The analysis of natural scenes is schematized in Figure 1. We collect an ensemble of image patches from the calibrated Penn natural image database (PIDB) (Tkačik et al., 2011). We preprocess the image patches as shown in Figure 1A. This involves first averaging pixel luminances over a square region of N × N pixels, which converts an image of size L₁ × L₂ pixels into an image of reduced size L₁/N × L₂/N pixels. Images are then divided into R × R square patches of these downsampled pixels and whitened (see ‘Materials and methods’, Image preprocessing, for further details). Since the preprocessing depends on a choice of two parameters, the block-average factor N and patch size R, we report results for multiple image analyses performed using the identical preprocessing pipeline but for various choices of N and R. After preprocessing, we binarize each patch to have equal numbers of black and white pixels (black = −1, white = +1). We characterize each patch by the histogram of 16 binary colorings (2^2×2) seen through a square 2 × 2 pixel glider (Figure 1B). Translation invariance imposes constraints on this histogram, reducing the number of degrees of freedom to 10 (Victor and Conte, 2012). These degrees of freedom can be mapped to a set of image statistic coordinates that separates correlations based on their order: (i) one first-order coordinate, γ, describes overall luminance, (ii) four second-order coordinates, $\{{\beta }_{|}$, ${\beta }_{-}$, ${\beta }_{/}$, ${\beta }_{\backslash }\}$, describe two-point correlations between pixels arranged vertically, horizontally, or diagonally, (iii) four third-order coordinates, $\{{\theta }_{\mathrm{⌞}}$, ${\theta }_{\mathrm{⌜}}$, ${\theta }_{\mathrm{⌝}}$, ${\theta }_{\mathrm{⌟}}\}$, describe three-point correlations between pixels arranged into ⌞-shapes of different orientations, and (iv) one fourth-order coordinate, α, describes the single four-point correlation between all four pixels in the glider (Figure 1C). The binarization step of the preprocessing pipeline forces γ to zero, leaving nine coordinates. Each image patch is thus characterized by a vector of coordinate values $\left\{{\beta }_{|},{\beta }_{-},{\beta }_{/},{\beta }_{\backslash },{\theta }_{\mathrm{⌜}},{\theta }_{\mathrm{⌝}},{\theta }_{\mathrm{⌟}},{\theta }_{\mathrm{⌞}},\alpha \right\}$, that is, a point within the multidimensional space of image statistics. Accumulating these points across patches yields a multidimensional probability distribution that characterizes the local correlations in natural scenes (schematized in Figure 1D). A total of 724 images (up to 249780 patches, depending on the choice of N and R), was used to construct this distribution.

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To summarize this distribution, we compute the degree of variation (standard deviation) along each coordinate axis (Figure 1E). As is shown, the degree of variation along different coordinate axes exhibits a characteristic rank-ordering, given by $\left\{{\beta }_{|},{\beta }_{-}\right\}>\left\{{\beta }_{/},{\beta }_{\backslash }\right\}>\alpha >\left\{{\theta }_{\mathrm{⌞}},{\theta }_{\mathrm{⌜}},{\theta }_{\mathrm{⌝}},{\theta }_{\mathrm{⌟}}\right\}$; that is, the most variable correlations are pairwise correlations in the cardinal directions, followed by pairwise correlations in the oblique directions, followed by fourth-order correlations. Interestingly, third-order correlations are the least variable across image patches. An analogous analysis performed on white noise yields a flat distribution with considerably smaller standard deviation values (See ‘Materials and methods’, Analysis variants for Penn Natural Image Database, and Figure 1—figure supplement 3 for comparison), and performing the analysis on a colored Gaussian noise (e.g. $1/f^k$ spectrum) would also yield a flat distribution because of the whitening stage in the image preprocessing pipeline. These (and subsequent) findings are preserved across different choices of image analysis parameters (shown in Figure 1E for block-average factors N = 2, 4 and patch sizes R = 32, 48, 64; see ‘Materials and methods’, Analysis variants for Penn Natural Image Database, and Figure 3—figure supplement 5A for a larger set of parameters) and also across other collections of natural images (see ‘Materials and methods’, Comparison with van Hateren Database, and Figure 3—figure supplement 5B for a parallel analysis of the van Hateren image dataset (van Hateren and van der Schaaf, 1998), which gives similar results).

To characterize perceptual sensitivity to different statistics, we isolated them in synthetic visual images and used a figure/ground segmentation task (Figure 2B). We used a four-alternative forced-choice task in which stimuli consisted of a textured target and a binary noise background (or vice-versa). Each stimulus was presented for 120ms and was followed by a noise mask. Subjects were then asked to identify the spatial location (top, bottom, left, or right) of the target. Experiments were carried out for synthetic stimuli in which the target or background was defined by first varying image statistic coordinates independently (Figure 2A shows examples of gamuts from which stimuli are built). Along each coordinate axis, threshold (1/sensitivity) was defined as the coordinate value required to support a criterion level of performance (Figure 2C, inset). We then performed further experiments in which the target or background was defined by simultaneously varying pairs of coordinates. For measurements involving each coordinate pair (to which we will refer as a ‘coordinate plane’), we traced out an isodiscrimination contour (Figure 2C) that describes the threshold values not only along the cardinal directions, but also along oblique directions. Measurements were collected for four individual subjects in each of 11 distinct coordinate planes (representing all distinct coordinate pairs up to 4-fold rotational symmetry; see ‘Materials and methods’, Psychophysical methods, for further details). Each subject performed 4320 judgements per plane, for a total of 47,520 trials per subject.

