An image reconstruction framework for characterizing initial visual encoding

Visual perception begins at the retina, which takes sensory measurements of the light incident at the eyes. This initial representation is then transformed by computations that support perceptual inferences about the external world. Even these earliest sensory measurements, however, do not preserve all of the information available in the light signal. Factors such as optical aberrations, spatial and spectral sampling by the cone mosaic, and noise in the cone excitations all limit the information available downstream.

One approach to understanding the implications of such information loss is ideal observer analysis, which evaluates the optimal performance on psychophysical discrimination tasks. This allows for quantification of the limits imposed by features of the initial visual encoding, as well as predictions of the effect of variation in these features (Geisler, 1989; Geisler, 2011). Ideal observer analysis separates effects due to the visual representation from inefficiencies in the processes that mediate the discrimination decisions themselves. Such analyses have often been applied to analyze performance for simple artificial stimuli, assuming that the stimuli to be discriminated are known exactly (Banks et al., 1987; Davila and Geisler, 1991) or known statistically with some uncertainty (Pelli, 1985; Geisler, 2018). The ideal observer approach has been extended to consider decision processes that learn aspects of the stimuli being discriminated, rather than being provided with these a priori, and extended to handle discrimination and estimation tasks with naturalistic stimuli (Burge and Geisler, 2011; Burge and Geisler, 2014; Singh et al., 2018; Chin and Burge, 2020; Kim and Burge, 2020). For a recent review, see Burge, 2020; also see Tjan and Legge, 1998 and Cottaris et al., 2019; Cottaris et al., 2020.

It is generally accepted that the visual system has internalized the statistical regularities of natural scenes, so as to take advantage of these regularities for making perceptual inferences (Attneave, 1954; Field, 1987; Shepard, 1987; Knill et al., 1996). This motivates interest in extending ideal observer analysis to apply to fully naturalistic input, while incorporating the statistical regularities of natural scenes (Burge, 2020). Here, we pursue an approach to this goal that, in addition, extends the evaluation of performance to a diverse set of objectives.

We developed a method that, under certain assumptions, optimally reconstructs images from noisy cone excitations, with the excitations generated from an accurate image-computable model of the front end of the visual system (Cottaris et al., 2019; Cottaris et al., 2020). (We use the term ‘image-computable’ here in contrast with observer models that operate on abstract and/or hypothetical internal representations.) The image reconstruction approach provides us with a unified framework for characterizing the information loss due to various factors in the initial encoding. In the next sections, we show analyses that: (1) use image reconstruction error as an information metric to understand the retinal mosaic ‘design’ problem, with one example examining the implications of different allocations of retinal cone types; (2) allow both visualization and quantification of information loss due to physiological optics and cone mosaic sampling and how this varies with eccentricity, as well as with different types of color deficiency; (3) combine the image reconstruction approach with analysis of psychophysical discrimination, thus providing a way to incorporate into such analyses the assumption that our visual system takes into account the statistical regularities of natural images.

We developed a Bayesian method to reconstruct images from sensory measurements, which we describe briefly here (see Materials and methods for details). We begin with a forward model that expresses the relation between an image and its visual representation at a well-defined stage in the visual pathway. Here that stage is the excitations of the photoreceptors of the retinal cone mosaic, so that our model accounts for blur in retinal image formation, spatial and spectral sampling by the cone mosaic, and the noise in the cone excitations. The approach is general, however, and may be applied to other sites in the visual pathways (see e.g. Naselaris et al., 2009; Parthasarathy et al., 2017). Our forward model is implemented within the open-source software package ISETBio (isetbio.org; Figure 1A–C) which encapsulates the probabilistic relationship between the stimulus (i.e. pixel values of a displayed RGB image) and the cone excitations (i.e. trial-by-trial photopigment isomerizations). ISETBio simulates the process of displaying an image on a monitor (Figure 1A), the wavelength-dependent optical blur of the human eye and spectral transmission through the lens and the macular pigment (Figure 1B), as well as the interleaved spatial and chromatic sampling of the retinal image by the L, M, and S cones (Figure 1C). Noise in the cone signals is characterized by a Poisson process. The forward model allows us to compute the likelihood function. The likelihood function represents the probability that an observed pattern of cone excitations was produced by any given image.

Figure 1

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To obtain a prior over natural images, we applied independent components analysis (ICA, see Materials and methods) to a large dataset of natural images (Russakovsky et al., 2015), and fit an exponential probability density function to the individual component weights (Figure 1D). The prior serves as our description of the statistical structure of natural images.

Given the likelihood function, prior distribution, and an observed pattern of cone excitations, we can then obtain a reconstruction of the original image stimulus by applying Bayes rule to find the posterior probability of any image given that pattern. We take the reconstructed image as the one that maximizes the posteriori probability (MAP estimate, see Materials and methods) (Figure 1D).

