Abstract
Visual clutter affects our ability to see. Objects that would be identifiable on their own may become unrecognizable when presented close together (“crowding”), but the psychophysical characteristics of crowding have resisted simplification. Image properties initially thought to produce crowding have paradoxically yielded unexpected results; for example, adding flanking objects can ameliorate crowding (Manassi, Sayim, & Herzog, 2012; Herzog, Sayim, Chcherov, & Manassi, 2015; Pachai, Doerig, & Herzog, 2016). The resulting theory revisions have been sufficiently complex and specialized as to make it difficult to discern what principles may underlie the observed phenomena. Here, a generalized formulation of simple visual contrast energy is presented, arising from straightforward analyses of center and surround neurons in the early visual stream. Extant contrast measures, such as root mean square contrast, are easily shown to fall out as reduced special cases. The new generalized contrast energy metric surprisingly predicts the principal findings of a broad range of crowding studies. These early crowding phenomena may thus be said to arise predominantly from contrast or are, at least, severely confounded by contrast effects. Note that these findings may be distinct from accounts of other, likely downstream, “configural” or “semantic” instances of crowding, suggesting at least two separate forms of crowding that may resist unification. The new fundamental contrast energy formulation provides a candidate explanatory framework that addresses multiple psychophysical phenomena beyond crowding.
An oft-cited view is that crowding arises from “critical spacing, independent of spatial frequency” (
Pelli, 2008) and specifically that “contrast” does not suffice as an explanatory mechanism (
Flom, Weymouth, & Kahneman, 1963;
Chung, Levi, & Legge, 2001;
Levi, Hariharan, & Klein, 2002); for a partial review, see
Strasburger, Rentschler, & Jüttner, 2011). Multiple contrast measures exist in the literature, such as Weber, Michelson, and root mean square (RMS) contrast (
Peli, 1990); correspondingly, “contrast energy” is typically defined as the integral of the square of the contrast over all dimensions in which it varies (e.g.,
Watson, Barlow, & Robson, 1983;
Kukkonen, Rovamo, Tiippana, & Nasanen, 1993).
Most such measures treat pixels as being independent of each other. We proffer a new, radially generalized account of contrast energy (of which RMS contrast and others are shown to be special cases). The new account, rather than evaluating contrast pixel by pixel in an image, instead formally evaluates radial regions corresponding to receptive fields, within which pixels may have interacting (rather than independent) effects, as viewed by a perceiver.
This new contrast measure arises from work unrelated to crowding. Studies of the visual dissimilarity between two similar images (such as an image and a degraded or compressed version of the image) led to the derivation and analysis of the primary new contrast measure that is also introduced here in
Equation 4, below (
Bowen, Rodriguez, Sowinski, & Granger, 2020).
The generalized contrast measure is shown to be specifically predictive of the essential results of several well-studied crowding effects from the literature. To reproduce those published results, the sole two steps are (a) measuring contrast energy and (b) mapping it to behavior (the subject's identification of the flanked target). The sole parameter simply maps contrast quantities directly onto behavioral performance by estimating the threshold at which the contrast has changed enough to begin generating identification errors.
The resulting straightforward measures surprisingly account for multiple instances of crowding across the published literature, including some exemplars that have thus far been resistant to simplification.
At minimum, this is evidence that many standard crowding effects are severely confounded by variations in the introduced contrast measure. We specifically propose that a substantial number of results attributed to crowding actually arise directly from contrast.
We also provide examples of crowding that are not predicted by contrast. Because there are many clear instances where crowding is predicted by contrast and instances where it is not, we suggest that attempts to unify crowding to a single phenomenon would presumably have to account for these instances in which contrast alone is explanatory.
Contrast-dependent crowding effects may arise extremely early in the visual stream. Other evidence indicates that certain other crowding effects may arise from later processing; our findings suggest that experiments may be profitably divided into at least two possibly distinct categories: those that are explained by contrast and those that are not. This may indicate that crowding phenomena are not all due to a unitary mechanism. The findings may also help determine which apparent crowding effects are precortical versus cortically dependent. In addition, the findings may aid in separating attributes of model neural architectures into characteristics that are needed for a particular effect (such as crowding) versus additional neural features that may not be required to explain these phenomena.