Figure 2

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Figure 2D shows perceptual sensitivities measured along each coordinate axis. For each of four subjects, a similar pattern emerges for sensitivities as was observed for variation in natural image statistics: sensitivities are rank-ordered as $\left\{{\beta }_{|},{\beta }_{-}\right\}>\left\{{\beta }_{/},{\beta }_{\backslash }\right\}>\alpha >\left\{{\theta }_{\mathrm{⌜}},{\theta }_{\mathrm{⌝}},{\theta }_{\mathrm{⌟}},{\theta }_{\mathrm{⌞}}\right\}$.

Note that the difference between the sensitivities in the horizontal and vertical directions (${\beta }_{-}$ and ${\beta }_{|}$) vs the diagonal directions (${\beta }_{\backslash }$ and ${\beta }_{/}$) is not simply an ‘oblique effect’, that is, a greater sensitivity to cardinally- vs obliquely-oriented contours (Campbell et al., 1966). Horizontal and vertical pairwise correlations differ from the diagonal pairwise correlations in more than just orientation: pixels involved in horizontal and vertical pairwise correlations share an edge, while pixels involved in diagonal pairwise correlations only share a corner. Correspondingly, the difference in sensitivities for horizontal and vertical correlations vs diagonal correlations is approximately 50%, which is much larger than the size of the classical oblique effect (10–20%) (Campbell et al., 1966).

Figures 1E and 2D show a rank-order correspondence between natural image statistics and perceptual sensitivities. This qualitative comparison can be converted to a quantitative one (Figure 3A), as a single scaling parameter aligns the standard deviation of natural image statistics with the corresponding perceptual sensitivities. In this procedure, each of the six image analyses is scaled by a single multiplicative factor that minimizes the squared error between the set of standard deviations and the set of subject-averaged sensitivities (see ‘Materials and methods’, Image preprocessing, and Figure 3—figure supplement 1 for additional details regarding scaling). The agreement is very good, with the mismatch between image analyses and human psychophysics comparable to the variability from one image analysis to another, or from one human subject to another.

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We quantify the correspondence between image analyses and psychophysical analyses by computing the scalar product between the normalized vector of standard deviations (extracted separately from each image analysis) and the normalized vector of subject-averaged sensitivities (extracted from the set of psychophysical analyses). A value of 1 indicates perfect correspondence, and 0 indicates no correspondence. This value ranges from 0.987 to 0.999 across image analyses and is consistently larger than the value measured under the null hypothesis that the apparent correspondence between statistics and sensitivities is chance (p ≤ .0003 for each image analysis; see Tables 1–2 and ‘Materials and methods’, Permutation tests, for details regarding statistical tests).

These findings support our hypothesis that human perceptual sensitivity measured along single coordinate axes (assessed using synthetic binary textures) is predicted by the degree of variation along the same coordinate axes in natural scenes.

The correspondence shown in Figure 3A considers each image statistic coordinate in isolation. However, it is known that image statistics covary substantially in natural images (as diagrammed in Figure 1D) and also that they interact perceptually (as diagrammed in Figure 2C). When pairs of natural image statistics covary, thus sampling oblique directions not aligned with the coordinate axes in the space of image statistics, our hypothesis predicts that human perceptual sensitivity is matched to both the degree and the direction of that covariation (we are referring here to the orientation of a distribution in the coordinate plane of a pair of image statistics, and not to an orientation in physical space). To test this idea, we proceeded as follows.

First, we fit the distribution of image statistics with a multidimensional Gaussian. When projected into pairwise coordinate planes, the isoprobability contours of this Gaussian capture the in-plane shape and orientation of the covariation of the distribution. Along single coordinate axes, the variation in natural image statistics predicts human perceptual sensitivities, as we have shown (Figure 3A). More generally, we would predict that sensitivity should be be high along directions in which the distribution of natural image statistics has high standard deviation, because in those directions, the position of a sample cannot be guessed. Within coordinate planes, the quantitative statement of this idea is that the inverse covariance matrix, or precision matrix, predicts perceptual isodiscrimination contours. Sensitivity is expected to be low (and therefore threshold high) along directions in which the precision matrix has a high value and the position of a sample can be guessed a priori.

Results in each coordinate plane are shown in Figure 3B. Across all subjects and all coordinate planes, we find that the shape and orientation of perceptual isodiscrimination contours (red ellipses) are predicted by the distribution of image statistics extracted from natural scenes (blue ellipses). As in Figure 3A, the correspondence is very good, with mismatch that is comparable to the variability observed across image analyses and across subjects.

To quantify the correspondence between natural image and psychophysical analyses, we describe each ellipse by a single vector $\overrightarrow{\omega }$ that combines information about shape (eccentricity) and orientation (angular tilt), and we compute the scalar product between the image analysis vector ${\overrightarrow{\omega }}_{\text{NI}}$ and the subject-averaged psychophysical vector ${\overrightarrow{\omega }}_{\text{PP}}$. This value, averaged across coordinate planes, ranges from 0.953 to 0.977 across image analyses. We compared this correspondence to that obtained under the null hypotheses that (i) the apparent correspondence between image statistic covariances and isodiscrimination contours is chance, or (ii) the apparent covariances in image statistics are due to chance. The observed correspondence is much greater than the value measured under either null hypothesis (p ≤.0003 for each image analysis under both hypotheses; see ‘Materials and methods’, Analysis of image statistics in pairwise coordinate planes, and Figure 3—figure supplement 2 for comparisons of eccentricity and tilt, and Tables 1–3 and ‘Materials and methods’, Permutation tests, for statistical tests).