Basic properties of the reconstructions

To understand the consequences of initial visual encoding, we need to study the interaction between the likelihood function (i.e. our model of the initial encoding) and the statistics of natural images (i.e. the image prior). There are strong constraints on the statistical structure of natural images, such that natural images occupy only a small manifold within the space of all possible images. The properties of the initial encoding produce ambiguities with respect to what image is displayed when only the likelihood function is considered, but if these can be resolved by taking advantage of the statistical regularities of the visual environment, they should in principle, not prohibit effective visual perception. To illustrate this point, consider the simple example of discrete signal sampling: Based on the sampled signal, one cannot distinguish between the original signal from all its possible aliases (Bracewell, 1986). However, with the prior knowledge that the original signal contains only frequencies below the Nyquist frequency of the sampling array, this ambiguity is resolved. In the context of our current study, the role of the natural image prior comes in several forms, as we will demonstrate in Results. First, since the reconstruction problem is underdetermined, the prior is a regularizer, providing a unique MAP estimate; Second, the prior acts as a denoiser, counteracting the Poisson noise in the cone excitation; Lastly, the prior guides the spatial and spectral demosaicing of the signals provided via the discrete sampling of the retinal image by the cone mosaic.

To highlight the importance of prior information while holding the likelihood function fixed, we can vary a parameter $\gamma$ that adjusts the weight of the log-prior term in the reconstruction objective function (see Materials and methods). Explicitly manipulating $\gamma$ reveals the effect of the prior on the reconstruction (Figure 2). When $\gamma$ is small, the reconstruction is corrupted by the noise and the ambiguity of the initial visual encoding (Figure 2A and B). When $\gamma$ is large, the prior leads to desaturation and over-smoothing (Figure 2E) in the reconstruction. For the rest of our simulations, the value of $\gamma$ is determined on the training set by a cross-validation procedure that minimizes the reconstruction error, unless specified otherwise (Figure 2C).

Figure 2

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To further elucidate properties of the Bayesian reconstruction, especially the interaction between the likelihood and prior, we plotted a few representative images in a log-prior, log-likelihood coordinate system, given a particular instance of cone excitations (Figure 3). The optimal reconstruction, taken as the MAP estimate, has both a high prior probability and likelihood value as expected (Figure 3A). In fact, for our reconstruction algorithm, there should not exist any image above the $\gamma x+y=c$ line that goes through A (solid line, Figure 3), otherwise the optimization routine has failed to find the global optimum. The original image stimulus (ground truth) has a slightly lower likelihood value, mainly due to noise present in the cone excitations, and also a slightly lower prior probability, possibly due to the fact that our prior is only an approximation to the true natural image distribution (Figure 3B). The detrimental effect of noise becomes prominent in a maximum likelihood estimate (MLE, Figure 3C): Noise in the cone excitations is interpreted as true variation in the original image stimulus, thus slightly increasing the likelihood value but also creating artifacts. Such artifacts are penalized by the prior in other reconstructions. Furthermore, even without the presence of noise, other features of the initial visual encoding (e.g. Figure 1B and C) cause loss of information and ambiguity for the reconstruction. This is illustrated by a set of images that lie on the equal likelihood line with the MAP reconstruction (Figure 3D): There exist an infinite set of variations in the image (stimulus) that have no effect on the value of the likelihood function (i.e. variations within the null space of the linear likelihood render matrix, see Materials and methods). Thus, the cone excitations provide no information to distinguish between images that differ by such variations. However, as with the case of noise, variations inconsistent with natural images are discouraged by the prior. (Another implication of the existence of the null space is that the MLE solution to the reconstruction problem is actually underdetermined, as an entire subspace of images can have the same likelihood value. In the figure we show one arbitrarily chosen MLE estimate.) Other corruptions of the image, such as addition of white noise in the RGB pixel space, are countered by both the likelihood and prior (Figure 3E). Lastly, for illustrative purposes, we can increase the prior probability of the reconstruction relative to the optimal by making it spatially or chromatically more uniform (Figure 3F), but doing so decreases the likelihood.

Figure 3

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Optimal allocation of retinal photoreceptors

Within the Bayesian reconstruction framework, the goal of the visual front end can be characterized as minimizing the average error in reconstruction across the set of natural images. In this context, we can ask how to choose various elements of the initial encoding, subject to constraints, to minimize the expected reconstruction error under the natural image prior (Levin et al., 2008; Manning and Brainard, 2009). More formally, we seek the ‘design’ parameters $\mathrm{\theta }$ of a visual system:

$\theta =\underset{\theta }{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}{E}_{p(x)}[E_{p(m|x;\theta )}[L(\hat{x}(m;\theta ),x)]],$

where ${\displaystyle \hat{x}(m;\theta )={\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}_{x}p(m|x;\theta )p(x)}$. Here, $x$ represents individual samples of natural images, $m$ represents instances of cone excitation (i.e. sensory measurements), and ${\displaystyle p(m|x;\theta )}$ is our model of the initial encoding (i.e. likelihood function). The particular features under consideration of the modeled visual system are indicated explicitly by the parameter vector $\mathrm{\theta }$. The MAP image reconstruction is indicated by ${\displaystyle \hat{x}\left(m;\theta \right)}$ , and $L\left(\bullet ,\bullet \right)$ is a loss function that assesses reconstruction error. In practice, the expectations are approximated by taking the average over large samples of natural images and cone excitations. (For simplicity in the development here, we did not include the parameter $\gamma$ that we incorporated into our reconstruction algorithm in the equations above. It was included in the actual computations that investigated the reconstruction performance. Also note that the MAP estimate is not in general the one that minimizes the expected loss. We use the MAP estimate as a computationally tractable proxy for the loss-minimizing estimate.)

