This relationship means that for an orientation in which the average extension of real-world surfaces is relatively less—which increases the empirical rank of a given line length and therefore the apparent length of the line see above —the density of luminance contrast transitions would be relatively high.

Thus, the reason for the correlation between perceived line length and the density of luminance contrast transitions is straightforward; by the same token, however, variation of the density of zero-crossings is not the cause of the apparent variation in line length with orientation.

SUMMARY The observed variation in the apparent length of lines as a function of their projected orientation agrees remarkably well with the percentile ranks of lines on the relevant empirical scales of line length derived from the probability distributions of the physical sources of line projections. Thus, this otherwise puzzling peculiarity about one of the simplest aspects of perceived geometry can be neatly explained as a particular manifestation of a broader visual strategy that generates percepts according to the probability distributions of the possible sources of inherently ambiguous stimuli.

Chapter 4 Angles A second fundamental aspect of the physical arrangement of objects in space and the perceptions of this geometry is the angle made between two lines that meet—either explicitly or implicitly—at a point. Like the apparent length of lines, an intuitive expectation about the perception of angle subtense is that such a basic feature of what we see should scale with the dimensions of the angles projected in retinal images.

It has long been known, however, that this is not what people see. What follows is a review of the evidence that the peculiar way we see angles is, in fact, a further manifestation of the way the visual system contends with the inverse optics problem. Even in the absence of the rich contextual information that biases the probable sources of the projected angles in this obvious way, observers tend to overestimate the magnitude of acute angles and underestimate obtuse ones by a few degrees Figure 4.

In fact, each of the objects subtends exactly the same right angle in the image see inset. The several parallel vertical lines in this presentation appear to be tilted away from each other, again in directions opposite the oblique orientation of the contextual line segments. A further well known permutation of such effects is the Hering illusion mentioned earlier see Figure 1.

## An Empirical Explanation of the Speed-Distance Effect

The issue considered here is whether the statistical relationship between images and sources in natural scenes can also rationalize the perception of angle subtense, as well as more complex effects elicited by angular stimuli. The evidence presented below indicates that the perception of angles and all the various effects shown in Figure 4. Perceiving angles in this way allows observers to contend with the inevitable ambiguity of projected angles.

A Psychophysical results Nundy et al. B The tilt illusion. In the standard presentation of this effect, the vertical test lines gray appear tilted in directions opposite to the orientations of the contextual lines black. D The Hering illusion. The two vertical lines gray appear bowed when presented in the context of radiating lines. If the set of points underlying the reference line template in the image corresponded to physical points that formed a straight line in 3-D space, the physical points were accepted as a valid source of the reference line.

A The pixels in an image region are represented diagrammatically by the grid squares. B The set of white points overlaid on the image indicates a valid sample for the reference line in A. After Howe and Purves, a When a valid physical source of the reference line was found in a region of a scene, the probability of occurrence of a second line forming an angle with the reference line in the same region was then determined.

As illustrated in Figure 4. The results of sampling range images in this manner are shown in Figure 4. This outcome applies both to scenes that are fully natural and those scenes in the database that contained some or mostly human artifacts cf. The three columns represent the physical sources found using a horizontal left , oblique middle or vertical right reference line, as indicated by the icons below the graphs.

The upper row represents the results obtained from fully natural scenes and the lower row from environments that contained some or mostly human artifacts. As noted in the last chapter, almost all straight lines in the real world are components of planar surfaces.

Figure 4. When compared to a cumulative distribution derived from a hypothetical probability distribution in which the probability for the physical sources of all angles is uniform indicated by the black line in Figure 4. If the probability of the sources of all angle subtenses were uniform, the ranks of all angles would be evenly distributed on this empirical scale, i. The actual distribution of the occurrences of the physical sources of angles derived from the image database, however, is not uniform see Figure 4.

As is apparent in Figure 4. A The gray curve is the cumulative probability distribution of the physical sources of angles derived from the probability distribution pooled from the 6 distributions shown in Figure 4. The black line, in contrast, indicates the cumulative probability distribution derived from a hypothetical distribution in which the probability of any given source is the same see inset.

B The predicted perceptions of angle subtenses follow from the empirical ranks of angles in the cumulative probability distribution and are indicated by the gray curve; the dashed black line indicates the actual subtenses of the stimuli. C The magnitude of angle misperception predicted by the analysis i.

