the geometry of visual space

The Geometry of Visual SpaceAuthor(s): Robert FrenchSource: Noûs, Vol. 21, No. 2 (Jun., 1987), pp. 115-133Published by: WileyStable URL: http://www.jstor.org/stable/2214910 .

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The Geometry of Visual Space ROBERT FRENCH

CENTRAL MICHIGAN UNIVERSITY

INTRODUCTION

In this paper I shall investigate the geometrical structure of phenomenal visual space. I shall assume at the outset that visual perception is indirect, and thus that we are never immediately aware of even the surfaces of physical objects. Instead I will hold that our immediate visual awareness consists of a phenomenal field of colors, visual space, whose geometrical and qualitative features are determined by causal connections with the physical objects being "seen;" that is by light rays reflected from these objects being focused on the retina, and the various subsequent neural events which take place in the brain. I have defended the causal theory of perception in detail elsewhere (French, 1987), but in order to keep this paper within manageable length, while still doing justice to the geometrical arguments involved, I shall not summarize those arguments here.

While there may be various structural resemblances between portions of visual space and the corresponding physical objects being "seen," many of which being given by the laws of projective geometry, it remains possible that both the topological and metric structure of visual space is quite different from that of physical space. Thus, an attitudinal shift is required here away from direct realism, which numerically equates the two, to a more neutral attitude where a geometrical analysis can be made of our visual experience in and of itself. I shall attempt to undertake such a geometrical analysis in this paper. Thus, I shall now turn to an investigation of the topology of visual space, to be followed by an investigation of its metric structure.

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THE TOPOLOGY OF VISUAL SPACE

While it may not be possible to rigorously deduce the topology of visual space, as Blank (1958) notes "since the concept of open set is not readily applicable to immediate visual experience," I believe that it is still possible to strongly motivate a particular topological structure, such as by the following considerations. My discussion will first concern the plausibility of considering visual space to be a continuous space, and I will then investigate its dimensionality.

With regard to the issue of the continuity of visual space, it is very important to distinguish between the concept of continuity and that of homogeneity. There is a limit to the spatial acuity associated with visual perception, which is partially due to the finite sizes of the photoreceptors (rods and cones) of the retina. In portions of visual space correlated with the fovea of the retina, the limit of this spatial acuity is approximately one minute of arc, but it decreases rapidly towards peripheral regions of the space. Thus, there are units of visual space, which Berkeley called "minimum visibles," which cannot be resolved into smaller units possessing different colors, and thus these units must be homogeneous. The fact that these units are homogeneous though does not prove that they are not also continuous, but only that any smaller portions of visual space must all be of the same color. Bertrand Russell (1922, p. 156) for example makes this point as follows:

We must do one of two things: either declare that the world of one man's sense-data is not continuous, or else refuse to admit that there is any lower limit to the duration and extension of a single sense- datum. The latter hypothesis seems untenable, so that we are apparently forced to conclude that the space of sense-data is not continuous; but that does not prevent us from admitting that sense-data have parts which are not sense-data, and that the space of these parts may be continuous.

Thus, it is at least possible that in spite of the limited spatial acuity of visual perception, visual space itself still may be continuous, or at least be piecewise continuous - i.e. possess only a finite number of discontinuities. In fact, in my section on the metric structure of visual space, I shall argue that it usually will possess a finite number of discontinuities. Still, in view of the fact that visual space introspectively appears to be a continuum, and not just an aggregate of indivisible sense data, it would not seem to be unreasonable to conclude that it in fact is always at least piecewise continuous.

I shall now turn to my analysis of the dimensionality of visual space. This is a controversial topic, inasmuch as one group of



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philosophers and psychologists, including Berkeley (1963), Thomas Reid (1970), and Helmholtz (1962), has argued that it is two dimensional, while another group, including Poincare (1963), William James (1890), Rudolph Luneburg (1947), and J. J. Gibson (1950), has argued that it is three-dimensional. Most of these writers have connected the issue of whether or not visual space possesses a third dimension, with the issue of whether or not we are immediately aware of visual depth. For example, Berkeley argued that we are never immediately aware of visual depth, inasmuch as configurations of objects possessing different physical depths can project iden-. tical images onto the retina, and he thus argued that visual space is only two dimensional. Berkeley's argument here is invalid, since a sufficient amount of information is presented from the two eyes, notably involving the binocular depth cue of retinal disparity, for the reconstruction of the depths of objects being "seen" to take place in the brain. Such monocular depth cues as perspective, motion parallax, occulusion, relative sizes of the objects being "seen," and textural gradients are also important in this reconstruction. The question arises though as to whether or not the presence of visual depth perception shows that visual space itself is three-dimensional. I wish to argue that it does not.