One intriguing design problem is the allocation of cone photoreceptor types: The maximum number of photoreceptors (cones) per unit area is bounded due to biological constraints. How should the visual system assign this limited resource across the three different types of cones? It has been observed in human subjects that there is a relatively sparse population of S cones, while large individual variability exists in the L/M cone ratio (Hofer et al., 2005). Previous research has used information-theoretical measures combined with approximations to address this question (Garrigan et al., 2010). Here, we empirically evaluated a loss function (i.e. we used root sum of squares distance in the RGB pixel space as well as the S-CIELAB space) on the reconstructed images, while systematically changing the allocation of retinal cone types (Figure 4).

Figure 4 with 2 supplements

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Figure 4—figure supplement 2

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Figure 4—figure supplement 1

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Interestingly, we found that large variations (nearly a 10-fold range) in the assignment of L and M cones have little impact on the average reconstruction error (Figure 4A). Only when the proportion of L or M cones becomes very low is there a substantial increase in reconstruction error, as the modeled visual system approaches dichromacy. On the other hand, the average reconstruction error as a function of the proportion of S cones shows a clear optimum at a small S-cone proportion (~10%; Figure 4B).

Our results are in agreement with a previous analysis in showing that the empirically observed allocation of retinal photoreceptor type is consistent with the principle of optimal design (Garrigan et al., 2010; also see Levin et al., 2008; Manning and Brainard, 2009; Sampat et al., 2015; Jiang et al., 2017). The indifference to L/M ratio can be explained by the large spatial and chromatic correlations present in natural images, together with the high overlap in L- and M-cone spectral sensitivities. This leads to a high correlation in the excitations of neighboring L and M cones in response to natural images, allowing cones of one type to be substituted for cones of the other type with little effect on reconstruction error (see the next paragraph for additional analysis on this point). Additional analysis (Figure 4—figure supplement 1) revealed that the sensitivity to S cone proportion is due to a combination of two main factors: (1) chromatic aberrations, which blur the retinal image at short wavelengths and reduce the value of dense spatial sampling at these wavelengths; and (2) S cones mainly contribute to the estimation of pixel values in the B-pixel plane, whereas L and M cone contribute to both the R- and G-pixel planes (see Figure 4—figure supplement 1). This makes L and M cones more informative than S cones, given the particular loss functions we employ to evaluate reconstruction error. To further validate our conclusion, we have also replicated our analysis with a dataset of hyperspectral (as opposed to RGB) images (Nascimento et al., 2002; Chakrabarti and Zickler, 2011), with a loss function applied directly to the whole spectrum, and have obtained similar results (Figure 4—figure supplement 2, also see Materials and methods).

To further study the role of statistical regularities in the optimal allocation of photoreceptor type, we repeated the L-cone proportion analysis above, but on different sets of synthetic image datasets for which the spatial and chromatic correlations in the images were manipulated explicitly (see Materials and methods). The dependence of the average reconstruction error on the L-cone proportion decreases as the chromatic correlation in the signal increases (Figure 5). A decrease of spatial correlation has little impact on the shape of the curves, but increases the overall magnitude of reconstruction error (Figure 5; to highlight the shape, the scale of the y-axis is different across rows and columns. See Figure 5—figure supplement 1 for the same plot with matched y-axis scale). When both the chromatic and spatial correlation are high, there is a large margin of L-cone proportion within which the reconstruction error is close to the optimal (minimal) point (Figure 5, shaded area). This analysis highlights the importance of considering visual system design in the context of the statistical properties (prior distribution) of natural images, as it shows that the conclusions drawn can vary with these properties (Barlow, 1961; Derrico and Buchsbaum, 1991; Barlow and Földiàgk, 1989; Atick et al., 1992; Lewis and Zhaoping, 2006; Levin et al., 2008; Borghuis et al., 2008; Garrigan et al., 2010; Tkacik et al., 2010; Atick, 2011; Burge, 2020). Natural images are thought to have both high spatial and high chromatic correlation (Webster and Mollon, 1997; Nascimento et al., 2002; Garrigan et al., 2010), making the results shown in Figure 5 consistent with those in Figure 4.

Figure 5 with 1 supplement

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Figure 5—figure supplement 1

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Visualization of color deficiency with image reconstruction

In addition to quantification, the reconstruction framework also provides a method for visualizing the effect of information loss in the initial visual encoding. We know that extreme values of L:M cone ratio create essentially dichromatic retinal mosaics, and from the analysis above we observed that these lead to high reconstruction error. To understand the nature of this error, we can directly visualize the reconstructed images.

Figure 6A shows reconstructions of a set of example images from different dichromatic retinal mosaics. While the spatial structure of the original images is largely retained in the reconstructions, each type of dichromacy creates a distinct pattern of color confusions and shifts in the reconstructed color. Note that in the case where there is no simulated cone noise (as in Figure 6), the original image has a likelihood at least as high as the reconstruction obtained via our method. Thus, the difference between the original images and each of the corresponding dichromatic reconstructions is driven by the image prior. On the other hand, the difference in the reconstructions across the three types of dichromacy illustrates how the different dichromatic likelihood functions interact with the prior.