These findings confirm that the shape and orientation of human isodiscrimination contours, measured across all pairwise combinations of coordinates, can be quantitatively predicted from the covariation of image statistics extracted from natural scenes. The observed correspondence is maintained within the full 9-dimensional coordinate space (see ‘Materials and methods’, Analysis of the full 9-dimensional distribution of image statistics, and Figure 3—figure supplement 3 for principal component analyses, and Tables 1–3 and ‘Materials and methods‘, Permutation tests, for statistical tests), confirming that our hypothesis describes human sensitivity in the full 9-dimensional space of local image statistics extracted from natural scenes.

Although we did not record cortical responses directly, several lines of evidence indicate that that the perceptual thresholds we measured are determined by cortical processes. First, the stimuli had high contrast (100%) and consisted of pixels that were readily visible (14 arcmin), so retinal limitations of contrast sensitivity and resolution were eliminated. Second, the task requires pooling of information over wide areas (100–200 pixels, that is, a region whose diameter is 10–15 times the width of an image element; see Figure 7 in Victor and Conte, 2005). Retinal receptive fields are unlikely to do this, as the ratio of their spatial extent (surround size) to their resolution (center size) is typically no more than 4:1 (Croner and Kaplan, 1995; Kremers et al., 1995). Third, to account for the specificity of sensitivity to three- and four-point correlations, a cascade of two linear-nonlinear stages is required (Victor and Conte, 1991); retinal responses are quite well-captured by a single nonlinear stage (Nirenberg and Pandarinath, 2012), and cat retinal populations show no sensitivity to the four-point correlations studied used here (Victor, 1986) while simultaneous cortical field potentials do. Conversely, macaque visual cortical neurons (Purpura et al., 1994), especially those in V2, manifest responses to three- and four-point correlations (Yu et al., 2013).

Successive stages of sensory processing share the same broad goals: invest resources in encoding stimulus features that are sufficiently informative, and suppress less-informative ones. In the periphery, this is exemplified by the well-known suppression of very low spatial frequencies; in cortex, this is exemplified by insensitivity to high-order correlations that are predictable from lower-order ones. Previous work has shown that such higher-order correlations can be separated into two groups—informative and uninformative—and only the informative ones are encoded (Tkačik et al., 2010). We used this finding to select an informative subspace for the present study, and we asked how resources should be efficiently allocated amongst features within this informative subspace.

A simple model of efficient coding by neural populations is shown in Figure 4A (details in ‘Materials ans methods’, Two regimes of efficient coding). Here, to enable analytical calculations, we used linear filters of variable gain and subject to Gaussian noise to model a population of neural channels encoding different features. The optimal allocation of resources to maximize information transmitted by the population depends on the amount of input noise, the amount of output noise, the input signal variability, and the total resources available to the system, here quantified as a constraint on the total output power (i.e., sum of response variances) in the neural population. The constrained output power and the output noise together determine the ‘bandwidth’ of the system—that is, the expressive capacity of its outputs. Consider a neural population with input noise, output noise, and a fixed amount of output power. We find that when input signal variability is sufficiently large compared to the input noise, the gain of neurons should decrease with the variance of the input (regions to the right of the peaks in the right-hand panel of Figure 4A). This is a regime where the output bandwidth is low compared to the input range, and efficient coding predicts that signals should be ‘whitened’ by equalizing the variance in different channels. Conversely, consider input signals with a smaller range, which are thus more disrupted by input noise. In this case, the gain of neurons should increase with the variance of the input (regions to the left of the peaks in the right-hand panel of Figure 4A). This is a regime where the input noise dominates, and efficient coding predicts that the system should invest more resources in more variable, and hence more easily detectable, input signals. The relative sizes of input and output noise (controlled by $\text{Λ}$ in Figure 4A) determines the input ranges over which the two qualitatively different regimes of efficient coding apply.

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To make these abstract considerations concrete, we first considered coding in the sensory periphery. A common strategy employed in the periphery is ‘whitening’, where relatively fewer resources are devoted (yielding lower gain) to features with more variation (Olshausen and Field, 1996). As an example, within the spatial frequency range that the retina captures well, sensitivity is greater for high spatial frequencies than for low ones, that is, sensitivity is inversely related to the degree of variation in natural scenes (the well-known $1/f^2$ power spectrum [Olshausen and Field, 1996]). Figure 4B illustrates how this strategy can emerge from the simple efficient coding scheme discussed above as applied to peripheral sensory processing. Spatiotemporal correlations of light undergo filtering before passing through the optic nerve bottleneck (a constraint on bandwidth). Such a constraint on bandwidth is equivalently understood as a regime where output noise is relatively large compared to input noise. In this limit, where output noise dominates over input noise, the optimal strategy is whitening (See Srinivasan et al., 1982 and Figure 4A). Of course, real neural systems contend with both input and output noise; indeed recent work has shown that simply whitening to deal with output noise underestimates the optimal performance that the sensory periphery can achieve (Doi and Lewicki, 2014).

An alternative regime arises when input noise limits performance. In this regime, relatively more resources are devoted to features with more variation. This regime was discussed in early work of van Hateren, (1992a) and was also recognized in (Doi and Lewicki, 2011, 2014), although it has received much less attention than the ‘whitening’ regime. Our results suggest that this is the regime is relevant to cortex, where it predicts the relative allocation of resources to higher-order image statistics. Figure 4C illustrates the simple efficient coding scheme in this context. We use our image preprocessing pipeline to mimic early visual processing, and we consider the downstream coding of higher-order image features. Because these features must be sampled from a finite patch of an image, they are subject to input noise arising from fluctuations in statistical estimation. When such input noise is limiting, the ability to detect a signal from noise increases with the variability of that signal. In this limit, efficient coding predicts that resources should be allocated in proportion to feature variability (Figure 4C). This captures the intuition that when signal reliability is in question, more reliable signals warrant more resources. Furthermore, if two or more channels have covarying signals, resources should be devoted in relation to the direction and degree of maximum covariance (see ‘Materials and methods’, Two regimes of efficient coding, Figure 4—figure supplement 3, and Figure 4—figure supplement 4).