A short digression is needed here into dimension theory. Two quite different types of analyses of "dimension" have historically been given. One type of analysis is in terms of the minimum number of independent variables needed in order to mathematically characterize a given situation. Such an analysis is often given of various "abstract" spaces, such as in quantum mechanics, or even economics. Using it, one could even argue that visual space is five- dimensional, if for example color and brightness are counted as dimensions. Obviously in such an analysis spatial dimensions are conflated with non-spatial ones; that is the color and brightness dimensions. Thus, an analysis of "dimension" is needed here which at least wards against such conflations taking place. This point is important, since it may not be obvious before undertaking such an analysis, as to which category "visual depth perception" belongs to.

Such an analysis of "dimension" has been developed in a nonrigorous form by Poincare (1963), and then rigorously by a group of mathematicians including Karl Menger (1943). In this analysis, "dimension" is defined recursively in terms of the minimum number of dimensions required of a space in order for that space to "cut" or give boundaries to the space whose dimensionality is being tested. The bounding space will then possess one fewer dimension than



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the space being bounded, as for example a one-dimensional space, such as a circle, can bound a two-dimensional space, or a two- dimensional space, such as the surface of a sphere, can bound a three- dimensional space. This is a synopsis of Poincare's analysis. Menger et al. added the requirement that the space whose dimensionality is being tested, must be capable of being given boundaries in each of its infinitely small neighborhoods, by a space of one fewer dimension, in order to guard against such counterexamples to Poincare's analysis as two cones meeting at a point, which could be bounded at the point of intersection by a space of zero dimensions (a point). However, it seems to be clear that visual space bears little in common with any counterexamples like this, and thus it would seem that one could use Poincare's criterion for determining the dimensionality of visual space; that is by determining the minimum number of dimensions needed by a space in order for it to give boundaries to a section of visual space.

Consider the following example given by Sir William Hamilton (1877, Vol. II, p. 165), of how the contrast between two colors constitutes a line, which may come back on itself and thus constitute a figure:

It is admitted by all that we have by sight a perception of colours, consequently a perception of the difference of colours. But a perception of the distinction of colours necessarily involves the perception of a discriminating line; for if one colour be laid beside or upon another, we can only distinguish them as different by perceiving that they limit each other, which limitation necessarily affords a breadthless line-a line of demarcation. One colour laid upon another, in fact, gives a line returning upon itself, that is a figure.

I will now show that it is plausible to hold that such a division of visual space into two regions where one region possesses a homogeneous color which is sharply contrasted with the color of a surrounding region, is sufficient to define a "boundary" in the topological sense of the word. In topology, a "boundary" is defined as the set of points contained in both the closure of the set being bounded and in the closure of the complement of that set. It is true that since closure is defined in terms of properties of neighborhoods in the space, and since these neighborhoods are defined in terms of the containment of open sets, as Blank points out there is no way to rigorously test whether visual space possesses boundaries in this sense. However, inasmuch as the border between color regions in Hamilton's example is defined as the demarcation between the regions, it would seem to share the essence of the topological prop- erty of not being contained within either region, although being



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contained within the analogue of the closures of both. It should also be emphasized that this boundary is not a border possessing a third color, and thus as Hamilton points out is a breadthless line, and therefore truly one-dimensional. It follows then that an expanse of a given color in visual space can be bounded by a one-dimensional space, the line constituted by the boundary between the given color and its background color, and thus by Poincare's analysis of dimensionality, visual space must be two-dimensional.