As indicated in Figure 4. A comparison of the angle misperceptions actually seen by subjects and those predicted by this analysis shows remarkably close agreement Figure 4. Consider, for example, the apparent tilt of a vertical line caused by the presence of an oblique line that intersects it see Figure 4. The position of a vertical line i. This prediction is again consistent with what observers see in response to this sort of stimulus see Figure 4. Similarly, the parallel lines in the Hering stimulus Figure 4. In this case, the upper and lower contextual lines tilt in opposite directions, causing the perceived orientation of corresponding components of the test lines to change progressively, resulting in the apparent bowing.

B Left panel shows the standard presentation of the tilt illusion; the gray line is vertical.

### Introduction

Right panel indicates the direction arrows and magnitude of the tilt effect predicted by the distribution in A. The solid gray line is vertical; the dotted line indicates the predicted shift in the apparent position of the vertical line in the left panel. The visual circuitry relevant to processing information from simple line and angle stimuli has been studied in considerable detail. In this conception, the cortical response to an angle differs from a simple summation of the pattern of activity elicited by each angle arm alone because the orientation domains co-activated by the two arms presumably inhibit each other.

The effect would be to shift the distribution of the resulting cortical activity towards orientation domains whose selectivity is more orthogonal than would otherwise be the case. As it turns out, the effect of a contextual line on the response to a target line can be inhibition, facilitation, or some combination thereof, depending on a host of factors e.

As a result, there is no consensus about the interpretation of such interactions, or how they are related to the perceptions of angles and line orientations. SUMMARY Although the perceptual distortions that occur when viewing acute or obtuse angles in the absence of other contextual information may seem trivial with respect to the success or failure of human behavior, the visual strategy they signify lies at the core of vision. Since the circuitry underlying orientation is one of the most thoroughly studied aspects of the brain, this evidence about the probabilistic nature of angle perception also suggests a way of beginning to understand how the visual system instantiates these statistics.

Chapter 5 Size Another fundamental aspect of the perception of geometry is object size. Although the apparent length of lines and the subtense of angles are certainly pertinent to size, here we consider the appearance of object size more generally. Historically, studies of this aspect of vision have focused on yet another broad category of classical geometrical illusions characterized by effects referred to as size contrast and size assimilation. Several standard presentations of size contrast and assimilation stimuli are illustrated in Figure 5.

These stimuli typically entail two identical forms surrounded by one or more larger or smaller forms, generally of the same type e. The effect of these juxtapositions is that the target surrounded by the larger circles appears a little smaller than the identical target surrounded by the smaller ones e.

Another aspect of the anomalous perception elicited by the Ebbinghaus stimulus is that when the diameter of the surrounding circles is kept constant, the apparent size of the central target circle decreases as the interval between the central and the surrounding circles increases Massaro and Anderson, ; Girgus et al. A Standard presentation of the Ebbinghaus illusion. Observers see the central circle surrounded by smaller circles as being appreciably larger than the identical circle surrounded by larger circles. B Even when the size of the surrounding circles in a stimulus is the same, the central circle looks smaller when the interval between the central and the surrounding circles is increased.

C A similar size contrast effect is generated by a concentric presentation, known as the Delboeuf illusion. E When, however, the diameter of the outer circle is much larger than the inner circle, the inner circle looks smaller than an identical single circle.

G The effect in F diminishes as the inner circle becomes progressively smaller relative to the outer circle. After Howe and Purves, the inner target circle of the concentric set, the inner circle appears a little larger than the single circle Obonai, ; Oyama, ; Howard et al. This effect is diminished, however, when the diameter of the surrounding circle is increased.

Equally puzzling is the observation that the outer circle of the concentric set appears smaller when compared to a single circle of the same size Figure 5. As with the phenomenology of line and angle perception, the variety and complexity of these effects has been resistant to any coherent explanation. In this conception, the identical targets appear different in size because the probability distributions of the possible sources of the targets, given their different contexts, are different. According to the general hypothesis being examined here, it is the statistical structure of this accumulated experience that determines the geometrical characteristics ultimately seen.

To examine the merits of this supposition, we sampled the range image database to identify the physical sources that could give rise to projections whose geometrical structure was the same or similar to the size contrast or assimilation stimulus of interest Figure 5. By computing the frequencies of occurrence of the physical sources of target circles embedded in each of the different contexts as a function of the projected size of the target circles, we could in this way generate the probability distribution of the sources of the targets in the context of interest.

Each of these distributions thus provides the basis for constructing an empirical scale that ranks the size of a target circle in a particular context. As in the case of lines and angles, the rank of a target in a given context indicates the percentage of the physical sources of target circles that generated projections smaller than the size of the given target, and the percentage that generated larger targets.