The difference between these two efficient coding regimes is a consequence of the form of noise—output vs input noise—that is limiting. Our finding that cortex operates in a different regime than the well-known peripheral whitening reflects the fact that different stages and kinds of processing can face different constraints. While information transmission by the visual periphery is limited by a bottleneck in the optic nerve, cortex faces no such transmission constraint. Furthermore, while faithful encoding may be an immediate goal of early visual processing, cortical circuits have to interpret image features from a complex and crowded visual scene and perform statistical inference. For example, to discriminate between various textures, the cortex cannot perform pixel-by-pixel comparisons, but must rely on the estimation of local correlations (image statistics) instead. Because these correlations must be sampled from a finite patch of the visual scene, any estimate will be limited by sampling fluctuations.

Sampling fluctuations constitute a source of input noise, the magnitude of which depends on the size of the sampled region. For natural images, this gives rise to a tradeoff: small regions lead to large fluctuations in the estimated statistics, while large regions blur over local details. This blurring may obscure the boundaries between objects with different surface properties. While the brain must implement such sampling, the size, scale, and potentially dynamic nature of the sampling region is not known. Interestingly, our predictions of human sensitivities do not change substantially over a wide range of spatial scales and image patch sizes, perhaps reflecting a scaling property of natural images (Stephens et al., 2013). An avenue for future research is to determine whether there is an optimal region size, and if so, whether it could be estimated from images themselves.

Sampling limitations alone do not suffice to account for the observed differential sensitivity of the brain to local image statistics. Were sampling limitations the only consideration, perceptual sensitivity would be the same along each coordinate axis, and perceptual isodiscrimination contours would be circular in each coordinate plane. This follows from an ideal observer calculation (See Appendix B of Victor and Conte, 2012). In contrast, we find that human observers have a severalfold variability in sensitivity along different coordinate axes (Figure 3A) and have isodiscrimination contours that are elongated in oblique directions (Figure 3B). The efficient coding principle can account for these findings by taking into consideration the fact that a real observer has finite processing resources. In this context (finite resources and substantial input noise), the efficient coding principle predicts that resources are invested in relation to the range of signal values that are typically present (van Hateren, 1992a), as we find. Interestingly, resource limitations seem to play an important role in the cortex despite the vast expansion in the number of neurons compared to the optic nerve. Presumably, this reflects the large number of complex features that could be computed and the corresponding need for a large overrepresentation of the stimulus space (Olshausen and Field, 1997).

While we find a close match between the variation in natural image statistics and human psychophysical performance, some aspects of the distribution of natural image statistics do not match psychophysical data.

These differences are not readily apparent when we examine the variances and covariances (Figure 3) of the distribution of natural image statistics but emerge only when one considers its detailed shape (see ‘Materials and methods’, Asymmetries in distributions of natural image statistics). For example, the distribution of α-coordinate values has a longer tail in the positive vs negative direction (see Figure 3—figure supplement 9 and (Tkačik et al., 2010)). In contrast, human perceptual sensitivity is symmetric, or very nearly so (within $\sim 20\text{\%}$), for positive vs negative values of α (Victor et al., 2005; Victor and Conte, 2012; Victor et al., 2013). This suggests that limitations imposed by ‘neural hardware’ force the system to use heuristics instead of matching the natural image distribution exactly. For example, an opponent mechanism responsible for detecting variations along, example, the α coordinate, might be a useful and easy (although imperfect) way to process the asymmetric distribution of four-point correlations found in natural scenes. Such a mechanism could be matched to the variance of the natural image distribution along the α coordinate, but not to its skew or other odd moments. An opponent mechanism would necessarily give rise to equal sensitivities to positive vs negative values of α, as observed in psychophysical results. Further study of deviations from a perfect match to the distribution of natural image statistics might provide additional insight into these or other possible neural mechanisms, and into the goals of the computations. Independently, our results also raise an interesting theoretical question about the optimal representation of non-gaussian, multidimensional signals under resource-limited conditions.

Looking forward, we hypothesize that the principle of efficient coding might apply to cortical processing at higher levels. For example, more complex image features, such as shapes, are represented as conjunctions of contour fragments (Brincat and Connor, 2004), where each contour fragment is a local image object defined by particular multi-point correlations. We might speculate that the joint statistics of contour fragments in natural scenes can predict, through appropriate formulation of the same efficient coding principle used here, the properties of neurons in area IT (Hung et al., 2012; Yau et al., 2012) or the associated perceptual sensitivities of human observers.

Finally, although we have focused on perception of image statistics, we do this with the premise that this process is in the service of inferring the materials and objects that created an image and ultimately, guiding action. Thus, it is notable that we found a tight correspondence between visual perception and natural scene statistics without considering a specific task or behavioral set. Indeed, the emergence of higher-order percepts without explicit task specification was the original hope of the efficient coding framework as first put forward by Barlow and Attneave (Attneave, 1954; Barlow, 1959, 1961). Doubtless, these ‘top-down’ factors also influence the visual computations that underlie perception, and the nature and site of this influence are an important focus of future research.