One possible objection to the foregoing account is that inasmuch as sharp boundaries between different color regions are not always present in visual space (an extreme example of which would be the case of viewing a cloudless sky which appears to be a homogeneous blue color), the preceding analysis cannot always be applied to visual space. In reply to this objection, it can be noted that a visual space constituted under such homogeneous conditions, does not seem to differ in any relevant way from one in which lines are clearly constituted, and thus even though the dimensionality of a visual space may not be directly testable for under such conditions, there is no reason to believe that the dimensionality has in any way changed from the two-dimensional case. In fact, it is particularly noteworthy that in the one case where prima facie a change in dimensionality might be thought possible, the case where objects constituted in one's visual space possess different physical depths, Poincare's analysis clearly does apply. One need then only consider the boundary between the doser object and the more distant background object (in fact the existence of such an "edge" often serves as a cue for visual depth), in which case it is clear that the closer object, as constituted in visual space, is bounded by a one-dimensional space, and thus that visual space again must be two-dimensional.

Further considerations pointing towards the two-dimensionality of visual space include the facts that under normal circumstances we only "see" the surfaces of objects being viewed and not their interiors (if visual space was topologically three-dimensional one would have to be able to see these interior points), and that the space in between "us" and objects being "seen," isn't itself seen, even though we do experience a sense of "outness" with respect to these objects. Of course one could point out that in the case of viewing semitransparent objects, some of their interior points are "seen," but even here the usual case does not involve the actual apprehension of the surface of the object separately from its interior, but rather a transformation (such as a change in color) of interior portions brought about by the semitransparency. In rare cases, however, we may actually see the surface of the semitransparent



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object as being separate from its interior, in which case visual space would have to "split" into two separate layers here (one layer for the surface of the object and one for its interior), but still each layer would be only two-dimensional. Still another class of potential counterexamples involves cases of viewing changing configurations of objects, where a one-time boundary as constituted in visual space disappears over time. The point to be made here is that the test for the dimensionality of visual space applies only to instantaneous configurations in the space, and that even though the qualitative and spatial arrangements of objects constituted in the space may change over time, the space itself remains two-dimensional.

If visual depth perception does not involve adding another dimension, in the topological sense, to visual space, the question arises as to how visual depth is sensed phenomenally. In the case of the monocular visual depth cues, it could be argued that they possess phenomenal two-dimensional correlates, such as changes in the sizes of the objects constituted in visual space as they become more distant, the occlusion of the image of one object by the image of another, phenomenal textural patterns, and motion parallax as evidenced by the different rates at which objects constituted in visual space move across the space. But what about the binocular visual depth cue of retinal disparity? It has been conclusively shown by the inven- tion of the stereoscope and Julesz' (1971) work with random dot stereograms, that this cue results in the phenomenal impression of visual depth, even when no monocular visual depth cues are present, and the question thus arises as to what is the phenomenal correlate to this cue. Certainly there is no obvious manner in which it could resemble its physical stimulus, as was the case with the monocular visual depth cues, inasmuch as the two disparate retinal images are fused into a single phenomenal visual space. I shall attempt to show how changes in the internal geometry of visual space can serve as such a phenomenal cue to visual depth shortly, but in order to do this, I must first turn to my analysis of the metric structure of visual space.

THE METRIC STRUCTURE OF VISUAL SPACE

As with the case of its topology, there has also been a controversy as to the nature of the metric structure of visual space. In fact, there have been advocates for each of the geometries of constant curvature as constituting this metric structure. Most of the proponents of the claim that visual space possesses a Euclidean metric have been philosophers rather than psychologists; most notably Kant (1929), and such modern commentators on Kant as Strawson (1976)



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and Ewing (1938). Thomas Reid (1970) and recent commentators on his work such as Norman Daniels (1974), and R. B. Angell (1974) have claimed that visual space possesses a spherical metric, while members of the recent Luneburg school (Luneburg 1947, 1950, Blank 1958, 1959) have held that it possesses a constant negative curvature, i.e. is hyperbolic. I shall later show that it may be possible to recon- cile at least much of the evidence in favor of these last two geometries by means of the geometry of a two-dimensional manifold possessing a variable curvature, but first I shall briefly summarize the evidence often cited in favor of each of the aforementioned geometries as being that of visual space.