Figure 6

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One might speculate as to whether the reconstructions predict color appearance as experienced by dichromats. To approach this, we compare the reconstructions with two other methods that have been proposed to predict the color appearance for dichromats (Brettel et al., 1997; Jiang et al., 2016). To determine an image based on the excitations of only two classes of cones, any method will need to rely on a set of regularizing assumptions to resolve the ambiguity introduced by the dichromatic retinas. Brettel et al., 1997 started with the trichromatic cone excitations of each image pixel, and projected these onto a biplanar surface, with each plane defined by the neutral color axis and an anchoring stimulus identified through color appearance judgments made across the two eyes of unilateral dichromats. The resulting trichromatic excitations were then used to determine the rendered RGB values (Figure 6B). Jiang et al., 2016 also adopted a reconstruction approach, but one that reconstructed the incident spectrum from the dichromatic cone excitations at each pixel. They then projected the estimated spectra onto trichromatic cone excitations, and used these to render the RGB values (Figure 6C). In their method, a spectral smoothness constraint was introduced to regularize the spectral estimates, which favors desaturated spectra. In this sense, their prior is similar to ours: The sparse prior we used is centered on the average image, which is desaturated, and also encourages achromatic content due to the high correlations across color channels. One noticeable difference between our method and the other two is that ours takes into account the spatial structure of the image.

Interestingly, although there are differences in detail between the images obtained, in many cases the different methods produce visualizations that are quite similar. We find the general agreement between the reconstruction-based methods and the one based on subject reports an encouraging sign that the reconstruction approach can be used to predict aspects of appearance.

Anomalous trichromacy is another form of color deficiency that is commonly found in human observers. For example, in deuteranomaly, the spectral sensitivity of the M cones is shifted toward that of the L cones (Figure 7B). Since the three cone spectral sensitivity functions are linearly independent of each other, in the absence of noise we should be able to obtain a trichromatic reconstruction from the excitations of the deuteranomalous mosaic. However, in the presence of noise, we expect that the high degree of overlap between M and L spectral sensitivities will result in a lower signal-to-noise ratio (SNR) in the difference between M- and L-cone excitations, compared to that of a normal trichromatic observer, and thus lead to worse reconstructions. We performed image reconstructions for a normal trichromatic (with a peak spectral sensitivity of M cone at 530 nm) and a deuteranomalous (with a peak spectral sensitivity of M cone at 550 nm) 1 deg foveal mosaic at different overall light intensity levels (Figure 7). Due to the nature of Poisson noise, the higher the light intensity, the higher the SNR of the cone excitations. At high light intensities, the reconstructions are similar for the normal and deuteranomalous mosaics (first row). At lower intensities, however, the deuteranomalous reconstruction lacks chromatic content still present in the normal reconstruction (second and third row). The increase in noise also reduces the amount of spatial detail in the reconstructed images, due to the denoising effect driven by the image prior. Furthermore, a loss of chromatic content is also seen for the reconstruction from the normal mosaic at the lowest light level (last row). This observation may be connected to the fact that biological visual systems that operate at low light levels are typically monochromatic, potentially to increase the SNR of spatial vision at the cost of completely disregarding color (e.g. the monochromatic human rod system; see Manning and Brainard, 2009 for a related and more detailed treatment; also see Walls, 1942; Rushton, 1962; van Hateren, 1993; Land and Osorio, 2003).

Figure 7 with 1 supplement

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Figure 7—figure supplement 1

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Effect of physiological optics and mosaic spatial sampling

So far, our visualizations have focused on chromatic information loss due to a reduced number of cone types or a shift in cone spectral sensitivity. However, imperfection in the physiological optics, combined with the spatial sampling of retinal mosaic, also introduces significant loss of information. Furthermore, the interleaved nature of the mosaic means that color and pattern are entangled at the very initial stage of visual processing (Brainard, 2019). To highlight these effects, we reconstructed natural images from 1 deg patches of mosaics at different retinal eccentricities across the visual field, with (1) changes in optical aberrations (Polans et al., 2015); (2) increases in size and decreases in density of the photoreceptors (Curcio et al., 1990); and (3) decreases in the density of the macular pigment (Nolan et al., 2008; Putnam and Bland, 2014). The degradation in the quality of the reconstructed images can be clearly observed as we move from the fovea to the periphery (Figure 8; See Figure 8—figure supplement 1 for an enlarged view of the mosaic and optics). For some retinal locations, the elongated point-spread function (PSF) also introduces a salient directional blur (Figure 8E and F). For a simple quantification of the average reconstruction error as a function of visual eccentricity, see Figure 8—figure supplement 2.