Foregrounds, backgrounds, and differences: From early anatomy to early perception
In the following published crowding experiments, subjects were presented with an image in which each object is designated by the experimenter as either target or distractor, and subjects were instructed to identify the target, with or without the presence of distractors. We test the hypothesis that, for these experiments, high values of the contrast energy metric predict improved target identification performance in the presence of the distractor, whereas low contrast energy predicts worse target identification performance. The latter is what is referred to in the experiments as a crowding effect.
In general, formal expressions that predict behavior contain at least one experimental parameter that corresponds to the mapping of a psychophysical calculation to explicit behavioral measures. Many such experiments contain multiple parameters, each of which is in some way fitted to observed data. In the present model, we wish to identify as closely as possible those response accuracies that can be said to arise directly from the contrast metric; thus, we wish to map the (internal) contrast calculation to the (overt) behavioral measure of correct target identification, via as few parameters as possible. It should be noted that this will have the effect of (a) directly implicating the contrast metric in the observed behavior (while avoiding overfitting), and (b) possibly demonstrating some behaviors that are not predicted by contrast but presumably by other psychophysical operations outside the scope of the present study. We will show instances of both behaviors that are, and behaviors that are not, accounted for by contrast, with the aim of separately characterizing these distinct psychophysical behaviors.
We thus introduce a single parameter, Eα, the contrast energy value at which a subject's ability to identify the target among distractors begins to become impeded by the flanking distractors in an image. This is the point at which a given experiment exhibits the subject's sensitivity to the effects of the distractors on the target. We construct a Gaussian that maps contrast to proportion of correct subject response to the target alone, and a Gaussian for when the target is presented with flanking distractors.
In isolation, a target has a given measurable contrast value, μτ. We assume that a 10% change to that target contrast value will yield a recognition error rate of 0.01 (i.e., 1% misidentifications of the altered target). More empirically fitted figures would normally be arrived at from experimental findings. As emphasized below, we do not consider any features or configurations of images whatsoever, nor are we attempting to match exact performance. We simply assume a tight range of contrast around which the target is identifiable, and that identification errors begin to arise at 10% contrast change. These straightforward simplifying assumptions are highlighted in order to spotlight the surprising ability of these simple contrast metrics to predict certain crowding findings.
These values are used to calculate a target-alone Gaussian distribution,
Gτ, with a mean of µ
τ and a standard deviation (
SD) of σ
τ (defined below), such that the value of the distribution drops to 1% at a point that is at either 0.9μ
τ or 1.1μ
τ (i.e., 10% from μ
τ) (
Figure 4, green Gaussian). These assumptions for
Gτ are fixed parameters that do not depend on the experiment or on any curve fitting.
\begin{equation*}{G_\tau }(x) = {e^{ - {{(x - {\mu _\tau })}^2}/2\sigma _\tau ^2}}\end{equation*}
For
x = 0.9μ
τ or for
x = 1.1μ
τ, then
\({G_\tau }(1.1{\mu _\tau }) = {e^{ - {{(1.1{\mu _\tau } - {\mu _\tau })}^2}/2\sigma _\tau ^2}}\).
For the conditions when the target is presented with flanking distractors, a target-plus-flankers Gaussian distribution is calculated,
Gϕ (see
Figure 4a), which shares the mean μ
τ and has a
SD of σ
ϕ such that the value of the Gaussian drops to 1% at whatever point in the experiment that a flanker elicits no crowding. In other words, we assume that, for that particular stimulus image, that the target will be identified 99% of the time.
\begin{equation*}{G_\phi }(x) = {e^{ - {{(x - {\mu _\tau })}^2}/2k_\phi ^2\sigma _\tau ^2}}\end{equation*}
where
kϕ = σ
ϕ/σ
τ. That point, again, constitutes the single parameter in the model that arises from the experiment itself: the contrast value
Eα, at which a subject's ability to identify the target first begins to be impeded by distractors (see
Figure 4a).