Claims that visual space possesses a Euclidean metric structure stem predominantly from direct realist or idealist (as in Kant's "transcendental idealism") ontologies, together with the considera- tion that physical space, which has thus been equated with visual space, possesses at least a locally Euclidean metric. For the pur- poses of this paper, I am unwilling to just assume the truth of direct realism though, and thus my investigation of whether or not visual space possesses a Euclidean metric will be based on introspection. Interestingly enough when this is done, there is strong evidence that visual space possesses a spherical metric instead, as for example J. R. Lucas (1969) argues as follows:

Ewing (1938) and Strawson (1966) have attempted to save Kant's account of geometry by maintaining that it is a priori true at least of phenomenal geometry - the geometry of our visual experience - that it is Euclidean. But this is just what the geometry of appearances is not. Let the reader look up at the four corners of the ceiling of his room, and judge what the apparent angle at each corner is; that is, at what angle the two lines where the walls meet the ceiling appear to him to intersect each other. If the reader imagines himself sketching each corner in turn, he will soon convince himself that all the angles are more than right angles, some considerably so. And yet the ceiling appears to be a quadrilateral. From which it would seem that the geometry of appearances is non-Euclidean, with the angles of a quadrilateral adding up to more than 3600.

It might be noted that the phenomenon which Lucas refers to occurs only when figures are constituted over relatively wide areas of visual space, and steadily diminishes as smaller areas are subtend- ed; that is, visual space seems to approach a Euclidean metric in the small. But this is consistent with the claim that visual space possesses a spherical metric, since it points both to the non-existence of similar figures in visual space, and also to its being locally Eucli- dean in the metric sense that at each of its points its metric tensor is approximated to first order by the metric of a Euclidean two-dimen-



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sional space. Both of these properties are characteristic of non- Euclidean spaces of constant curvature, and the fact that the sums of the angles of the squares are greater than 3600 would constitute evidence that the space possesses a spherical metric. Of course, it could be questioned whether the sides of Lucas' ceiling are really seen as being straight, and I shall address that issue shortly, after first reviewing some further evidence that visual space possesses a spherical metric.

The first philosopher to claim that visual space possesses a spherical metric seems to have been Thomas Reid (1970), who in his chapter "Geometry of Visibles" took note of a number of theorems of spherical geometry which he claimed apply to visual experience. These include the claims that straight lines correspond to great circles (geodesics) in visual space, that such a space is finite, being cut into two equal parts by a straight line which comes back on itself, that two straight lines can enclose a space, and thus that there are no parallel lines, and that triangles will possess more than 1800 although there will be no similar triangles. The difficulty with Reid's claims here, other than the one that triangles in visual space possess more than 1800, which parallels Lucas'- ceiling example, is that in order for the effects in question to take place, head movement is required. The field of view of vision is only approximately 1700 by 1200, which corresponds to something less than a hemisphere, instead of the full sphere which would be required for the applicability of most of the theorems which Reid points to.

Nevertheless, a powerful argument can still be made that visual space possesses a spherical metric. Consider the case of marginal distortions in wide-angle photography. In these "distortions," objects in peripheral regions of the photograph are disproportionately enlarged in comparison with objects near the center. This is due to the fact that when an image is projected onto a plane (such as

A

Yr

B C

Figure 1



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the film of a camera), different parts of the plane subtend different solid angles, as illustrated by Figure 1. It can be seen by trigonometry, that the relation between these solid angles and parts of the plane being projected onto, is given by:

dx/dO ac sec 0 (1)

It can further be noted that such "distortions" do not occur with a projection onto a sphere (r in Figure 1), inasmuch as there equal areas do subtend equal solid angles. Inasmuch as the field of view of vision is quite wide in comparison with most wide-angle photographs, the fact that these "distortions" do not occur in vision constitutes strong evidence that it possesses a spherical metric. This point is sometimes taken into account in photography when very wide angles of view are involved. For instance, in panoramic photography often the film is curved in the back of the camera, and similarly cinemas with wide screens often curve these screens, an extreme example of which is the planetarium, where extreme wide-angle effects are projected onto a dome. It should also be emphasized that any continuous mapping of the surface of a sphere onto a plane involves distortions, those present in the Mercator projection of the globe onto a flat map, and that it thus is impossible to embed a Cartesian coordinate system onto the surface of a sphere, without affecting the metrical relations of those coordinates.