If the probabilistic framework outlined earlier is correct, then the different rankings of the size of a target on these empirical scales of target size should predict the different apparent sizes of identical targets in different contexts. As is apparent in Figure 5. A As in earlier chapters, pixels in an image region are represented by grid squares; the black pixels indicate a template of the surrounding circles in an Ebbinghaus stimulus overlaid on the image.

The pixels covered by the template thus comprise a potential sample of the contextual elements of the Ebbinghaus stimulus. If the set of physical points corresponding to the pixels comprising each of the circles formed a geometrical plane in 3-D space, the set was accepted as a valid sample of the physical source of the contextual circles. B The pixels underlying the four template circles on the left are an example of a valid sample of this sort; pixels underlying the template on the right are a valid sample of the contextual circle the inner circle in this case in a Delboeuf stimulus.

C Blowups of the areas delineated by the boxes in B. After identifying physical sources capable of giving rise to projections that appropriately matched the contextual component of the stimulus of interest the white circles , we determined the frequencies of occurrence of the sources of all possible target circles, given the presence of the contextual components.

To this end, a series of target templates of various sizes colored circles was sequentially overlaid on the same image region only 3 such templates are shown here as examples. The set of pixels underlying each target template was then examined according to the same geometrical criterion applied to the contextual circles; the physical points corresponding to these pixels were counted as a physical source of the target circle only if this criterion was met.

After Howe and Purves, sources of target circles vary systematically as a function of the size of the surrounding circles. Given these different distributions, why then should two targets identical in size be perceived differently when surrounded by contextual circles of different sizes, as in the standard Ebbinghaus stimulus? To answer this question, consider a target circle Size 51 Figure 5. The probability of the physical sources of the target circle is plotted as a function of the diameter of the target in the 2-D images. The dashed line indicates, as an example, the position where the diameter of the target circle equals 14 pixels.

D Examples of the sorts of regions in natural scenes from which a sample of large contextual circles left or small contextual circles right was likely to be obtained. After Howe and Purves, 14 pixels in diameter as an example. The dashed line in Figure 5. Since the ratio of the area on the left of the dashed line to the area on the right is greater for the probability distributions derived in the presence of the smaller contextual circles than the larger ones, a target circle 14 pixels in diameter occupies a different relative position in each of these probability distributions.

As in the preceding chapters, the parameter that best describes these different relative positions is the cumulative probability derived from each distribution. As shown in Figure 5. This relationship means that when the surrounding circles are small, a target 14 pixels in diameter ranks relatively high on the empirical scale of target size; conversely, when the surrounding circles are large, the same target size ranks relatively low on the pertinent empirical scale. Thus, if the apparent size of the identical targets in the Ebbinghaus stimulus is determined by these empirical ranks, a given target circle presented in the context of small circles should appear larger than the same target in the context of large surrounding circles.

This is, of course, what observers see in response to this stimulus see Figure 5. Thus, just as lines and angles arise from planar surface patches, the elements of the Ebbinghaus stimulus will typically have arisen from planes in the 3-D world. It follows that the probability of encountering the physical source of a target circle in the database decreases as the size of the target circle increases see Figure 5. A further question is why the presence of contextual circles of various sizes modulates the occurrence of the sources of target circles differently.

Since larger surrounding circles necessarily arise from larger planes in the physical world, the relevant region of the scene is less likely to contain depth discontinuities. A region with a smoother physical structure is more likely to contain a larger planar area capable of giving rise to the projection of a larger central target circle.

In short, the presence of larger contextual 53 Size circles increases the probability of occurrence of the physical sources of larger target circles. As a result, the probability distribution of the sources of target circles varies according to the size of contextual circles, as indicated in Figures 5.

Figure 5. The range of possible target sizes differs among these distributions because the largest possible target size increases as the contextual circles are further separated. This difference in the maximum target size naturally causes the same target to have different relative locations in the three probability distributions, as indicated by the dashed line an example in which the diameter of the central circle is again 14 pixels. When the cumulative probability for a target circle 14 pixels in diameter is determined from each of these distributions, it is apparent that the empirical rank of the target decreases as the interval between the target and the surrounding circles increases Figure 5.

The dashed line indicates, as an example, a target circle 14 pixels in diameter. After Howe and Purves, 54 Perceiving Geometry: Geometrical Illusions Explained by Natural Scene Statistics high on the empirical scale of target size associated with that interval; conversely, when the interval is large, the same target will rank relatively low on the relevant empirical scale. This statistical relationship correctly predicts that the apparent size of the same target circle decreases as the interval between the target and the surrounding circles in the Ebbinghaus stimulus increases.