Figure 8 with 4 supplements

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Figure 8—figure supplement 4

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Figure 8—figure supplement 3

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Figure 8—figure supplement 2

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Figure 8—figure supplement 1

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The consequences of irregular spatial sampling by the cone mosaic have been previously studied with the framework of signal processing (Snyder et al., 1977; Yellott, 1983). Our results highlight that optimizing the initial visual encoding depends in rich ways on the interplay between the cone sampling and the optics. While less information (i.e. at more eccentric locations) does lead to overall lower quality reconstructions (Figure 8—figure supplement 2), exactly which aspects of the reconstructions are incorrect can vary in subtle ways. Concretely, in Figure 8, we observe a trade-off across visual eccentricity between spatial and chromatic vision. In the image of the dragonfly, for example, the reconstructed colors are desaturated at intermediate eccentricities (e.g. Figure 8C and D), compared with the fovea (Figure 8A) and more eccentric locations (Figure 8E and F). The desaturation is qualitatively consistent with the literature that indicates a decrease in chromatic sensitivity at peripheral visual eccentricities, at least for the red-green axis of color perception and for some stimulus spatial configurations (Virsu and Rovamo, 1979; Mullen and Kingdom, 1996; but see Hansen et al., 2009). To further elucidate this richness, in an additional analysis, we systematically varied the size of the PSF for a fixed peripheral retinal mosaic. This revealed that (Figure 8—figure supplement 3): (1) A larger PSF does lead to better estimate of chromatic content, albeit eventually at the cost of spatial content. (2) In general, an appropriate amount of optical blur is required to achieve the best overall image reconstruction performance, presumably due to its prevention of aliasing. We will treat the issue of spatial aliasing further in the next section.

Lastly, to emphasize the importance of the natural image prior, we performed a set of maximum likelihood reconstructions with no explicit prior constraint, which resulted in images with less coherent spatial structure and lower fidelity color appearance (Figure 8—figure supplement 4). Thus, the prior here is critical for the proper demosaicing and interpolation of the information provided by the sparse cone sampling at these peripheral locations.

Spatial aliasing

As we have alluded to above, the retinal mosaic and physiological optics can also interact in other important ways: Both in humans and other species, it has been noted that the optical cut-off of the eye is reasonably matched to the spacing of the photoreceptors (i.e. the mosaic Nyquist frequency), enabling good spatial resolution while minimizing spatial aliasing due to discrete sampling (Williams, 1985; Snyder et al., 1986; Land and Nilsson, 2012). In contrast to our work, these analyses did not take into account the fact that the cone mosaic interleaves multiple spectral classes of cones (but see Williams et al., 1991; Brainard, 2015), and here we revisit classic experiments on spatial aliasing for a trichromatic mosaic using our reconstruction framework.

Experimentally, it has been demonstrated that with instruments that bypass the physiological optics and present high contrast grating stimuli directly on the retina, human subjects can detect spatial frequencies up to 200 cyc/deg (Williams, 1985). For foveal viewing, subjects also report having a percept resembling a pattern of ‘two-dimensional noise’ and/or ‘zebra stripes’ when viewing those high spatial frequency stimuli (Williams, 1985). For peripheral viewing, high frequency vertical gratings can be perceived as horizontal (and vice-versa; Coletta and Williams, 1987). We explored these effects within our framework as follows: We reconstructed a set of vertical chromatic grating stimuli from the cone excitations of a foveal and a peripheral mosaic. To simulate the interferometric experimental conditions of Williams, 1985, we used diffraction-limited optics with no longitudinal chromatic aberration (LCA), allowing high-frequency stimuli to reach the cone mosaic directly. For gratings that are above the typical optical cut-off frequency, we obtained reconstructions that (1) are quite distinct from a uniform field, which would allow them to be reliably detected in a discrimination protocol; and (2) lack the coherent vertical structure of the original stimulus (Figure 9). Concretely, the reconstructions recapitulate the ‘zebra stripe’ percept reported at approximately 120 cyc/deg in the fovea (Figure 9A); as well as the orientation-reversal effect at an appropriate spatial frequency in the periphery (Figure 9B). Both results corroborate previous theoretical analysis and psychophysical measurements (Williams, 1985; Coletta and Williams, 1987), but now taking the trichromatic nature of the mosaic into account. On the other hand, with full optical aberrations, the reconstructed images became mostly uniform at these high spatial frequencies (Figure 9—figure supplement 1). Since our method accounts for trichromacy, we have also made the prediction that for achromatic grating stimuli viewed under similar diffraction-limited conditions, while the spatial aliasing pattern will be comparable, additional chromatic aliasing should be visible (Figure 9—figure supplement 2; also see Williams et al., 1991; Brainard, 2015).

Figure 9 with 2 supplements

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Figure 9—figure supplement 2

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Figure 9—figure supplement 1

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Contrast sensitivity function

Our framework can also be adapted to perform ideal observer analysis for psychophysical discrimination (threshold) tasks, which have been used previously to evaluate the information available in the initial encoding. Here, we use the reconstructed images as the basis for discrimination decisions. This is potentially important since even the early post-receptoral visual representation (e.g. retinal ganglion cells), on which downstream decisions must be based, is likely shaped by the regularities of our visual environment (Atick et al., 1992; Borghuis et al., 2008; Karklin and Simoncelli, 2011; Atick, 2011). Our method provides a way to extend ideal observer analysis to incorporate these statistical regularities.

Concretely, we predicted and compared the diffraction-limited spatial contrast sensitivity function (CSF) for gratings with a half-degree spatial extent (see Materials and methods). First, we applied the classic signal-known-exactly ideal observer to the Poisson distributed excitations of the simulated cone mosaic. We computed CSFs for both achromatic (L + M) and chromatic (L - M) grating modulations, with matched cone contrast measured as the vector length of the cone contrast vector. As expected, the ideal observer at the cone excitations produces nearly identical CSFs for the contrast-matched L + M and L - M modulations; also, as expected, these fall off with spatial frequency, primarily because of optical blur (Figure 10A).