\begin{equation*}{G_\phi }({E_\alpha }) = {e^{ - {{({E_\alpha } - {\mu _\tau })}^2}/2k_\phi ^2\sigma _\tau ^2}}\end{equation*}
To compute the value of σ
τ, the SD of the Gaussian
Gτ, we recall the assumption that a 10% change to the target will correspond to 1% successful identification of the target, and we solve for σ
τ:
\begin{equation*}\begin{array}{@{}l@{}} {G_\tau }(1.1{\mu _\tau }) = {e^{ - {{(1.1{\mu _\tau } - {\mu _\tau })}^2}/2\sigma _\tau ^2}}\\ 0.01 = {e^{ - {{(1.1{\mu _\tau } - {\mu _\tau })}^2}/2\sigma _\tau ^2}}\\ \ln (0.01) = - {(1.1{\mu _\tau } - {\mu _\tau })^2}/2\sigma _\tau ^2\\ \sigma _\tau ^2 = - {(1.1{\mu _\tau } - {\mu _\tau })^2}/2\ln (0.01) \end{array}\end{equation*}
We define a constant
h = 2ln(0.01) (
h < 0); then,
\begin{equation*}\sigma _\tau ^{} = {\left[ { - {{(1.1{\mu _\tau } - {\mu _\tau })}^2}/h} \right]^{1/2}}\end{equation*}
For σ
ϕ, similarly,
\begin{equation*}\begin{array}{@{}l@{}} {G_\phi }({E_\alpha }) = {e^{ - {{({E_\alpha } - {\mu _\tau })}^2}/2\sigma _\phi ^2}}\\ 0.01 = {e^{ - {{({E_\alpha } - {\mu _\tau })}^2}/2\sigma _\phi ^2}}\\ \ln (0.01) = - {({E_\alpha } - {\mu _\tau })^2}/2\sigma _\phi ^2\\ \sigma _\phi ^2 = - {({E_\alpha } - {\mu _\tau })^2}/h\\ \sigma _\phi ^{} = {\left[ { - {{({E_\alpha } - {\mu _\tau })}^2}/h} \right]^{1/2}} \end{array}\end{equation*}
Having computed σ
τ and σ
ϕ, we can express σ
ϕ in terms of multiples of σ
τ by defining
kϕ such that
kϕ = σ
ϕ/σ
τ. The values used for each of these parameters, for each of the experiments analyzed, appear later in
Table 1.
In total, the predicted proportion correct (target-alone and target-plus-flanker trials) is calculated from examination of the table in
Figure 4b:
\begin{equation}p = {G_\tau } + (1 - {G_\phi })\end{equation}
It is worth noting that the Gaussians used for purposes of estimating subject response are not related to the Gaussians that comprise the center-surround organization in the early visual pathway that are used to derive the generalized contrast metric.
In sum, low values of this formula predict a low percentage of target identifications by the subject; high values predict a higher proportion of identification success. In the experiments reported here, all values are scaled from percentage of 0% to a maximum percentage correct of 85%; that is, all results (p) are multiplied by 0.85. This arises solely from the data that appear throughout the cited experiments from the literature; those experiments are apparently calibrated such that subjects tend never to achieve 100% correct recognition rates, but, rather, their ceiling occurs at roughly 85% empirically. These results may arise from other factors such as resolution, distance, or brightness, all of which contribute to the ultimate ability of the subject to perform the task correctly; the experimenters may have calibrated the tasks so as to avoid ceiling effects. Empirically, in the experiments, the 85% ceiling is thus not related to crowding effects per se but rather is the best that the subjects can do when the targets are not crowded. Our model thus is simply calibrated to that empirical ceiling from the literature (as with typical parameters of other models), in order to compare model results to subject results.
We provide this somewhat extensive derivation of
Equation 5 to make it clear that the only experiment-derived parameter that appears anywhere in the calculations is that of
Eα and that
Equation 5 is then derived according to usual principles of classification using a contingency table as shown in
Figure 4b.
Equation 5 then straightforwardly maps the contrast energy of an image to the proportion of a subject's correct responses. Note that the full code for computing all of the calculations in this article and reproducing all of the material for the figures is available on github (
https://github.com/DartmouthGrangerLab/Contrast/).
The relevance of these estimations of subjects’ response accuracies can be intuitively understood by noting that recognizing is not simply perceiving; rather, it is matching the perceived entity against some stored version. That is the difference, for example, between perceiving that there are pixels present versus recognizing that they take a form that has previously been seen by the subject. Every crowding experiment presented here (possibly all such experiments in general) rests on the presupposition that the subject can identify whether or not the seen target entity is the “same” as some previously seen entity, whether that was long-ago acquired (e.g., a typed alphabetic letter) or indicated to the subject in the instructions (e.g., target Landolt C angle vs. flanking Landolt C angle).