When the two sets of concentric circles are compared, the inner target circle within the relatively small contextual circle appears larger than the identical target in the relatively large outer circle. This effect can be rationalized in the same framework used to explain the perceptual responses elicited by the standard Ebbinghaus stimulus and its variants. The probabilities of occurrence of the physical sources of an inner target circle, given the presence of an outer concentric circle, are shown in Figure 5. The cumulative probabilities again show that a target circle of a given diameter ranks higher on the empirical scale associated with a smaller outer circle than on the scale associated with a larger one Figure 5.

This statistical relationship thus predicts that the target within the smaller outer circle should appear larger than the same target in the larger outer circle, as is the case. A The probability of occurrence of the physical sources of inner target circles in the presence of an outer contextual circle as a function of the diameter of the target circle in the image plane. Probability distributions associated with contextual circles of two different sizes are shown. The dashed line indicates, as an example, the position in the two distributions of a target circle 24 pixels in diameter.

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B Cumulative probabilities for a target circle 24 pixels in diameter i. After Howe and Purves, Size 55 Figure 5. B The maximum target length is not restricted by contextual lines in this variant of the standard Ponzo stimulus in which the illusory effect is diminished.

Of course, the explanation of size contrast offered is not limited to circles; the same argument applies to the apparent size of any geometrical form in some sort of spatial context. In the Ponzo stimulus, two identical horizontal lines look different in length in the context of two converging lines, the line closer the point of convergence of the contextual lines appearing longer. As in the case of a Delboeuf stimulus in which the size of the outer circle limits the maximal size of the inner target circle, the converging context lines in the Ponzo stimulus limit the maximum possible length of the horizontal target lines.

The maximum length of a target near the convergent end of the contextual lines is thus shorter than the maximum target length near the opposite end Figure 5. This difference causes the length of a given line to rank higher on the empirical scale of target length near the convergent end compared to the ranking of the same length on the scale for targets nearer the opposite end. Accordingly, the same line near the convergent end of the contextual lines in the Ponzo stimulus should look longer than the other target line, as it does.

Recall that when the horizontal lines in the standard Ponzo stimulus are rotated such that they become vertical attachments to one of the contextual lines, they no longer appear very different in length see Figure 1. Thus the empirical scale of target length near the convergent end will not differ much from the empirical scale associated with the opposite end. Accordingly, a vertical target near the convergent end of the stimulus would have about the same 56 Perceiving Geometry: Geometrical Illusions Explained by Natural Scene Statistics empirical rank as the identical target near the opposite end, meaning that the targets should appear about the same length, as they do.

This argument, however, should not obscure the fact that multiple empirical factors determine the relevant probability distributions for the perception of any of these stimuli. In the Ponzo stimulus, for example, the space between the target lines and the convergent lines is only a major factor in the effect; other categories of information, such as the length of the converging lines or perspective, are also relevant factors in the complete statistics that underlie these perceptions.

Conversely, it appears smaller than the single circle when the diameter of the outer circle is more than 4 or 5 times that of the inner circle see Figure 5. For the reasons already given, the probability of encountering the real-world source of a projected circle decreases as the projected size of the circle increases. Now compare the probability distribution for a single circle to the distribution of the physical sources of inner target circles in the presence of an outer contextual circle that is either relatively small Figure 5.

The cumulative probabilities associated with the target 24 pixels in diameter derived from these two distributions Figure 5. This relationship predicts that when the ratio of the diameter of the outer circle to the target circle is less than 2 it is 1. The cumulative probabilities in this case indicate that the rank of the target is now lower in the context of the outer circle than on the empirical scale for single circles Figure 5.

This further statistical relationship predicts that when the ratio of the diameter of the contextual circle to the target circle 57 Size A B 0. A The probability of occurrence of the physical sources of a single circle as a function of the diameter of the circle in the image plane. B The probability distribution of the sources of single circles in A superimposed on the probability distribution of the sources of inner target circles, given the presence of an outer circle 32 pixels in diameter.

The dashed line indicates a target circle 24 pixels in diameter as an example. C The probability distribution of the sources of single circles superimposed on the probability distribution of the sources of inner circles, given the presence of an outer circle pixels in diameter. Dashed line again indicates a target 24 pixels in diameter.

After Howe and Purves, is large, the target in the contextual circle should look smaller than an identical single target circle. Again, this is what is seen. Figures 5. Oddly, this ratio varies according to the absolute size of the target circle, decreasing as the visual angle subtended by the target increases. This fact presents yet another challenge to any explanation of size contrast and assimilation. In the empirical framework here, the point of perceptual transition should correspond to the ratio of the outer circle to the inner target circle at which the target has the same empirical rank in the context of the outer circle as when presented alone.