Figure 10 with 1 supplement

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Figure 10—figure supplement 1

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Next, we reconstructed images from the cone excitations produced by the grating stimuli. A template-matching observer based on the noise-free reconstructions was then applied to the noisy reconstructions (see Materials and methods). The image-reconstruction observer shows significant interactions between spatial frequency and chromatic direction. Sensitivity in the L + M direction is relatively constant with spatial frequency. Sensitivity in the L – M direction starts out higher than L + M at low spatial frequencies, but drops significantly and is lower than L + M at high spatial frequencies (Figure 10B). We attribute these effects to the role of the image prior in the reconstructions, which leads to selective enhancement/attenuation of different image components. In support of this idea, we also found that an observer based on maximum likelihood reconstruction without the explicit prior term produced CSFs similar in shape to the Poisson ideal observer (Figure 10—figure supplement 1).

It is intriguing that the CSFs from the reconstruction-based observer show substantially higher sensitivity for L - M than for L + M modulations at low spatial frequencies (with equated RMS cone contrast), but with a more rapid falloff such that the sensitivity for L + M modulations is higher at high spatial frequencies. Both of these features are characteristic of the CSFs of human vision (Mullen, 1985; Anderson et al., 1991; Chaparro et al., 1993; Sekiguchi et al., 1993). A more comprehensive exploration of this effect and its potential interaction with other decision rule used in the calculation awaits future research.

We developed a Bayesian image reconstruction framework for characterizing the initial visual encoding, by combining an accurate image-computable forward model together with a sparse coding model of natural image statistics. Our method enables both quantification and visualization of information loss due to various factors in the initial encoding, and unifies the treatment of a diverse set of issues that have been studied in separate, albeit related, ways. In several cases, we were able to extend previous studies by eliminating simplifying assumptions (e.g. by the use of realistic, large cone mosaics that operate on high-dimensional, naturalistic image input). To summarize succinctly, we highlight here the following novel results and substantial extensions of previous findings: (1) When considering the allocation of different cone types on the human retina, we demonstrated the importance of the spatial and spectral correlation structure of the image prior; (2) As we examined reconstructions as a way to visualize information loss, we observed rich interactions in how the appearances of the reconstruction vary with mosaic sampling, physiological optics, and the SNR of the cone excitations; (3) We found that the reconstructions are consistent with empirical reports of retinal spatial aliasing obtained with interferometric stimuli, adding an explicit image prior component and extending consideration of the interleaved nature of the trichromatic retinal cone mosaic relative to the previous treatment of these phenomena; (4) We linked image reconstructions to spatio-chromatic contrast sensitivity functions by applying a computational observer for psychophysical discrimination to the reconstructions. Below, we provide an extended discussion of key findings, as well as of some interesting open questions and future directions.

First, we cast retinal mosaic design as a ‘likelihood design’ problem. We found that the large natural variations of L- and M-cone proportion, and the relatively stable but small S-cone proportion, can both be explained as an optimal design that minimizes the expected image reconstruction loss. This is closely related to an alternative formalism, often termed ‘efficient coding’, which seeks to maximize the amount of information transmission (Barlow, 1961; Karklin and Simoncelli, 2011; Wei and Stocker, 2015; Sims, 2018). In both cases, the optimization problem is subject to realistic biological constraints and incorporates natural scene statistics. Previous work (Garrigan et al., 2010) conducted a similar analysis with consideration of natural scene statistics, physiological optics, and cone spectral sensitivity, using an information maximization criterion. One advance enabled by our work is that we are able to fully simulate a 1 deg mosaic with naturalistic input, as opposed to the information-theoretical measures used by Garrigan et al., which became intractable as the size of the mosaic and the dimensionality of the input increased. In fact, Garrigan et al., 2010 approximated by estimating the exact mutual information for small mosaic size ($N=1\dots 6$ cones) and then extrapolated to larger cone mosaics using a scaling law (Borghuis et al., 2008). The fact that the two theories corroborate each other well is reassuring and suggests that the results are robust to the details of the analysis.

Our approach could be applied to analyzing the retinal mosaic characteristics of different animals. Adult zebrafish, for example, feature a highly regular mosaic with fixed 2:2:1:1 R:G:B:U cone ratios (Engström, 1960). Since our analysis has highlighted the importance of prior statistics in determining the optimal design, one might speculate whether this regularity results from the particular visual world of zebrafish (i.e. underwater, low signal-to-noise ratio), which perhaps demands a more balanced ratio of different cone types to achieve the maximum amount of information transmission. Further study that characterizes in detail the natural scene statistics of the zebrafish’s environment might help us to better understand this question (Zimmermann et al., 2018; Cai et al., 2020). It would also be interesting to incorporate into the formulation an explicit specification of how the goal of vision might vary across species. One extension to the current approach to incorporate this would be to specify an explicit loss function for each species and find the reconstruction that minimizes the expected (over the posterior of images) loss (Berger, 1985), although implementing this approach would be computationally challenging. Related is the task-specific accuracy maximization analysis formulation (Burge and Geisler, 2011; see Burge, 2020 for a review).