A B

A' / - B '

Figure 2

A further point, closely related to the previous one, is that when physical parallel lines are viewed at right angles to the line of sight, they are seen to converge at both ends. This effect can be seen by viewing railroad tracks from directly overhead, or by looking at a long low building head-on, from a position around halfway along its length. The tendency towards convergence here can be explained in terms of Euclid's law that for an object of a given size, the size of its visual angle is inversely proportional to its distance away from the eyes. Thus, inasmuch as the portion of the railroad tracks or building directly in front of one will be closer to the eyes than either



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end as it passes the limits of vision, it follows that both of these ends will be seen to converge to some extent, although not com- pletely, due to the limited field of view of vision. It is obvious from Figure 2, where X represents the railroad tracks or building being viewed, that the formula for the magnitude of this effect will be the same as that for marginal distortion given by Equation 1. Also, the symmetry in Figure 2 explains why this effect does not occur in wide-angle photography, since the marginal distortion on X' exactly compensates for it. There would be a resultant loss in symmetry if X' was curved, and thus the effect does occur in vision. Similarly, Helmholtz' (1962, Vol. 3, p. 181) checkerboard figure, shown in Figure 3, appears to be composed or normal squares when

Figure 3

viewed up close to one eye, since the distortions in the board com- pensate for the effect just discussed.

Of course, in all of these effects, as I noted with Lucas' ceiling example, it can be questioned as to whether the lines involved are really seen as being straight. If it is held that visual space actually corresponds to the surface of a sphere, then it would be reasonable to hold that these lines are the straightest possible, e. g. are geodesics, in spite of their tendency to coverage together, and thus their lack of apparent straightness. It is just the case that there are no straighter lines in visual space. This ends my discussion of the evidence that visual space possesses a spherical metric.

The theory that visual space is hyperbolic stems from the work of Luneburg (1947, 1950), and the subsequent investigations of Blank (1958, 1959), and others. The evidence in favor of this theory is of quite a different nature than that just discussed, since it involves aspects of visual depth perception, instead of effects taking place in the frontal plane of vision. It should also be noted that Luneburg



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assumed that visual space is three dimensional, while I have argued that it is only two dimensional, which was also an assumption in my arguments that it possesses a spherical metric. I won't go into Luneburg's theory in any detail in this paper, and thus I won't do justice to all of the evidence which he cites in favor of it. In- stead, I merely wish to demonstrate that much of the evidence which is cited in favor of the Luneburg theory involves the size constancy tendency, that is the tendency for an object in visual space to appear the same size when it is seen from different depths. In fact, I wish to argue that the chief piece of evidence cited by Luneburg in favor of his theory, the Blumenfeld parallel alley experiment, can be analyzed in terms of the tendency towards size constancy.

In the Blumenfeld parallel alley experiment, subjects in a dark room are asked to construct both parallel and equidistant "alleys" by arranging a pair of arrays of lights located at various depths in the horizontal plane of vision, so as to appear either as parallel straight lines (parallel alleys) or as always equidistant from each other (equidistant alleys). While the subject's head is held in a fixed position, free eye movement is allowed. The reason for the dark room is to remove the various monocular depth cues, and hence to isolate the effects of binocular cues on depth perception. The interesting thing about results from these experiments is that while neither set of alleys is found to conform to physical parallel lines, the set of parallel alleys is consistently constructed so as to lie within the set of equidistant alleys, with the two alleys converging as they become more distant. Inasmuch as in Euclidean geometry, parallel lines are always equidistant from each other, this result was taken by Luneburg and Blank as evidence that visual space possesses a non-Euclidean metric, in particular a hyperbolic one, since parallel lines in visual space were shown to diverge in the experiment.

However, it is also possible to explain the results of the Blumenfeld parallel alley experiment in terms of the size constancy tendency, by noting that in size constancy experiments, different instructions result in quite different amounts of constancy being reported (Gilinsky, 1955). For example, if a subject is asked to judge an object's "actual size," the results are very close to absolute size constancy, i.e. no change in size with increased physical depth, while if he is asked to judge the "projective size," like an artist drawing a scene in perspective, the size constancy tendency is greatly reduced, although still being significantly greater than that given if it was directly proportional to the size of the retinal image. Thus, it is quite significant that typically quite different types of instructions are given to subjects in the two different alley experiments.



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In the equidistant alley experiment, the subject is typically asked to judge the "apparent width" of the alleys, a type of instruction which normally results in a large amount of size constancy being reported, while in the parallel alley experiment, the subject is typically asked to avoid the appearance of the alleys converging, as in the case of railroad tracks which seem to meet as they go off in the distance (Shipley, 1957). Thus, it is not surprising that the parallel alleys are constructed so as to physically diverge as they become more distant, which would give a result with much less size constancy than in the equidistant alley case.