As already shown in Figure 5. Accordingly, when the diameter of the outer circle reaches this value, a target circle x pixels in diameter within the outer circle should appear equal in size to an identical single circle. Thus, the variable point of perceptual transition as a function of the visual angle subtended by the target circles is also predicted by empirical ranking. This latter distribution shows that the context provided by an inner circle increases the probability of occurrence of the sources of outer circles when the size of the outer circle is not much larger than the size of the inner circle which is now the contextual circle.

This distribution, however, converges with the distribution of the sources of single circles as the outer circle becomes increasingly large. A target circle 48 pixels in diameter, for example dashed line in Figure 5. These statistics predict that the outer target circle presented with an inner contextual circle should appear smaller than a single circle of the same size. These same statistics also predict the perceptual responses observed as the inner circle becomes increasingly smaller.

In addition to the two probability distributions Size 59 Figure 5. A The probability distribution of the physical sources of single circles black is superimposed on the probability distribution of the sources of inner target circles, given an outer contextual circle of diameter D0. In this example, in which the target circle is 24 pixels in diameter, D0 must be 52 pixels for the cumulative probabilities of this target derived from the two distributions to be equal, predicting a ratio Ri at perceptual equality of 2.

Thus the ratio of the diameters of the outer circle to the inner circle at the perceptual transition decreases as the absolute size i. After Howe and Purves, in Figure 5. The cumulative probabilities derived from these distributions increase as the size of the inner circle decreases see Figure 5. In fact, when the inner circle is very small relative to the outer circle e. A The probability distribution of the sources of single circles black; see Figure 5.

The dashed line indicates a target circle 48 pixels in diameter, as an example. B The probability distribution of the sources of single circles superimposed on the probability distributions of the sources of outer circles, given the presence of an inner circle 32, 16 and 8 pixels in diameter, respectively. Inset shows the cumulative probabilities for an outer target circle 48 pixels in diameter derived from these four distributions.

After Howe and Purves, outer target circle in the context of the inner circle should appear about the same size as an identical single circle, as indeed it does see Figure 5. This argument, however, cannot explain the effect of altering the interval between the central and surrounding circles of the Ebbinghaus stimulus see Figure 5.

In particular, the theory is contradicted by size assimilation effects. For example, adaptation level theory predicts that the inner target circle in Figure 5. This prediction is opposite the perception elicited. In this conception, size contrast effects are taken to be the consequence of comparing the properties of an object with its context, whereas assimilation is considered to be the result of incorporating the properties of contextual elements into the percept of the target Coren, ; Coren and Enns, ; Rock, A problem with this way of thinking is the lack of any biological reason why such processes would be necessary, or even useful.

Moreover, it is not clear under what conditions the mechanisms of contrast versus assimilation would or should operate, or how contrast and assimilation could be investigated in these terms. The analyses and observations summarized here show that, like simpler stimuli that involve single lines, angles formed by two lines or combinations of lines that cause tilt effects, the variety of size contrast and assimilation effects that have been described over the years are well accounted for in terms of the probability distributions of the possible sources of the relevant stimuli.

Chapter 6 Distance The way we see lines, angles and the relative size of objects are all aspects of the way human beings perceive physical space. For many who have thought about this issue, an appealing intuition has been that the properties of visual space arise as the result of a transformation of the Euclidean properties of physical space Indow, ; Hershenson, Indeed, the anomalies already discussed provide critical clues to understanding how visual space is actually generated. A fundamental descriptor of visual space not yet considered—and certainly one that is pertinent to this concept—is the perception of egocentric distance i.

The purpose of this chapter is thus to consider whether the perception of this further aspect of space can also be explained in terms of the statistical relationship between images and their physical sources. As illustrated in Figure 6. B The equidistance tendency. C The perceived distance of objects at eye-level. D The perceived distance of objects on the ground. After Yang and Purves, distance of objects bears a peculiar relationship to their physical distance from the observer Sedgwick, ; Gillam, ; Loomis et al.

When subjects are asked to make judgments with little or no contextual information e. Third, when presented at or near eye-level, the distance of objects relatively near the observer tends to be overestimated, whereas the distance of objects that are further away tends to be underestimated Epstein and Landauer, ; Gogel and Tietz, ; Morrison and Whiteside, ; Foley, ; Philbeck and Loomis, Figure 6.

Although a variety of explanations have been proposed in the various studies cited, there has been little or no agreement about the basis of these unusual perceptions of egocentric distance. More often than not, the several tendencies illustrated in Figure 6. Given the ability of the probabilistic relationship between retinal images and sources to explain a variety of other geometrical percepts, it makes sense to ask whether the probability distributions of the possible sources of visual stimuli also determine apparent distance.