Second, we applied our framework to cone excitations of retinal mosaics with varying degrees of optical quality, photoreceptor size, density, and cone spectral sensitivity. The reconstructed images reflect accurately the information loss in the initial encoding, including spatial blur due to optical aberration and mosaic sampling, pixel noise due to Poisson variability in the cone excitations, and reduction of chromatic contrast in anomalous trichromacy. Although we have mainly focused on visualization of these effects in our current paper, it would be possible to perform quantitative analyses. In fact, our reconstruction algorithm could provide a natural ‘front-end’ extension to many image-based perceptual quality metrics, such as spatial CIELAB (Zhang and Wandell, 1997; Lian, 2020), structural similarity (Wang et al., 2004), low-level feature similarity (FSIM; Zhang et al., 2011), or neural network-based approaches (Bosse et al., 2018). Doing so would incorporate factors related to the initial visual encoding explicitly into the resulting image quality metrics.

In addition, when SNR is high, we found that we are able to fully recover color information even from an anomalous trichromatic mosaic. As SNR drops, this becomes less feasible. Although our analysis does underestimate the amount of total noise in the visual system (i.e. we only consider noise at cone excitations, but see Ala-Laurila et al., 2011 for a detailed treatment of noise in the retina), this nonetheless suggests that a downstream circuit that properly compensates for the shift in cone spectral sensitivity can, in principle, maintain relatively normal color perception in the low noise regime (Tregillus et al., 2021). This may potentially be related to some reports of less than expected difference in color perception between anomalous trichromats and color normal observers (Bosten, 2019; Lindsey et al., 2020).

Third, we speculate that image reconstruction could provide a reasonable proxy for modeling percepts in various psychophysical experiments. We found that images reconstructed from dichromatic mosaics resemble results generated by previously proposed methods for visualizing dichromacy, including one that uses explicit knowledge of dichromatic subjects’ color appearance reports (Brettel et al., 1997). We have also reproduced the ‘zebra stripes’ and ‘orientation reversal’ aliasing patterns when reconstructing images from cone excitations to spatial frequencies above the mosaic Nyquist limit, similar to what has been documented experimentally in human subjects (Williams, 1985; Coletta and Williams, 1987). In a similar vein, previous work has used a simpler image reconstruction method to model the color appearance of small spots light stimulus presented to single cones using adaptive optics (Brainard et al., 2008). Our method could also be applied to such questions, and also to a wider range of adaptive optics (AO) experiments (e.g. Schmidt et al., 2019; Neitz et al., 2020), to help understand the extent to which image reconstruction can capture perceptual behavior. More speculatively, it may be possible to use calculations performed within the image reconstruction framework to synthesize stimuli that will maximally discriminate between different hypothesis about how the excitations of sets of cones are combined to form percepts, particularly with the emergence of technology that enables precise experimental control over the stimulation of individual cones in human subjects (Harmening et al., 2014; Sabesan et al., 2016; Schmidt et al., 2019).

Last, we showed that our method can be used in conjunction with analysis of psychophysical discrimination performance, bringing to this analysis the role of statistical regularities of natural images. In our initial exploration, we found that the image-reconstruction based observer exhibits significant interaction between spatial frequency and chromatic direction in its contrast sensitivity function, a behavior distinct from its Poisson ideal observer counterpart, and is more similar to the human observer. Future computations will be needed to understand in more detail whether the reconstruction approach can account for other features of human psychophysical discrimination performance that are not readily explained by ideal-observer calculations applied to the cone excitations.

Our current model only considers the representation up to and including the excitations of the cone mosaic. Post-excitation factors (e.g. retinal ganglion cells), especially in the peripheral visual field, are likely to lead to additional information loss. In this regard, we are eager to incorporate realistic models of retinal ganglion cells into the ISETBio pipeline. Nevertheless, the value of the analysis we have presented is to elucidate exactly what phenomena can or cannot be attributed to factors up to the cone excitations, thus helping to dissect the role of different stages of processing in determining behavior. For example, we found there is desaturation of chromatic content in reconstructed images in the periphery, with the details depending on interactions between the physiological optics, cone mosaic sampling, macular pigment density, and the model of natural image statistics. This is in contrast to more traditional explanations of the decrease in peripheral chromatic sensitivity, which often consider it in the context of models of how different cone types are wired to retinal ganglion cells (e.g. Lennie et al., 1991; Mullen and Kingdom, 1996; Hansen et al., 2009; Field et al., 2010; Wool et al., 2018). Whether the early vision factors are sufficient to account for the full variation in chromatic sensitivity awaits a more detailed future study, but the fact that early vision factors can play a role through their effect on the available chromatic information is a novel insight that should be incorporated into thinking about the role of post-excitation mechanisms.

More generally, we can consider the locus of the signals analyzed in the context of the encoding-decoding dichotomy of sensory perception (Stocker and Simoncelli, 2006; Rust and Stocker, 2010). Here, we reconstruct images from cone excitations, thus post-excitation processing may be viewed as part of the brain’s implementation of the reconstruction algorithm. When we apply such an algorithm to, for example, the output of retinal ganglion cells, we shift the division. Our view is that analyses at multiple stages are of interest, and eventual comparisons between them are likely to shed light on the role of each stage.