Within a given task, a specific target is associated with a specific value of its measured contrast energy. If the target were presented alone, then potential targets that deviate from the target value of the task would be more difficult to identify, thus impairing performance whether the contrast value is increased or decreased (matching a target alone does not arise in the experiments modeled here). When target plus flankers are presented, the subject must identify which pixels in the image represent the instructed target and report on its characteristics (e.g., its name, “r,” or its gap angle, “90 degrees”). The more distant the overall contrast energy is from the target-alone contrast energy, the easier it is for the subject to distinguish the target within the interfering flankers. The closer the flankers are to the target, the closer the contrast energy of the overall image is, causing reduced predicted correct response rates.
For a given image from each of the experiments in the following section, the eccentricities of the pixels in the relevant peripheral region were calculated given the reported details of the experimental setup. The retinal coordinates for the corresponding image pixels were computed using the viewing distance and screen resolution used in the specified experiment. Midget cell diameters (
Dacey & Petersen, 1992) for those retinal coordinates were collected with a 17.2° width around the region of interest.
We again emphasize that the present work entails no analyses of any detailed feature or configuration characteristics of any kind, such as shapes or orientation. The sole quantity tested is the newly introduced contrast metric. Thus, no specific experimental results from the crowding literature are addressed with respect to description of a target object, its orientation, or other features. Rather, each treatment of an experiment simply proceeds by first computing the contrast energy for the images used (following the methods from the previous sections) and then using the mapping of
Equation 5 (
Figure 4) to calculate the efficacy with which the computed contrast energy can determine which visual entities constitute the target and which do not.
This procedure would thus seem to be utterly insufficient to capture crowding findings, which, after all, appear to entail subjects’ identifying detailed features of an image (such as the orientation angle of a Landolt C). It is thus illuminating to show that contrast accurately predicts the accuracies of subjects’ responses, despite the fact that contrast has no information whatsoever about orientation or other configural attributes of the image. This suggests at minimum a substantial role for contrast in these reported findings in the literature.
For each experiment below, both target-alone and target-plus-flanker images are viewed, and contrast is computed for each. The figures show (a) experimental stimuli, (b) calculation of contrast energy from flanker distance, (c) calculation of predicted identification accuracy from contrast energy, and (d) by combining (b) and (c), calculation of predicted information accuracy from flanker distance, which is then compared alongside corresponding measures from the reported literature.
Again, lacking any features, orientations, or other attributes of the images, the model produces no specific image characteristics; instead, it determines the calculated contrast energy of the image under the given viewing conditions and determines, solely from this value whether the image will be recognized.
It is further noted that, in addition to the direct predictability of the experimental results from the new contrast metric, several of the experiments are predictable even with the simpler standard RMS contrast quantity. For example, the experimental materials in
Pachai, Doerig, and Herzog (2016) create flanking Landolt Cs that surround a central C; as those surrounds grow larger, they add more pixels to the image, which increases generalized contrast and also increases simple RMS contrast in the image (
Table 2, below). Controlling for these factors is required to separate contrast-dependent from contrast-independent crowding effects.
Table 2. Instances of confounds and predictions in the analyzed experiments.
Table 2. Instances of confounds and predictions in the analyzed experiments.
The following sections detail the findings of specific instances of crowding studies. These provide a simple range of basic crowding results illustrating the dependency of the effect on flank distance and eccentricity across a span of visual images with different features. What is seen is that contrast energy (and thus contrast) alone is highly predictive of the recognizability of the crowded target objects.
Study by Flom et al. (1963)
Study by Manassi et al. (2012)
Studies by Harrison and Bex (2015), Harrison and Bex (2016), and Pachai et al. (2016)
The target image is an oriented Landolt C, as in
Flom et al. (1963), now with concentric surrounding flanker Landolt Cs with differing radii, differing gap orientations, and sometimes no gaps (
Harrison & Bex, 2015;
Harrison & Bex, 2016;
Pachai et al., 2016). The task is to report the angle of the target C in the face of sometimes conflicting angles from flanking Cs. As emphasized, the contrast metric computed here yields no information whatsoever about most visual features, such as shapes, gaps, and angles. The computations solely indicate the value of contrast energy (as calculated via
Equation 4), independent of feature configurations. Yet, as has been seen in previous examples, those contrast calculations repeatedly generate predictions of recognizability of images, despite doing so in the absence of the features of the images themselves, and we so far have seen several cases in which those predictions match empirical findings (
Figures 5,
6, and
7). This again suggests the possibility that extremely simple visual characteristics may be responsible for differences in reported image recognition errors in those experiments.