It would seem then to be possible to explain the Blumenfeld parallel alley experiment in terms of the size constancy tendency, and inasmuch as the tendency itself has been cited as constituting evidence in favor of the Luneburg theory (Luneburg, 1950), it would seem that an alternative explanation of the tendency would ipso fac- to also account for at least much of the evidence for the Luneburg theory. There remains some question as to what type of instructions are most relevant for determining the amount of size constancy actually present in visual space, rather than in judgments of the physical sizes of objects constituted in it. It is clear that instructions asking for judgments of the actual sizes of objects are not relevant here, and that ones in which the subject is asked to assume the viewpoint of an artist and judge the projected sizes of objects as they are constituted in visual space would be at least more appropriate. Inasmuch as it is the case that even these types of instructions result in a large amount of size constancy being reported, it would seem that some tendency towards size constancy is a primitive fact about visual perception, and must be accounted for in an adequate description of the geometry of visual space.

I shall now turn to a discussion of how it may be possible to account both for the size constancy tendency and the fact that globally visual space possesses a spherical metric, by means of the metric structure of a two-dimensional space of variable curvature. Thus, my conclusion that visual space possesses a two-dimensional topology will be accounted for along with at least much of the evidence in favor of the Luneburg theory, and the evidence that visual space is spherical. It should be noted that this type of analysis bears a fair amount in common with David Marr's (1982) "2 V2 D" sketch (where the "2 Y2 D" refers to the number of dimensions of the sketch), which describes the geometrical structure of a "functional" visual space, rather than a phenomenal one such as the visual spaces previously described. In that sketch, depth and surface orientation information is displayed in a retinocentric manner upon a two-dimen-



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sional continuum; the " Y2 D" referring to the depth information. However, Marr does not postulate a global spherical metric structure for his sketch, although it does share a two dimensional topology with my variable curvature theory. I shall now turn to my derivation of that theory.

The metric structure of the surface of a sphere in terms of the line element can be derived from the equation for marginal distortion (Equation 1), and is given by:

ds2 = [dO2 + sin2 d412]r2 (2)

where r is the radius of the sphere, its area being proportional to r2, and where 0 and 4 are spherical coordinates on the surface of the sphere. The problem then is how to define a transformation on Equation 2, dependent upon the direction being looked at 0,0, which can account for the size constancy tendency. Inasmuch as the visual size of an object depends both on its physical size and its distance away from the eyes (due to the size constancy tendency), it is useful to define a "depth function"

P = f(o,40,t) (3)

at this point, where f(0,O,t) refers to the magnitude of the physical distance from the "cyclopean eye" (an imaginary eye centered between the two eyes) to the object being seen in the direction 0,0, at time t. In normal circumstances, a unique object will be determined by these conditions, and thus Equation 3 will be a genuine function, but there are cases where such an analysis breaks down, such as when one looks at an object through a semitransparent substance. While I have argued that even in these special cases visual space remains two-dimensional, a more sophisticated analysis would be necessary in order to determine the resultant metric structure, and I shall not attempt such an analysis in this paper. Thus, the following analysis only applies to "normal" cases, where a unique object is determined by Equation 3.

The question thus arises as to how to use the depth function given by Equation 3, so as to define a transformation on the metric of a sphere given by Equation 2, which will account for the size constancy tendency. For one point, it can be noted that the size constancy tendency seems to be independent of the direction being looked at 0,4 (e. g. things look the same size when they are seen out of the corner of the eye as when they are seen straight ahead), although it is dependent on the distance away of the object being looked at, p. It follows that a visual space will be spherical when objects constituted in it are equidistant from the cyclopean eye, in



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particular when they are infinitely distant, a case approximated by looking at the sky. If two physical objects located at different distances away from the cyclopean eye subtend equal solid angles, the closer object will be constituted smaller in visual space than the more distant one, due to the size constancy tendency. Thus, inasmuch as the area on the surface of a sphere projected by a given solid angle is proportional to the square of the radius of the sphere, it follows that a visual space constituted from objects at the greater physical distance will possess a greater radius than a visual space constituted from objects at the closer distance. It also follows that a visual space constituted from objects infinitely distant will possess the greatest possible radius.

