Gestalt Isomorphism and the Primacy of the Subjective Conscious Experience: A Gestalt Bubble Model

Steven Lehar

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A serious crisis is identified in theories of neurocomputation marked by a persistent disparity between the phenomenological or experiential account of visual perception and the neurophysiological level of description of the visual system. In particular conventional concepts of neural processing offer no explanation for the holistic global aspects of perception identified by Gestalt theory. The problem is paradigmatic, and can be traced to contemporary concepts of the functional role of the neural cell, known as the Neuron Doctrine. In the absence of an alternative neurophysiologically plausible model, I propose a perceptual modeling approach, i.e. to model the percept as experienced subjectively, rather than the objective neurophysiological state of the visual system that supposedly subserves that experience. A Gestalt Bubble model is presented to demonstrate how the elusive Gestalt principles of emergence, reification, and invariance, can be expressed in a quantitative model of the subjective experience of visual consciousness. That model in turn reveals a unique computational strategy underlying visual processing, which is unlike any algorithm devised by man, and certainly unlike the atomistic feed-forward model of neurocomputation offered by the Neuron Doctrine paradigm. The perceptual modeling approach reveals the primary function of perception as that of generating a fully spatial virtual-reality replica of the external world in an internal representation. The common objections to this "picture-in-the-head" concept of perceptual representation are shown to be ill founded.

1 Introduction

Contemporary neuropscience finds itself in a state of serious crisis. For the deeper we probe into the workings of the brain, the farther we seem to get from the ultimate goal of providing a neurophysiological account of the mechanism of conscious experience. Nowhere is this impasse more evident than in the study of visual perception, where the apparently clear and promising trail discovered by Hubel and Wiesel leading up the hierarchy of feature detection from primary to secondary and to higher cortical areas, seems to have reached a theoretical dead-end. Besides the troublesome issues of the noisy stochastic nature of the neural signal, and the very broad tuning of the single cell as a feature detector, the notion of visual processing as a hierarchy of feature detectors seems to suggest some kind of "grandmother cell" model in which the activation of a single cell or a group of cells represents the presence of a particular type of object in the visual field. However it is not at all clear how such a featural description of the visual scene could even be usefully employed in practical interaction with the world. But the most serious indictment of contemporary feature detection theories is that they offer no hint of an explanation for the subjective experience of visual consciousness. For visual experience is more than just an abstract recognition of the features present in the visual field, but those features are vividly experienced as solid three-dimensional objects, bounded by colored surfaces, embedded in a spatial void. There are a number of enigmatic properties of this world of experience identified decades ago by Gestalt theory, suggestive of a holistic emergent computational strategy whose operational principles remain a mystery.

The problem in modern neuroscience is a paradigmatic one, that can be traced to its central concept of neural processing. According to the Neuron Doctrine, neurons behave as quasi-independent processors separated by relatively slow chemical synapses, with strictly segregated input and output functions through the dendrites and axon respectively. It is hard to imagine how such an assembly of independent processors could account for the holistic emergent properties of perception identified by Gestalt theory. In fact the reason why these Gestalt aspects of perception have been largely ignored in recent decades is exactly because they are so difficult to express in terms of the Neuron Doctrine paradigm. The persistent disparity between the neurophysiological and phenomenal levels of description suggests that either the subjective experience of visual consciousness is somehow illusory, or that contemporary concepts of neurocomputation are fundamentally in error, and that neural computation is in fact holistic in nature as suggested by Gestalt theory, and by the properties of the world of visual experience.

Pessoa et al. (1998) make the case for denying the primacy of conscious experience. They argue that although the subjective experience of filling-in phenomena is sometimes accompanied by some neurophysiological correlate, that such an isomorphism between experience and neurophysiology is not logically necessary, but is merely an empirical issue, for, they claim, subjective experiences can occur in the absence of a strictly isomorphic correlate. They argue that although the subjective experience of visual consciousness appears as a "picture" or three-dimensional model of a surrounding world, this does not mean that the information manifest in that experience is necessarily explicitly encoded in the brain. That consciousness is an illusion based on a far more compressed or abbreviated representation, in which percepts such as that of a filled-in colored surface can be explained neurophysiologically by "ignoring an absence" rather than by an explicit point-for-point mapping of the perceived surface in the brain.

In fact, nothing could be farther from the truth. For to propose that the subjective experience of perception can be more enriched and explicit than the corresponding neurophysiological state flies in the face of the materialistic basis of modern neuroscience. Every perceptual experience, whether a simple percept such as a filled-in surface, or a complex percept of a whole scene, has two essential aspects; the subjective experience of the percept, or the subjective percept, and the objective neurophysiological state of the brain that is responsible for that subjective experience, or the objective percept. Like the two faces of a coin, these very different entities can be identified as merely different manifestations of the same underlying structure, viewed from the internal first-person, v.s. the external third-person perspectives. The dual nature of a percept is analogous to the representation of data in a digital computer, where a pattern of voltages present in a particular memory register can represent some meaningful information, either a numerical value, or a brightness value in an image, or a character of text, etc. when viewed from inside the appropriate software environment, while when viewed in external physical terms that same data takes the form of voltages or currents in particular parts of the machine. While we cannot observe phenomenologically the physical medium by which perceptual information is encoded in the brain, we can observe the information encoded in that medium, expressed in terms of the variables of subjective experience. The same principle must also hold in perceptual experience, as originally proposed by Müller (1896), in the psychophysical postulate. Müller argued that since the subjective experience of perception is encoded in some neurophysiological state, therefore the dimensions of conscious experience cannot possibly be any higher than the dimensions of the corresponding neurophysiological state. Therefore it is possible by direct phenomenological observation to determine the dimensions of conscious experience, and thereby to infer the dimensions of the information encoded neurophysiologically in the brain.