Accordingly, the database of natural scene geometry described in Chapter 2 was used to test whether the anomalous perceptions of distance illustrated in Figure 6. This decay is presumably due to the fact that the farther away an object is, the less the area it spans in the image plane and the more likely it is to be occluded by other intervening objects.

The falloff at very near distances arises in large part because the range scanner was never placed directly in front of large objects that would have prevented the beam from scanning the majority of a scene see Chapter 2 ; as a result there is very little probability mass at distances 66 Perceiving Geometry: Geometrical Illusions Explained by Natural Scene Statistics Radial Eye level Height relative to eye level Elevation angle distan ce Horizontal distance Height Line o f sigh t Ground plane Figure 6. A second feature emerging from the analysis concerns how different physical locations in natural scenes are typically related to each other in terms of their radial distance from the observer.

The distribution of the absolute differences in the radial distance from the image plane of the scanner to any two physical locations whether separated horizontally or vertically is highly skewed, having a maximum near zero and a long tail Figure 6. A third statistical feature that emerges from the range data is that the probability distribution of horizontal distances from the scanner to physical locations i.

The probability distribution of physical distances at eye level has a maximum at about 4m and decays gradually over greater distances. The distributions of the horizontal distances of surface locations at different heights above and below eye level have roughly the same shape as the distribution at eye level. In terms of the empirical framework used to A 10 -1 Probability 10 -2 10 -3 10 -4 10 -5 10 -6 0 50 1 Distance difference m 10 Distance m B 10 0 Probability 0. A The probability distribution of distances from the center of the scanner to all surface locations.

B The probability distribution of the differences in the distances of two surface locations in physical space separated by three different visual angles in the horizontal plane vertical separations, which are not shown here, had a generally similar distribution. C Probability distributions of the horizontal distances of surface locations at different heights with respect to eye level see Figure 6.

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After Yang and Purves, 68 Perceiving Geometry: Geometrical Illusions Explained by Natural Scene Statistics rationalize the perceptual anomalies discussed in previous chapters, the probability distributions of distances illustrated in Figure 6. Thus, unlike stimuli examined in previous chapters where a measurable feature of the stimulus e.

Given these circumstances, the perceived distance should simply accord with the distances most frequently encountered in the past experience of human observers. As indicated in Figure 6. In other words, there are far more points in natural scenes associated with this range than with other distances. This prediction is in agreement with the evidence reported in psychophysical studies that observers tend to perceive objects to be at a distance of 2—4m under these experimental conditions see Figure 6. The Equidistance Tendency The similar distance of neighboring points perceived in the absence of additional information see Figure 6.

Since the probability distribution of the differences of the physical distances from the image plane to two locations with relatively small angular separations the black line in Figure 6. At large angular separations the colored lines in Figure 6. Accordingly, the tendency to see neighboring points at the same distance from the observer would be expected to decrease somewhat as a function of increasing angular separation.

This tendency has also been observed in psychophysical studies, although it has not been documented quantitatively Gogel, The agreement between this psychophysical evidence and the distribution of relative distances as a function of Distance 69 object separation is thus consistent with a probabilistic explanation of the equidistance tendency.

Perceived Distance of Objects at Eye Level The probability distribution of physical distances at eye level the black line in Figure 6. Based on this probability distribution, the perceived distance of an object located at eye level should, in the absence of other contextual information in the retinal image, be perceived to be about 4m away. Therefore, the distance of an object that is actually located closer than 4m would be overestimated and the distance of an object farther than 4m would be underestimated.

For instance, Philbeck and Loomis showed that the apparent distance of a dimly luminous object presented at eye level in an otherwise dark environment tends to remain at about 4m as the actual distance is varied between 2 and 5m subjects reported apparent distance in this case by walking blindfolded to the place they thought the object was, explaining the relatively small range of distances tested. As shown in Figure 6. Moreover, the distribution shifts toward nearer distances as the line of sight deviates increasingly from eye level, a tendency that is more pronounced below than above eyelevel.

These statistical differences as a function of the elevation of the line of sight are more obvious in Figure 6. The most likely distances from the eyes of an observer to surface locations at various elevations form a curve that is relatively near the ground at closer horizontal distances, but increases in height as the horizontal distance from the observer increases.

These further statistical characteristics of physical distances can thus account for the perceptual effects illustrated in Figure 6. A Contour plot of the probability distributions of distances at different elevation angles.

