sigma$, but checked that the results are similar when using other priors.

The results from these statistical tests are summarized in Appendix 4—table 1 for the various choices of downsampling factor ${\displaystyle N}$ and patch size ${\displaystyle R}$, and they show that the match we see holds across the explored range of ${\displaystyle N}$ and ${\displaystyle R}$. Interestingly, the 95% credible intervals obtained for the exponent ${\displaystyle \eta }$ in the exponent-estimation model do not include one for most choices of preprocessing parameters, suggesting that an additional nonlinearity might be at play here.

Overall, the natural-image predictions are in good agreement with the measured psychophysical thresholds. The discrepancies, when they occur, are not entirely random. These trends are shown in Figure 1. As in the main text, we focus on second-order correlations.

Appendix 5—figure 1

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As shown in Figure 1A, prediction errors in simple planes tend to be negative, while prediction errors in mixed planes tend to be slightly positive (medians −0.090 and +0.008; ${\displaystyle p=0.0014}$, Kolmogorov-Smirnov (K-S) test). Note that on balance, prediction errors will be close to zero, since we applied an overall scale factor to minimize the overall difference between predictions from natural images and psychophysical data. Thus, this finding means that perceptual performance when more than one kind of correlation is present is (slightly) disproportionately better than perceptual performance when only one kind of correlation is present. This suggests that the brain uses more resources to analyze mixed-plane correlations than our efficient-coding model would predict, or that processing of multiple correlations is in some sense synergistic.

Second-order correlations in the texture space employed here are defined using either sums or differences (modulo 3) of discretized luminance values. The wrap-around due to the modular arithmetic has different effects on these two kinds of correlations. Specifically, correlations due to modular sums imply a tendency for adjacent checks of a specific gray level to match (see main text Figure 1C), while correlations due to modular differences imply a tendency for checks to match their neighbor independent of gray level, or, for mini-gradients (black to gray to white) to be present (see main text Figure 1D). To test whether this distinction influenced prediction error, we split the data into planes (either simple or mixed) that use only modular sums, and planes (simple or mixed) that use only modular differences. Mixed planes such as ${\displaystyle ({\beta }_{++}[0],{\beta }_{+-}[0])}$ that use both sums and differences were excluded from this analysis.

Figure 1B shows that while there is no significant difference between the median log errors in sum versus difference directions, the full distributions are different (K-S ${\displaystyle p}$-value is $p=1.8\cdot {10}^{-4}$), largely due to the fact that the sum directions exhibit significantly larger absolute errors compared to the difference ones (median absolute log error 0.19 vs. 0.07, respectively). We suggest that this reflects the greater dependence of the sum-correlations on the gray-level discretization.

We next examined whether predictions were more or less accurate on the coordinate axes, by comparing prediction errors for on-axis correlations (i.e., correlations described by one of the $p_i=1$ corners of the probability simplex (see Figure 1); or one of the two axes in mixed planes [see Figure 2]) with all other measurements (off-axis). The distributions of prediction errors for on-axis vs. off-axis thresholds do not exhibit any noticeable differences (see Figure 1C).

Finally, we noted that in the mixed planes, some correlations involved only two checks—and thus were one-dimensional, while others involved overlapping dyads in orthogonal directions and thus were two-dimensional. Specifically, in a mixed plane such as ${\displaystyle ({\beta }_{++}[0],{\beta }_{+-}[0])}$, only a horizontally-oriented pair of checks is involved in both types of correlations, meaning that nearby rows in such textures are independent. In contrast, in the mixed plane ${\displaystyle ({\beta }_{++}[1],{\beta }_{\genfrac{}{}{0ex}{}{+}{-}}[0])}$, a horizontal pair of checks is used in the first dimension and a vertical one in the second, leading to correlations along both dimensions. This distinction was not associated with over- or under-prediction of thresholds (KS test $p=0.25$; Wilcoxon rank-sum test $p=0.96$, medians 0.042 and 0.004, respectively), but the distribution of errors was wider for the 1-D correlations than for the 2-D correlations (0.19 vs. 0.12, $p=0.01$, K-S test on absolute log errors).

Appendix 5—figure 2

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In directions where human subjects showed low sensitivity to textures, leading to large measured thresholds, the natural-image predictions were consistently lower than the measurements. This suggests that adding a nonlinearity to the model could improve the fit to the data. Indeed, as shown in Appendix 4, when fitting a model of the form ${\displaystyle \text{threshold}\propto (\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}{)}^{-\eta }}$, an exponent slightly smaller than one is most consistent with the data (95% credible interval for ${\displaystyle \eta }$ is $[0.81,0.98]$).

The match between psychophysical thresholds and efficient coding predictions is robust under a number of alterations in the analysis pipeline, including changing the parameters of the preprocessing and using different image databases.

As we showed in the main text, changing the patch size ${\displaystyle R}$ or the downsampling ratio ${\displaystyle N}$ does not significantly affect the match between experiment and theory. Another aspect of the preprocessing that we have varied is the ternarization procedure. In the main text, images were ternarized by splitting the entire dynamic range of each patch into three regions, each corresponding to an equal number of checks (up to one check, due to the fact that our patch sizes are not divisible by three). Here, we consider variants of the ternarization procedure, parameterized by the fraction $\rho$ of checks that are mapped to gray, while assuming that the remaining $1-\rho$ checks are equally distributed between black and white. Thus, the default ternarization procedure corresponds to $\rho =1/3$. The effects of varying $\rho$ are small in most texture planes, with an increase in prediction error as we move away from histogram equalization ($\rho =1/3$), as seen in the figure below.

Appendix 6—figure 1

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The accuracy of the natural image threshold predictions is also good when we use different image databases. Figure 2 shows the match between predictions and measurements in each plane when using the van Hateren database (van Hateren and van der Schaaf, 1998) with a downsampling ratio ${\displaystyle N=2}$ and patch size ${\displaystyle R=32}$.

Appendix 6—figure 2

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The distance measure from Equation (22) comparing these results to the results obtained from the Penn Image Database using the same preprocessing options was $D_{\text{Penn-vH}}=0.086$, showing that the two sets of measurements not only both agree with the psychophysics, but also agree with each other. Appendix 6—table 1 shows the results of the statistical tests for various preprocessing parameters for the van Hateren database.

As mentioned in the main text, the psychophysical thresholds show remarkable consistency across subjects, with most differences attributable to an overall scaling factor. The results are shown in Figure 3.

Appendix 6—figure 3

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The differences between different subjects were more pronounced in higher-order planes (Figure 4). Note that the predicted thresholds (shown as blue dots in the figure) are still not far from the measurements; in particular, the predictions reflect the fact that the thresholds in the higher-order planes are generally high.

Appendix 6—figure 4

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