The foregoing considerations suggest a possible "external" description of visual space, in which the two-dimensional visual space is embedded in a three-dimensional space, and its shape is determined by means of a function operating in that three-dimensional space. A mapping can then be defined between the physically defined coordinates, 0,4 and p, and a set of coordinates, 0 ,k' and p', used for the external description of visual space, with the correlations between the coordinates being defined as follows:

0' = 09 (4)

(5)

p'=C-1/ap (6)

C is the radius of a visual space in which all of the objects constituted in it are infinitely distant, and a is a proportionality constant. My hyphothesis now is that these transformations constitute an external description of the geometry of visual space, where p' gives the distance to a point in visual space in the direction #',X', from a point in the three-dimensional over-all space not in visual space (the center of the sphere in the special case where p' is the same in all directions). It can be noted that p' approaches 0 when ap approaches 1/C, but inasmuch as there exists a minimum threshold on the distance at which physical objects can be brought into sharp focus-about 6 inches from the eye-this result is consistent with the visual perception process. Also, it is possible that the transformation represented by Equation 6 would need to be more complex in order to fully account for the size constancy tendency, but due to the controversy as to what the precise nature of that tendency is, it would seem to be judicious to keep the transformation as simple as possible for the present. It should also be em- phasize that p' has no immediate phenomenal significance, inasmuch as it itself is not contained in visual space, and thus it does not,



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for example, correspond to perceived visual depth. I shall now turn to my derivation of the internal geometry of

visual space. Inasmuch as this discussion will assume that visual space is not only differentiable, but also locally Euclidean in the metric sense defined earlier, with the exception of a finite number of discontinuities, it should first be examined whether such assumptions are reasonable. It will turn out that there is no way to rigorously prove that visual space has either of these properties, and thus at various stages in the following arguments I shall just assume that it is at least piecewise smooth and hence differentiable, and later that it is locally Euclidean in the metric sense. The following considerations should help to motivate the claim that these assumptions are not unreasonable though.

Inasmuch as I have placed a boundary condition on the values allowed for p,

ap 21/C (7)

it follows that a visual space will be differentiable if the two-dimensional physical manifold represented by f(0,4,t) is. It should be further recognized that the transformations from physical space to visual space given by Equations 4 - 6, are somewhat of an idealization due to the finite spatial acuity of the eye, and thresholds in its ability to distinguish between different physical depths. Thus, the fact that from a microscopic perspective physical surfaces are not even piecewise smooth, and the fact that a very distant physical manifold, such as the night sky, may possess a large number of large physical

ds Surface of Sphere of Radius r

/ Visual Space

Figure 4

discontinuities, do not mitigate the claim that the corresponding sections of visual space are differentiable, due to these visual thresholds. It is only in cases where there are significantly large discontinuities or sharp corners in the spatial arrangements of



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relatively close physical objects, such as in cases where one object partially occludes the view of another, that visual space itself won't be differentiable, or will possess discontinuities. It should also be pointed out that a visual space thus defined seems to be locally Eucli- dean in the metric sense, since as I pointed out when I argued that the global metric structure of visual space is spherical, figures in the small in the space seem introspectively to possess a Euclidean metric.

Now consider the surface of a sphere of radius r centered at the same third-dimensional point as a given visual space (i.e. where p'=0), interesting a given point of the visual space, (thus at this point r = p') as in Figure 4. Working with differentials, the question is how does a deviation from the surface of the sphere by the visual space, dp', which of course is in turn determined by the depth functions f(O,O,t), cause a deviation in the metric structure of the visual space away from that of the sphere. If dl is a differential on the surface of the sphere, and assuming that visual space is differentiable here and also locally Euclidean in the metric sense, the differential in visual space ds will be determined by the Pythagorean theorem as being:

ds2 = dp '2 + d12 (8)

dp' is given by

dp = p dO'+ ap d+' (9) ao'a

and dl2 corresponds to the metric of a sphere, e. g.

d12 = [dO2 + sin2 0d42]r2 (10)

where 0 and 4 are spherical coordinates whose values are coordinated with 0' and 4'. In view of this coordination, I shall substitute 0 and 4 for 0' and A' from here on. It follows that:

dS2 =(P)2 dO2 + 2 a P a P' dda + (a P )2d2 + [dO2 + sin2Odo2]r2 ao ao a) a) (11)

Since we have been working with differentials, p' can be substituted for r, and thus the internal geometry of visual space in terms of the line element is given by:

ds2 = [p '2 + ( P \2]d2 +2 a P a P dOda + [(dP + p 2sin2o],2 ao /ao a a (12)

which corresponds to a deviation on the metric structure of a sphere determined by the depth functions f(0,4,t).