The reported empirical results shown in
Figure 8e indicate that a flanker containing its own gap causes more interference with correct target angle recognition than a flanker with no gap (a concentric circle). More intriguingly, the researchers found, in agreement with the findings of
Manassi et al. (2012), that, whereas recognition of a target is impeded by a flanking object, that interference paradoxically is lessened, not increased, by adding further flankers.
Note that these flankers have a characteristic not present in the flankers from other experiments. When an encircling flanker is moved farther from the target, the flanker becomes larger and thus contains more pixels. This increases the contrast energy of the overall image, and increased contrast energy predicts improved target recognition in the presence of crowding flankers; this potential confound may affect subjects’ success rates.
As with the previous analyses, the angle of the target is not calculated or reported here. What is predicted by the contrast calculation is the rate of failure of subjects to correctly report this feature (gap angle) of the target (or, possibly, any specific feature present in the target). As the authors of the studies show, empirically either the subjects recognized the target C and responded correctly or they failed to distinguish the target C, in which case they reliably reported instead the angle of the confounding flanker. These findings are consistent with the mapping process reported here (
Equation 5), in which the formula distinguishes between contrast characteristics of the target alone versus contrast characteristics of other image constituents.
The contrast predictions (
Figure 8d) somewhat predict the direction and shape of the reported empirical findings from the published article (reproduced in
Figure 8e), although the quantities clearly differ.
Figure 8e also shows a good fit to the empirical findings of a model introduced by the authors. That model is further tested in
Figure 8f, which shows the effects of different numbers of flankers on human and model recognition. We reproduce their results together with the predictions made by the contrast energy metric. Human recognition (light blue) empirically exhibits lower error (less crowding) for zero and for five flankers than for a single flanker; the bars show mean and
SD, and the individual results of the four subjects are shown as individual small symbols.
Pachai et al. (2016) showed that the model introduced by
Harrison and Bex (2015) erroneously predicts that crowding will be roughly the same whether there is a single flanker or five flankers (red bars). The contrast metric generates predictions that are more in line with empirical findings, in that five flankers elicit less crowding than a single flanker (orange). This is simply because the added flankers substantially increase the generalized contrast of the overall image to be viewed by the subject, and higher contrast predicts better recognition.
Once again, it is not clear whether this seemingly too simple explanation is accurately reflecting the vision mechanisms of the viewing subject, but it clearly shows that contrast alone does match these particular aspects of the empirical findings. Contrast is an explanation at best, a confound of the experiment at worst.
We emphasize the unexpected nature of these findings. The contrast energy model presented here intentionally omits any information about object shapes, angles, and gaps. This, of course, should be expected to prevent the model from predicting the experimental findings, as the features that are specifically being measured in those experiments (such as Landolt C gap angle) are being entirely ignored in the contrast energy model.
Yet, what we find is that the simple contrast energy model produces predicted response accuracies that appear to qualitatively correspond with subjects’ response accuracies, even though the model is clearly not, and cannot be, carrying out the task that the subjects are purportedly accomplishing. Rather, contrast energy alone predicts reduced response accuracies in a way that tracks subjects’ response accuracies, suggesting that either the human experimental results may actually be due to something other than they are intended to or at least they may be confounded by these contrast measures.
As seen in
Figure 8f, response accuracies are reduced (errors increased) by the addition of one flanker to the Landolt C, but those reduced accuracies are ameliorated by the addition instead of five (rather than one) flankers. The leftmost graph shows this for human subjects (
Pachai et al., 2016). The next graph shows that use of the model of
Harrison and Bex (2015) by Pachai et al. does not predict this non-monotonic effect; the Harrison and Bex model predicts that five flankers will yield roughly as many errors as one flanker. Both of those graphs exhibit “perceptual error” on the
y-axis in the form of the
SD of a von Mises function fit to the distribution of errors across trials, corresponding to subjects’ reports of the orientation of the target Landolt C gap (
Harrison & Bex, 2015). The rightmost graph, however, exhibits no reference to the C gap orientation, but rather reports the percent error that is predicted solely from the calculation of contrast energy values in the visual materials of each of the experiments. Yet, these measures appear to qualitatively correspond to the human measures, in that five flankers produce less crowding interference than one flanker.
Many features may cause crowding, but the present model proceeds from the prediction that contrast will itself affect response accuracy, independent of other features. We are proposing the possibility that this is a confound that may affect the interpretation of the identified results; that is, it is not yet ruled out that the magnitude of the angle error differences may be due wholly or in part to contrast energy differences, which change with the flanker features in these specific experiments.