We determined perceptual sensitivity to local image statistics via a texture segmentation paradigm adapted from (Chubb et al., 2004), and in standard use in our lab (Victor et al., 2005; Victor and Conte, 2012; Victor et al., 2013); we describe it briefly here. These measurements were carried out in parallel with the natural scene analysis described above. Some of the psychophysical results have been previously reported (Victor and Conte, 2012; Victor et al., 2013); see ‘Subjects’ below.

In pairwise coordinate planes, our hypothesis predicts that the inverse covariance matrix, or precision matrix, matches human isodiscrimination contours. A precision matrix is represented by the contour lines of its inverse (the covariance matrix M); these are the points (x, y) at which $M_{xx}{x}^2+2M_{xy}xy+M_{yy}{y}^2=$ constant. A short distance of this contour from the origin thus indicates a large value of M and a small value of the precision matrix. This in turn denotes a direction in which prior knowledge of the image statistic is imprecise.

Figure 3B shows a correspondence between contours of the precision matrix (extracted from natural images) and human isodiscrimination contours. This is shown again here in Figure 3—figure supplement 2A for subject-specific (lower half grid) and subject-averaged (upper half grid) isodiscrimination contours. This correspondence can be made quantitative by computing the angular tilt (Figure 3—figure supplement 2B) and eccentricity (Figure 3—figure supplement 2C) of each ellipse. Across all 36 pairwise coordinate planes, we find a detailed quantitative match between the shape and orientation of precision matrix contours and human isodiscrimination contours.

Our results, shown in Figure 3 for single coordinates and pairwise coordinate planes, and extended to the full 9-dimensional distribution in Figure 3—figure supplement 3, show a consistent match between the variation in natural image statistics and psychophysical sensitivities. We quantify this match by first assigning vectors to the quantities shown in Figure 3 and Figure 3—figure supplement 3, and then computing the overlap between natural image vectors and the corresponding psychophysical vectors. We consider the following vector quantities:

1. Single coordinates: We describe the range of variation in natural image statistics by the normalized 9-component vector of standard deviations ${\overrightarrow{\sigma }}_{\text{NI}}/\left|\left|{\overrightarrow{\sigma }}_{\text{NI}}\right|\right|$, where $\left|\left|\overrightarrow{v}\right|\right|$ denotes the L2 norm $\frac{1}{N}{{{\displaystyle \sum }}^{\text{}}}_{i=1}^N{v}_i^2$ of a vector $\overrightarrow{v}$. Similarly, we describe the set of perceptual sensitivities by the normalized vector ${\overrightarrow{s}}_{\text{PP}}/\left|\left|{\overrightarrow{s}}_{\text{PP}}\right|\right|$. In both cases, the vector components are measured with respect to the coordinates $\left\{{\beta }_{|},{\beta }_{-},{\beta }_{\backslash },{\beta }_{/},{\theta }_{\mathrm{⌜}},{\theta }_{\mathrm{⌝}},{\theta }_{\mathrm{⌟}},{\theta }_{\mathrm{⌞}},\alpha \right\}$.

2. Pairwise coordinate planes: We describe each ellipse by the unit vector $\overrightarrow{\omega }$ that is a combined measure of eccentricity (∈) and tilt (δ). We define $\overrightarrow{\omega }$ on one quarter of the unit sphere: $\overrightarrow{\omega }=\text{sin\hspace{0.17em}}\alpha \text{\hspace{0.17em}cos\hspace{0.17em}}\delta \widehat{\mathbf{x}}+\text{sin\hspace{0.17em}}\alpha \text{\hspace{0.17em}sin\hspace{0.17em}}\delta \widehat{\mathbf{y}}+\text{cos\hspace{0.17em}}\alpha \widehat{\mathbf{z}}$, where $\mathit{ϵ}=\text{sin\hspace{0.17em}}\alpha$ and $\text{cos\hspace{0.17em}}\delta$ are defined on the interval [0,1] (the second follows from the 180^° rotational symmetry of ellipses). Note that this definition of $\overrightarrow{\omega }$ captures the ellipse property that when $\mathit{ϵ}=\text{sin\hspace{0.17em}}\alpha =0$ (circular ellipses), δ is not defined. See Figure 3—figure supplement 4 for a schematic of this representation.

3. Principal components: We consider two related measures for describing principal components. As shown in Figure 3—figure supplement 3, we describe each principal component $\left\{{\overrightarrow{\xi }}^{\left(i\right)}\right\}$ by the normalized vector ${\overrightarrow{f}}^{\left(i\right)}/\left|\left|{\overrightarrow{f}}^{\left(i\right)}\right|\right|$, which measures the fractional contribution of sets of statistics to the principal components ${\overrightarrow{\xi }}^{\left(i\right)}$. For a more detailed comparison, we can similarly describe each principal component by the normalized vector ${\overrightarrow{F}}^{\left(i\right)}/\left|\left|{\overrightarrow{F}}^{\left(i\right)}\right|\right|$, where ${\overrightarrow{F}}^{\left(i\right)}=[f_{{\beta }_{|}}^{\left(i\right)},f_{{\beta }_{-}}^{\left(i\right)},f_{{\beta }_{\backslash }}^{\left(i\right)},f_{{\beta }_{/}}^{\left(i\right)},f_{{\theta }_{\mathrm{⌜}}}^{\left(i\right)},f_{{\theta }_{\mathrm{⌝}}}^{\left(i\right)},f_{{\theta }_{\mathrm{⌟}}}^{\left(i\right)},f_{{\theta }_{\mathrm{⌞}}}^{\left(i\right)},f_{\alpha }^{\left(i\right)}]$. This measures the fractional contribution of individual statistics (rather than sets of statistics) to the principal components ${\overrightarrow{\xi }}^{\left(i\right)}$.