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In order to determine the magnitude of distances in visual space, ds must be integrated, and the degree of difficulty in evaluating the resulting integral will depend upon the nature of the depth function f(O,O,t), which obviously can possess very odd properties. Never- theless, in regions where visual space is continuous, it will be generally possible to evaluate this integral using a computer program, but I shall not attempt any such integrations in this paper.

Having shown how the internal metric structure of visual space follows from the assumptions that it is a two-dimensional space whose global metric structure is spherical, and whose local metric structure possesses a variable curvature governed by a depth function so as to accomodate the size constancy tendency (although of course this global structure is fully determined by integrating the local structure), the question arises as to whether or not other phenomenal aspects of vision can also be explained by such an analysis. For one point, I have already noted that such an analysis predicts the occurrence of discontinuities in visual space when there are sharp discontinuities in the depth function for relatively close objects, and these discontinuities would seem to be capable of explaining the phenomenon of seeing an "edge" when one relatively close physical object partially occludes the view of another. While there may be more to the "edge" phenomenon in some cases, such as when color contrast is involved, the fact that edges are clearly distinguishable when an object is seen against a qualitatively identical background- as demonstrated by many of Julesz' (1971) random dot stereograms-clearly shows that it also involves a geometrical pro- perty, such as that which I have just explained.

Another phenomenal characteristic of vision which my theory seems to b-> at least in principle capable of explaining is the shape constancy tendency; that is the tendency for the visual shapes of objects seen at slants to vary not in accordance with the laws of geometrical perspective, but instead to remain closer to the shapes projected when these objects are seen head-on. For example, a circular object seen at a slant will appear less elliptical than its retinal image, but not circular either, as Thouless (1931 A) and many more recent perceptual studies have shown. The main point to be made here is that if an object is seen at a slant, then one edge of it must be nearer to the viewer than the other edge, and thus both p and p' will possess different values for these two edges, and the object will be constituted in visual space at a slant also. However, due to the inverse nature of Equation 6 relation p to p', the area of the object as constituted on this slant in visual space, will not be sufficiently great so as for the object to retain the same shape as



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when it is seen head-on, and so it will instead be perceived at a compromise shape in between that shape and the one given by the laws of perspective, thus agreeing, at least qualitatively, with Thouless' results.

I shall now turn to a discussion of how my theory can account for the aspect of phenomenal visual depth perception which is enhanced by binocular vision. My claim here is that one can ap- prehend to some extent the internal metric structure of visual space, as for example by noticing the presence of a "corner," the presence of convexity or concavity, or the presence of a discontinuity, e. g. the phenomenal "edge" which I discussed earlier. Certainly visual space does not appear as being "flat" when the objects constituted in it possess different physical depths, and one can use this "lack of flatness," that is the apprehension of an internal curvature, as a phenomenal cue for visual depth.

While I claim that the foregoing phenomenal visual depth cues are present to some extent in monocular vision, and certainly the facts that one can reverse a "Necker cube" or see movies in depth with one eye point towards monocular visual depth perception oc- curring, there is also a great deal of evidence that both the size and shape constancy tendencies are greatly enhanced binocularly, and thus these tendencies will have greater effects then upon the internal metric structure of visual space. For example, Thouless (1931 B) has shown that perceived shape is closer to the shape of the retinal image in monocular vision than it is binocularly, and this effect can be readily confirmed by closing one eye, and then seeing how the shape of a given object constituted in visual space changes. Also, a large number of perceptual psychologists, including Boring (1942) and Chalmers (1952) have shown that the size constancy tendency is much greater in binocular vision than it is monocularly.

It would seem then that the binocular depth cue of retinal disparity enhances monocular phenomenal depth effects by means of increasing the value of the proportionality constant a in Equation 6:

p' =C -1/ap

Thus, due to this enhancement in binocular vision of the effects of the depth function on the metric structure of visual space, that structure is more readily apprehended phenomenally, resulting in enhanced visual depth perception.

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