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Probabilities are indicated by colorcoding bar on the right. The vertical axis is the height relative to eye level; the horizontal axis is horizontal distance from the image plane. The blue line indicates the position of the ground plane at 1. These statistics predict that the apparent distance of an object on the ground more than a few meters away in a darkened environment should appear closer and higher than the actual location of the object.

Others have suggested 71 Distance that visual space is a computed composite, based on more or less independent information derived from cues such as perspective, texture gradients, binocular disparity, and motion parallax Gillam, ; Loomis et al. Gibson argued that since human beings are terrestrial, the ground is the key factor in determining the perception of geometry. Thus whereas each of these approaches has some merit, in terms of the present argument they are in varying degrees off the mark.

If visual space is generated by the statistical relationship between images and their sources, then explaining the relevant perceptual phenomenology will inevitably depend on the statistical properties of natural visual environments with respect to visual observers. Absent the empirical information about image-source relationships derived from a range image database, any explanation of visual space is bound to be inadequate, as evidenced by the fact that none of these previous suggestions about visual space has been able to rationalize the full spectrum of discrepancies between physical and apparent distance.

SUMMARY The ability to explain a variety of subtle anomalies in the perception of distance based on the statistics of the physical distances of object surfaces from the observer in natural scenes offers further evidence that rationalizing perceived geometry in the probabilistic framework outlined in previous chapters is a powerful way of understanding visual space. For stimuli that generate the best known of the classical geometrical illusions see, for instance, Figures 1.

These stimuli and their variants emphasize that to understand the perceptual effects elicited by even relatively simple geometrical stimuli requires going beyond intuitions; the explanations that emerge from the analysis of the database are, as it turns out, quite subtle. As is apparent in Figure 7. Despite this substitution, the illusory effect remains much the same. C A similar effect is generated by a variant in which the central shafts are missing. In Figure 7. As discussed in Chapter 3, however, this approach is neither feasible nor conceptually appropriate.

Thus, an analysis of the database carried out in this way would yield only a small number of samples that would have commensurately little statistical meaning. As indicated in Figure 7. If this criterion was met, the points were accepted as a valid sample of the physical source of the conditional adornment. The white dots on the squares indicate the reference edge of these adornments. The length L of the target shaft line or interval was designated positive when the complementary template was on the right side of the conditional adornment, and negative when the complementary template was on the left.

B The sampling procedure applied to an image. The red template indicates a valid sample of the conditional adornment that met the geometrical criteria described in the text; a series of complementary templates was then overlaid at successively greater distances from the conditional adornment to assess the presence of valid samples of the complementary adornment, as indicated in the blow-up. The length of the shaft or the corresponding interval was varied incrementally, negative values indicating a complementary template to the left of the conditional adornment, and positive values to the right.

As for the conditional adornment, the physical points corresponding to each straight line in the complementary template were evaluated to see whether they also formed a straight line in 3-D space. We could then ask how the probability distributions obtained in this way vary according to how a shaft or interval is adorned at its ends.

The distribution indicated in black was derived from a sampling procedure in which the apex of the conditional adornment pointed to the right; the distribution in gray was derived using a conditional adornment whose apex pointed to the left see diagram in Figure 7. Thus the left half of the distribution in black where L 0 corresponds to shafts with arrow tails; the opposite is true for the distribution shown in gray.

Furthermore, for each value of L less than 0, the distribution represented in black gives a higher probability than the gray distribution, whereas the opposite applies for all values of L larger than 0. In the diagram above, the conditional adornments are indicated by solid lines, and the complementary adornments by dotted lines. B The cumulative probability distributions derived from the probability distributions in A the dotted portions of the curves are extrapolated.

C Superimposition of the two cumulative probability functions in B. D Examples of two shafts 50 pixels in length, one adorned with arrow tails and the other with arrowheads upper panel. The left adornments are arbitrarily designated the conditional adornments, and are indicated by the solid elements at position 0 in the lower panel. Given each of these conditional adornments, the probability distributions shown in A—C indicate that the complementary adornment and shaft dotted lines occur at different positions with different probabilities.

This statistical fact means that a complementary adornment at position 50, given a conditional adornment extending to the left of position 0, is further to the right in the empirical range of possible positions for complementary adornments than a complementary adornment at position 50 given a conditional adornment extending to the right.

After Howe and Purves, b corresponding cumulative probability distributions Figures 7. Take one of the adornments on each shaft, the one on the left for instance, as the conditional adornment, and the position of its apex as 0 the same argument of course applies if the right adornment is selected. The complementary adornments are thus at position Figure 7. There is no obvious difference among the probability distributions obtained from this subset of the range data and the distributions derived from fully natural scenes see Figure 7.