For each vector quantity ($\overrightarrow{\sigma }$, $\overrightarrow{\omega }$, $\overrightarrow{f}$, and $\overrightarrow{F}$), we compute the scalar product between a given image analysis vector and the subject-averaged psychophysical vector. We then report the overlap values (scalar products) measured for the six image analyses considered Figures 1 and 3 (N = 2, 4 and R = 32, 48, 64). In computing the scalar product between ${\overrightarrow{\omega }}_{\text{NI}}$ and ${\overrightarrow{\omega }}_{\text{PP}}$, we report the overlap averaged over all 36 pairwise coordinate planes. Similarly, in computing the overlap between ${\overrightarrow{f}}_{\text{NI}}$ and ${\overrightarrow{f}}_{\text{PP}}$ and between ${\overrightarrow{F}}_{\text{NI}}$ and ${\overrightarrow{F}}_{\text{PP}}$, we report the overlap averaged over all 9 principal components. Note that, for each vector $\overrightarrow{\sigma }$, $\overrightarrow{\omega }$, $\overrightarrow{f}$, and $\overrightarrow{F}$, the maximum overlap is 1.

We find that natural image analyses show consistently high overlap with the set of psychophysical results (see Tables 1–3). The overlap, as measured across image analyses, ranges from 0.988 to 0.999 for single coordinates ($\overrightarrow{\sigma }$), from 0.953 to 0.977 for pairwise coordinate planes ($\overrightarrow{\omega }$), from 0.987 to 0.993 for fractional principal axes ($\overrightarrow{f}$), and from 0.829 to 0.917 for the full principal axes ($\overrightarrow{F}$). We test the significance of this overlap by comparing our results to the following two null models:

• 1A. Shuffled coordinate labels: sets of coordinates. This model (and model 1b) tests the null hypothesis that the apparent correspondence between image statistic covariances and isodiscrimination contours is chance. We examine the 23 permutations of the sets of coordinates $\left\{{\beta }_{|-},{\beta }_{\backslash /},\theta ,\alpha \right\}$. We apply these permutations to the psychophysical data, as human subjects are equally sensitive to coordinates within each set ($\left\{{\beta }_{|},{\beta }_{-}\right\}$, $\left\{{\beta }_{\backslash },{\beta }_{/}\right\}$ and all θ's). This shuffling creates a new set of subjects whose second-order cardinal, second-order oblique, third-order, and fourth-order coordinate values are randomly permuted (transforming the original vector $[{\beta }_{|-},{\beta }_{\backslash /},\theta ,\alpha ]$ into, example, the shuffled vector $[{\beta }_{\backslash /},\theta ,{\beta }_{|-},\alpha ]$). If the correspondence between quantities derived from image analysis and psychophysics is statistically significant, we expect that the shuffled vectors $\overrightarrow{\sigma }$, $\overrightarrow{\omega }$, $\overrightarrow{f}$, and $\overrightarrow{F}$ will show less overlap with the image analysis vectors than do the original psychophysical vectors (note that the limited number of permutations restricts the minimum p-value to be 0.04).

• 1B. Shuffled coordinate labels: individual coordinates. Here, we expand the test described in 1a to randomly shuffle the full set of coordinate labels $\{{\beta }_{|},{\beta }_{-},{\beta }_{\backslash },$ ${\beta }_{/},{\theta }_{\mathrm{⌜}},{\theta }_{\mathrm{⌝}},{\theta }_{\mathrm{⌟}},{\theta }_{\mathrm{⌞}},\alpha \}$. In an analogous manner to that described in 2A, we expect that the shuffled vectors $\overrightarrow{\sigma }$, $\overrightarrow{\omega }$, $\overrightarrow{f}$, and $\overrightarrow{F}$ will show less overlap with the image analysis vectors than do the original psychophysical vectors if the correspondence between quantities derived from image analysis and psychophysics is statistically significant.

• 2. Shuffled patch labels. This model tests the null hypothesis that the apparent covariances in image statistics are due to chance. For each coordinate, we randomly shuffle image patch labels. This shuffling creates a new set of null patches whose second-, third-, and fourth-order coordinate values are randomly drawn from a subset of the original image patches (e.g. a given null patch can be described by a ${\beta }_{/}$-value measured from patch m but an α value measured from patch n). This shuffling destroys correlations between coordinate values measured within individual patches. Note that this shuffling does not alter the range of variation measured along single coordinate axes and will therefore not alter the values of precision matrix ellipses measured along coordinate axes. As a result, this test is not applicable to $\overrightarrow{\sigma }$, which measures natural image variation and human sensitivities along individual coordinate axes. However, shuffling will destroy correlations along oblique directions in coordinate planes, thereby aligning each ellipse along a single coordinate axis. Note that the eccentricity of each ellipse (in, e.g., the A-B plane) is then trivially related to the ratio of variances ${\sigma }^2$ measured along the corresponding coordinate axes: $\mathit{ϵ}=\sqrt{1-{\sigma }_A^2/{\sigma }_B^2}$. We therefore expect that this shuffling will most strongly affect the tilt and eccentricity in pairwise planes in which ellipses are oriented along oblique directions (${\beta }_{|}{\beta }_{-}$, ${\beta }_{\backslash }{\beta }_{/}$, and $\theta \theta$ planes). Finally, in destroying correlations between pairs of coordinates, this shuffling creates a diagonal covariance matrix, such that principal components are aligned with single coordinate axes. If the correspondence between quantities derived from image analysis and psychophysics is statistically significant, we expect that the shuffled vectors $\overrightarrow{\omega }$, $\overrightarrow{f}$, and $\overrightarrow{F}$ will show less overlap with the psychophysical vectors than do the original image analysis vectors.