Our current model also does not take into account fixational eye movements, which displace the retinal image at a time scale shorter than the integration period we have used here (Martinez-Conde et al., 2004; Burak et al., 2010). It has been shown that these small eye movements can increase psychophysically-measured visual acuity relative to that obtained with retinally-stabilized stimuli (Rucci et al., 2007; Ratnam et al., 2017). An intuition behind this is that fixational eye movements can increase the effective cone sampling density, if the visual system can sensibly combine information obtained across multiple fixation locations. This intuition is supported by computational analyses that integrate information across fixations while simultaneously estimating the eye movement path (Burak et al., 2010; Anderson et al., 2020). In their analysis, Burak et al., 2010 showed the effectiveness of their algorithm depended both on the integration time of the sensory units whose excitations were processed, and also on the receptive field properties of those units. In addition, consideration of the effects of fixational eye movement might also benefit from an accurate model of the temporal integration that occurs within each cone, as a consequence of the temporal dynamics of the phototransduction cascade (Angueyra and Rieke, 2013). ISETBio in its current form implements a model of the phototransduction cascade as well as of fixational eye movements (see Cottaris et al., 2020). Future work should be able to extend our current results through the study of dynamic reconstruction algorithms within ISETBio.

Since our framework is centered on image reconstruction, one may naturally wonder whether we should have applied the more ‘modern’ technique of convolutional neural networks (CNNs), which have become the standard for image processing-related tasks (Krizhevsky et al., 2012). For our scientific purposes, the Bayesian framework offers an important advantage in its modularity, namely, the likelihood and prior are two separate components that can be built independently. This allows us to easily isolate and manipulate one of them (e.g. likelihood) while holding the other constant (e.g. prior), something we have done throughout this paper. In addition, building the likelihood function (i.e. render matrix $R$, see Materials and methods) is a forward process that is computationally very efficient. Performing a similar analysis with the neural network approach (or supervised learning in general) would require re-training of the network with a newly generated dataset (i.e. cone excitations paired with the corresponding images) for every condition in our analysis.

However, the ability of neural networks to represent more complex natural image priors (Ulyanov et al., 2018; Kadkhodaie and Simoncelli, 2021) is of great interest. Currently, we have chosen a rather simple, parametric description of natural image statistics, which leads to a numerical MAP solution. Previous work has proposed methods that alternate, within each iteration, between regularized reconstruction and denoising, which effectively allow for transfer of the prior implicit in an image denoiser (e.g. a deep neural network denoiser) to be applied to any other domain with a known likelihood model (Venkatakrishnan et al., 2013; Romano et al., 2017). More recently, Kadkhodaie and Simoncelli, 2021 developed a related but more explicit and direct technique to extract the image prior (a close approximation to the gradient of the log-prior density, to be precise) from a denoising deep neural network, which could be applied to our image reconstruction problem. We think this represents a promising direction, and in the future plan to incorporate more sophisticated priors, to evaluate the robustness of our conclusions to variations and improvements in the image prior.

To conclude, we believe our method is widely applicable to many experiments (e.g. adaptive optics psychophysics) designed for studying the initial visual encoding, for modeling the effect of changes of various components in the encoding process (e.g. in clinical conditions), and for practical applications (e.g. perceptual quality metric) in which the initial visual encoding plays an important role.

The problem of reconstructing images from neural signals can be considered in the general framework of estimating a signal $x$, given an (often lower-dimensional and noisy) measurement $m$. We take a Bayesian approach. Specifically, we model the generative process of measurement as the conditional probability $p\left(m\right|x)$ and the prior distribution of the signal as the probability density $p\left(x\right)$. We then take the estimate of the signal, ${\displaystyle \hat{x}}$, as the maximum a posteriori estimate ${\displaystyle argmaxp\left(m|\hat{x}\right)p\left(\hat{x}\right)}$. We next explain in detail how each part of the Bayesian estimate is constructed.

The MATLAB code used for this paper is available at: https://github.com/isetbio/ISETImagePipeline, (copy archieved at swh:1:rev:72e7296dcaf8ebdcca35776d7a98026c8f041427). In addition, the curated RGB and hyperspectral image datasets, parameters used in the simulation including display and cone mosaic setup, as well as the intermediate results such as the learned sparse priors, likelihood functions (i.e., render matrices), are available through: https://tinyurl.com/26r92c8y.

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Author details

1. Ling-Qi Zhang

   Department of Psychology, University of Pennsylvania, Philadelphia, United States

   Contribution

   Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review and editing

   For correspondence

   lingqiz@sas.upenn.edu

   Competing interests

   Funding provided by Facebook Reality Labs

   "This ORCID iD identifies the author of this article:" 0000-0001-8468-7927

2. Nicolas P Cottaris

   Department of Psychology, University of Pennsylvania, Philadelphia, United States

   Contribution

   Data curation, Methodology, Software, Validation, Visualization, Writing – review and editing

   Competing interests

   Funding provided by Facebook Reality Labs

   "This ORCID iD identifies the author of this article:" 0000-0003-1829-6340

3. David H Brainard

   Department of Psychology, University of Pennsylvania, Philadelphia, United States

   Contribution

   Conceptualization, Funding acquisition, Methodology, Project administration, Resources, Supervision, Validation, Writing – review and editing

   Competing interests

   Funding provided by Facebook Reality Labs

   "This ORCID iD identifies the author of this article:" 0000-0001-9827-543X

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