The generalized contrast calculation introduced here is calculated from measures of dendritic radii in midget cells in the retina; corresponding center-surround phenomena also occur in the thalamus and cortex. Further experiments will pursue the question of which of the many Gaussian and/or center-surround operations along visual pathways may participate in these metric calculations.
There are other formulations of contrast that may play a similar role. Contrast response functions from the literature combine a measure of psychophysical contrast plus a behavioral mapping function; one such measure of center–surround interactions calculated the contrast values of a target and a surround (
CS,
Ct) and then fitted a response model:
\begin{equation*}{R_t} = {\textstyle{{k(1 + {W_e}C_s^{pe})C_t^p} \over {(1 + aC_t^q + {W_i}C_s^{qi})}}}\end{equation*}
by estimating seven parameters:
p,
q,
a,
We,
Wi,
pe, and
qi (
Xing & Heeger, 2001). Such models may, after suitable fitting, also account for crowding data of the kind addressed here. It will be of interest to pursue these possibilities in further studies.
The examples in the present paper arise from simple center–surround interactions at apparently relatively early stages of the visual stream. Higher level configural effects that also appear in the crowding literature may arise from completely different sources than contrast, or possibly could arise from combinations of successive center–surround operations, compositing these into more complex organizations. It may prove possible to distinguish early contrast-dependent effects from other more downstream crowding effects; if so, this perhaps may enable the nomenclature of crowding to be revised to reflect such a distinction.
Recognizing is not simply perceiving; it is, further, matching the perceived entity (e.g., a letter, a target C) against some specific stored memory or template from the experiment's instructions. Subjects still perceive the existence of, say, a crowded “r,” but they fail to identify what letter those perceived pixels connote; that is, they fail to match the perceived image against some predetermined knowledge of an “r” versus an “n” or “h.” Small receptive fields enable foveal pixels to be processed within minute regions, retaining the relative positions of different parts of an image (e.g., an “r”) and maintaining separate processing of the target versus neighboring pixels from flankers.
As receptive field size increases with eccentricity, acuity is reduced. That peripheral acuity is still quite sufficient to recognize a letter in isolation. What fails is that the subject fails to recognize the letter, even though its pixels are perceived. In peripheral larger RFs, pixels within a given RF are more likely to be processed as part of a single entity, rather than separately as they would be in smaller RFs. The boundaries between pixels of the target versus flanker are lessened; thus, the target is still perceived, but its identity may be obscured by interactions among pixels within too-large RFs (
Figures 5b and
6b). The radially generalized contrast energy introduced here essentially predicts simply that recognition is assisted by large contrast differences. This predicts that crowding should continue to occur for any RF size, even foveal, as long as the closeness of the flankers is scaled according to RF size. It is worth noting that evidence for this has been provided in the literature; see, for example,
Coates et al. (2018).
If early crowding were actively preventing recognition of target objects, then information would be lost and presumably could not be available for further (downstream) processing; yet, some experiments appear to show just such downstream availability (e.g.,
Manassi & Whitney, 2018). The results presented here suggest that information about an object is not “lost”; it simply is one type of information (contrast) that can become input to a (downstream) behavioral mapping process. When the data are queried, the information may be insufficient to answer certain questions yet may still provide an otherwise unimpeded stream of feature information that is available to other decision queries that may occur. These operations, too, are of interest as topics of further study.
Of related interest is recent work showing that standard feedforward convolutional neural networks (ffcnns) are in principle incapable of producing global shape computations that are shown to be used in human visual processing; some alternative models to ffcnns that avoid the ffcnn pitfalls are in extended development (
Doerig, Bornet, Rosenholtz, Francis, Clarke, & Herzog, 2019;
Doerig et al., 2020).
The authors offer their sincere gratitude for assistance with formal notation from Damian Sowinski and Eli Bowen, and their thanks to Eli Bowen, Eva Childers, and Annie Brown for advice and contributions to ongoing experimental work on visual psychophysics.
Supported in part by Grants from the Office of Naval Research (N00014-16-1-2359, N00014-19-1-2434).
Commercial relationships: none.
Corresponding author: Richard Granger.
Email: richard.granger@gmail.com.
Address: Department of Psychological and Brain Sciences, Dartmouth College, Hanover, NH, USA.