Each null model is constructed by randomly selecting permuted indices that independently shuffle coordinate labels for subject-averaged psychophysical data (Null Model 1) and independently shuffle image patch labels for a given statistic (Null Model 2). For null model 1a, we perform the full set of 23 non-identity permutations. For models 1B and 2, we perform 10,000 permutations.

For each permutation, we compute a set of shuffled vectors $\left\{\overrightarrow{\sigma },\overrightarrow{\omega },\overrightarrow{f},\overrightarrow{F}\right\}$, and we measure the overlap (defined as the scalar product ${\overrightarrow{\left(\text{*}\right)}}_{\text{NI}}\cdot {\overrightarrow{\left(\text{*}\right)}}_{\text{PP}}$) between each shuffled vector and the corresponding subject-averaged psychophysical vector. Note that, when assigning shuffled principal components to symmetry classes, no hand-tuning was performed. However, as described previously, such hand-tuning was only applied to a very small fraction of components for select image analyses.

When repeated for many permutations, this procedure yields a distribution of shuffled overlap values against which we measure the significance of the true (observed) overlap. Significance values (p-values) are estimated by computing the fraction of permutations for which the shuffled overlap exceeds the true overlap.

We find that the original image analyses show significantly higher overlap with psychophysical data than do the analyses produced by either of the null models. Results are significant for each measure of overlap and for each of the six analyses presented in Figures 1 and 3 (p <0.0005, or as small as possible given the number of possible permutations, in all cases); see Tables 1–3 for full results.

In Figures 1 and 3, we reported results using image analyses with varying values of the block-average factor N (N = 2, 4) and patch size R (R = 32, 48, 64). In Figure 1—figure supplement 3, we show that the relative variation in different image statistics (first shown in Figure 1E) is not an artifact of our image analysis pipeline, as the pattern of variation is destroyed if white-noise image patches are instead used. In Figures 3–figure supplement 5-3–figure supplement 8, we show that the comparison between natural image and psychophysical analyses is consistent across a wider range of image preprocessing parameters: N = 2, 4, 8, 12, 16, 20 and R = 32, 48, 64, 80, 128. Note that sampling limitations restrict some combinations of N and R (e.g. for sufficiently large N, we must choose sufficiently small R to have a statistically significant number of image patches).

All analyses reported in Results and shown in Figures 1 and 3 were performed on a set of images from the UPenn Natural Image Database (Tkačik et al., 2011). Here, we extend our analyses to a set of 2300 images from the van Hateren image database (van Hateren and van der Schaaf, 1998), using the same set of parameters used to analyze images from the UPenn database, with block-average factors N = 2, 4, 8, 12, 16, 20 and patch sizes R = 32, 48, 64, 80, 128. Note that we are able to perform a larger number of analyses (specific combinations of N and R) than was performed using the Penn database, as we have a larger selection of images and therefore do not face the same sampling limitations. Figures 3—figure supplement 5-3–figure supplement 8 confirm that our results are consistent across image databases.

We find systematic asymmetries in the distributions of natural image statistics when examined beyond their second moments. Figure 3—figure supplement 9 shows the distributions of single coordinates for the image analysis N = 2, R = 32. All distributions are shifted toward positive coordinate values, and there is larger variation in positive vs negative coordinate values. We assess this asymmetry in natural image analyses by computing the ratio of the standard deviations measured along positive vs negative coordinate axes. We similarly assess asymmetry in psychophysical analyses by computing the ratio of human sensitivities to positive vs negative deviations of coordinate values. This comparison is shown in Figure 3—figure supplement 9. The mismatch provides potential clues for the neural mechanisms responsible for processing local image statistics (See Discussion).

In this section, we illustrate how two contrasting regimes emerge from the efficient coding principle: (i) the well-known transmission-limited regime, in which ‘whitening’ is optimal, and (ii) the sampling-limited regime, which is the focus of this paper. To enable exact calculations of optimal behavior, we consider a simplified scenario, in which all signals and noises are Gaussian, and all filters are linear.

We consider a set of channels dedicated to processing independent signals of varying sizes. The channels, which are indexed by k, are abstract and general. For example, each k can represent a different spatial or temporal frequency in the input, as in the traditional analysis of visual coding in the periphery. Here, we take the signal on each channel k to represent a complex image feature, that is the result of a specific local nonlinear transformation applied to the input image.

Figure 4—figure supplement 1 shows the setup of a single channel dedicated to processing the signal $s_k$. Sampling noise, which is assumed to be identical for each channel, is added to this signal; without loss of generality, we can take its value to be unity. Note that for the parametrization of local image statistics used here, sampling noise is in fact identical for each parameter at the origin of the parameter space (see Equations B19-B20 in Victor and Conte, 2012).

The result is passed through a linear filter $L_k$, characterized by a gain $\left|L_k\right|$. The output of $L_k$ then has intrinsic channel noise added, and the total dynamic range of all channels is constrained. All channels are assumed to have the same intrinsic noise. Again, without loss of generality, we take this value to be unity (as any scale associated with this noise can be absorbed into an overall multiplier for the filters $L_k$ and the constraint on total dynamic range of the channels).