The probability distributions of the physical sources of these several variants are shown in Figure 7. Thus the similar perceptual effect elicited by each of these variants can be rationalized in the same statistical framework as the standard stimulus. C Variant with no shafts. D Variant comprising only dots. The icons indicate the templates used to sample the database for each variant, in the same manner shown in Figure 7. What, then, is the nature of the differences?

The opposite is true when the conditional adornment extends to the right of the starting position the relevant planes are indicated by the dashed black lines. The reason is that as the complementary adornment becomes further separated from the conditional adornment, the physical points corresponding to the complementary component are less likely to be in the same plane as the physical source of the conditional adornment.

The opposite is of course true in the presence of a conditional adornment extending to the right of the starting position Figure 7. A Diagram of these two types of 3-D corners. B Probability distributions of the distance from the image plane of the central edges of concave and convex corners in the database. After Howe and Purves, b persist in the absence of eye movements Evans and Marsden, ; Bolles, The assimilation theory argues that the length of the central shaft is misperceived because the visual system cannot successfully isolate parts from wholes. This explanation is contradicted, however, by the size contrast effects described in Chapter 5.

Recall that a target embedded in a large surround as in the Ebbinghaus stimulus, for example appears smaller than the same target in a less extensive surround. SUMMARY The results summarized here further support the idea that visual percepts are determined by the statistical relationship between retinal images and their possible real-world sources. In the format in Figure 8. Figure 8. A similar effect is elicited when the oblique line is interrupted by a space delineated by two parallel horizontal lines instead of two vertical lines Figure 8. In this presentation, the right segment appears to be shifted to the right from the position of collinearity with the left segment.

In the standard presentation of the Poggendorff stimulus, such as shown in Figures 8. A particularly puzzling aspect of the phenomenon is that the effect is largely abolished when only the acute components of the standard Poggendorff stimulus are shown. However, the effect remains if only the obtuse components are present Day, Figure 8. B A similar effect occurs when the orientation of the interrupted line is reversed. In this case, the collinear extension on the right appears to be shifted upward.

C When an oblique line is interrupted by a space delineated with parallel horizontal lines, the oblique line segments appear to be shifted horizontally with respect to each other. D The magnitude of the effect increases when the interrupted line becomes closer to vertical. E The magnitude also increases as the width of the interruption increases.

F The illusion is largely abolished when only the acute components of the stimulus are presented, but maintained when only the obtuse components are shown. After Howe et al. The oblique line segments in the standard Poggendorff presentation in Figure 8. In other words, given a line segment on one side of an implied occluder, the perceived position of another line segment in the same orientation should be determined by the relative probability of occurrence of the physical sources of line segments that have projected in that orientation across the occluder.

The determination of the physical sources of the stimulus involved two steps. Then, in the same region of the scene, we determined the frequency of occurrence of the physical sources of possible right line segments, i. A template for sampling the left oblique line segment and the two vertical lines is shown in Figure 8. As described in previous chapters, the points in the image underlying each straight line in the template were examined to determine if the corresponding physical points formed a straight line in 3-D space. This template was moved vertically in sequential applications to determine the occurrence of all the possible physical sources of the right oblique segment at different vertical locations relative to an extension of the left oblique segment.

A Examples of the templates for sampling the lines comprising the standard Poggendorff stimulus see Figure 8. The pixels in an image patch are represented diagrammatically by the grid squares; black pixels indicate the template for the left oblique line segment and the two vertical lines; the red pixels indicate a series of templates for sampling the right line segment at various vertical locations relative to the left oblique segment.

As in previous analyses, the points underlying a template were accepted as a valid sample only if they formed a straight line in 3-D space. B The solid white lines indicate a valid sample of the left oblique line segment and the two vertical lines. Blowup of the boxed portion of the scene shows the subsequent application of a series of templates red used to test for the presence of right oblique segments at different vertical positions.

In these further examples, the black lines are equivalent to the black template in A , and the red lines equivalent to the red templates. The physical sources of the variants of the Poggendorff stimulus illustrated in Figure 8. The Poggendorff illusion explained by natural scene geometry.

Natural-scene geometry predicts the perception of angles and line orientation. DOI: Size contrast and assimilation explained by the statistics of natural scene geometry. Journal of Cognitive Neuroscience. The statistical structure of human speech sounds predicts musical universals. Size contrast explained by the statistics of scene geometry Journal of Vision. Vision and the perception of music have a common denominator Journal of Vision.

Range image statistics can explain the anomalous perception of length. A probabilistic explanation of perceived line length and orientation Journal of Vision. Howe CQ , Purves D. PMID Show low-probability matches.