We seek to find the optimal set of gains $\left\{\left|L_k\right|\right\}$ that maximize the mutual information ${\displaystyle {\sum }_k{H}_k}$ between the signals $\left\{s_k\right\}$ and the channel input, subject to a constraint Q on total output power. Using a Lagrange multiplier $\text{Λ}$ for the constraint, the problems translates into extremizing $P={\displaystyle {\sum }_k{H}_k}+\text{Λ}Q$ by setting $\partial P/\partial L_k=0$.

The solution can be found in Equation 8 of van Hateren, (1992a), noting the following correspondences between the setup of Figure 4—figure supplement 1 and the scenario considered in that paper. Referring to the notation in (van Hateren, 1992a), the input and channel noises, N_p and N_c, respectively correspond here to the sampling and channel noises (both taken to be unity). The prefiltered stimulus power S_p corresponds here to signal variance $s_k^2$. The power transfer function p_n of the neural filter corresponds here to the filter power $|L_k{|}^2$. Finally, the negative Lagrange multiplier $-\lambda$ corresponds here to the positive Lagrange multiplier $+\text{Λ}$. With these correspondences, the optimal filter for channel k has a gain $\left|L_k\right|$ given by:

provided that the above quantity is non-negative, and has a gain of zero otherwise. The range of values of $s_k$ for which the above quantity is $\le 0$ corresponds to signals that are not worthwhile to code, because the signal-to-noise is too small given the constraint on the channel dynamic range. More specifically, the above quantity is positive (and hence $\left|L_k\right|$ is nonzero) provided that $s_k>\sqrt{\text{Λ}/\left(1-\text{Λ}\right)}$. Note that this critical value becomes infinite as $\text{Λ}$ approaches one from below, indicating that $\text{Λ}$ near one is the transmission-limited regime. Conversely, the critical value of $s_k$ approaches zero as $\text{Λ}$ approaches zero from above, indicating that this is the sampling-limited regime. We further discuss these regimes below.

We interpret the gain $\left|L_k\right|$ as representing the amount of resources devoted to a given signal $s_k$. Since it is a direct measure of signal-to-noise for a unit-size input, it therefore corresponds to perceptual sensitivity.

In the psychophysical experiments here, we measure sensitivity for each of the image statistic coordinates $\left\{{\beta }_{|},{\beta }_{-},{\beta }_{/},{\beta }_{\backslash },{\theta }_{\mathrm{⌜}},{\theta }_{\mathrm{⌝}},{\theta }_{\mathrm{⌟}},{\theta }_{\mathrm{⌞}},\alpha \right\}$, using a highly artificial set of stimuli. As predicted from the sampling-limited regime, we find that gains $\left|L_k\right|$ are larger for the channels in which the natural environment provides larger values of the signal $s_k$.

While this analysis provides a rigorous identification of a regime in which gain increases with signal strength, we caution that it is an asymptotic analysis of a simplified model of feature coding. It therefore stops short of making the quantitative prediction that gain (sensitivity) is proportional to the square root of the signal strength of each image statistic.

On the other hand, the analysis does translate into a quantitative prediction about perceptual axes (i.e., about the orientations of the isodiscrimination contours). As shown in Figure 3 (blue contours), the image statistic coordinates $\left\{{\beta }_{|},{\beta }_{-},{\beta }_{/},{\beta }_{\backslash },{\theta }_{\mathrm{⌜}},{\theta }_{\mathrm{⌝}},{\theta }_{\mathrm{⌟}},{\theta }_{\mathrm{⌞}},\alpha \right\}$ have substantial covariances. A rotation of the coordinates will thus yield a new set of coordinates with zero covariance and independent sampling errors. If these new coordinates are independently coded, then the perceptual axes will share the same axes as the image statistics which is what we find (Figure 3B).

$0\le \text{Ξ}\ll \text{Σ}$ (dominating channel noise). The optimal strategy is decorrelation by whitening (Figure 4—figure supplement 4A); the tilt of the filter relative to the signal is $\pi /2$, and the eccentricities are equal (i.e., the small eigenvalue of ${\mathbf{L}}^{\ast }{\mathbf{L}}^{\ast T}$ is proportional to the inverse of the large eigenvalue of S and vice versa, indicating that the gain scales as the inverse of the input power).

$0<\text{Ξ}\ll 1,\text{Σ}=0$ (zero channel noise, small sampling noise). The optimal strategy is still decorrelation (Figure 4—figure supplement 4B) with signal components of higher power being suppressed by the gain, but the suppression does not follow the inverse law as above.

$\text{Ξ}\ge 1,\text{Σ}=0$ (zero channel noise, large sampling noise). The tilt of the filter matches the tilt of the signal, and the gain scales with input power. For high sampling noise and zero channel noise, the gain scales as the square-root of the input power (Figure 4—figure supplement 4C).

$\text{Ξ}>\Sigma >0$ (dominating sampling noise). In a broad regime of noise strengths where sampling noise dominates over non-zero channel noise, the tilt of the gain matches the tilt of the signal, and the gain roughly scales with the input power (Figure 4—figure supplement 4D). This regime is consistent with the correspondence that we observe between the natural scenes statistics and the psychophysical measurements.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Jason Prentice and Eizaburo Doi for valuable discussions. This work was supported by NIH EY07977, NSF PHY-1058202, FWF P25651, NEI Vision Training Grant 5-T32-EY007035-32, and the Fondation Pierre Gilles de Gennes.

Human subjects: The human subjects research (visual psychophysics) was approved by the Institutional Review Board of the Weill Cornell Medical College, and was in accord with the World Medical Association Declaration of Helsinki. Informed consent was obtained from each subject prior to the experimental sessions, and consent to publish was obtained from Mary Conte (MC), the one subject who is potentially identifiable by the initials since she is also an author.

© 2014, Hermundstad et al.

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