The "bottom-up" approach that works upwards from the properties of the individual neuron, and the "top- down" approach that works downwards from the subjective experience of perception are equally valid and complimentary approaches to the investigation of the visual mechanism. Eventually these opposite approaches to the problem must meet somewhere in the middle. However to date, the gap between them remains as large as it ever was. Both approaches are essential to the investigation of biological vision, because each approach offers a view of the problem from its own unique perspective. The disparity between these two views of the visual representation can help focus on exactly those properties which are prominently absent from the conventional neural network view of visual processing.

2 The Gestalt Principle of Isomorphism

The Gestalt principle of isomorphism states explicitly what is merely implied by Müller's psychophysical postulate. For in the case of structured experience, equal dimensionality between the subjective and objective percept implies similarity of structure or form. For example the percept of a filled-in colored surface, whether real or illusory, encodes a separate and distinct experience of color at every distinct spatial location within that surface to a particular resolution. In other words the experience is extended in (at least) two dimensions, and therefore the neurophysiological correlate of that experience must also encode (at least) two dimensions of perceptual information. However the isomorphism required by Gestalt theory is not a strict structural isomorphism, i.e. a literal isomorphism in the physical structure of the representation, but merely a functional isomorphism, i.e. a behavior of the system as if it were physically isomorphic (Köhler 1969, p 92). For the exact geometrical configuration of perceptual storage in the brain cannot be observed phenomenologically any more than the configuration of silicon chips on a memory card can be determined by software examination of the data stored within those chips. Nevertheless the mapping between the stored perceptual image and the corresponding spatial percept must be preserved, as in the case of the digital image also, so that every stored color value is meaningfully related to its rightful place in the spatial percept.

Now a functional isomorphism must also preserve the functional transformations observed in perception, and the exact requirements for a functional isomorphism depend on the functionality in question. For example when a colored surface is perceived to translate coherently across perceived space, the corresponding color values in the perceptual representation of that surface must also translate coherently through the perceptual map. If that memory is discontinuous, like a digital image distributed across separate memory chips on a printed circuit board, then the perceptual representation of that moving surface must jump seamlessly across those discontinuities in order to account for the subjective experience of a continuous translation across the visual field. In other words a functional isomorphism requires a functional connectivity in the representation as if a structurally isomorphic memory were warped, distorted, or fragmented while preserving the functional connectivity between its component parts. Therefore an argument for structural isomorphism is an argument of representational efficiency and simplicity, rather than of logical necessity, whereas a functional isomorphism must hold in order to account for the properties of the perceptual world as observed subjectively.

Furthermore, even for a representation which is functionally but not structurally isomorphic, a description of the spatial transformations evident in perceptual processing are most simply expressed in their structurally isomorphic form, just as a panning or scrolling function in image data is most simply expressed as a spatial shifting of image data, even when that shifting is actually performed in hardware in a non-isomorphic memory array. The principle of isomorphism therefore validates a perceptual modeling approach, i.e. modeling the information apparent in the subjective experience of visual perception as opposed to the neurophysiological mechanism by which that experience is supposedly subserved. In the present discussion therefore, our concern will be chiefly with the functional architecture of perception, i.e. a description of the spatial transformations observed in perception, whatever form those transformations might take in the physical brain, and those transformations are most simply described as if taking place in a physically isomorphic space.

3 The Dimensions of Conscious Experience

The phenomenal world is composed of solid volumes, bounded by colored surfaces, embedded in a spatial void. Every point on every visible surface is perceived at an explicit spatial location in three- dimensions, and all of the visible points on a perceived object like a cube or a sphere, or this page, are perceived simultaneously in the form of continuous surfaces in depth. The perception of multiple transparent surfaces, as well as the experience of empty space between the observer and a visible surface, reveals that multiple depth values can be perceived at any spatial location. I propose to model the information in perception as a computational transformation from a two-dimensional colored image, (or two images in the binocular case) to a three-dimensional volumetric data structure in which every point can encode either the experience of transparency, or the experience of a perceived color at that location. The appearence of a color value at some point in this representational manifold corresponds by definition to the subjective experience of that color at the corresponding point in phenomenal space. If we can describe the generation of this volumetric data structure from the two-dimensional retinal image as a computational transformation, we will have quantified the information processing apparent in perception, as a necessary prerequisite to the search for a neurophysiological mechanism that can perform that same transformation.

It is not necessary for the perceptual model to actually become colored to represent the experience of color at some point in space, any more than the electrical voltages that encode the pixel values of a colored digital image are in any sense colored in themselves. All that is required is that the information content in the representation of color be equal to the informational content of the perceptual experience. Although this solution leaves unresolved the fundamental philosophical issue of the ultimate nature of consciousness, it does allow us to quantify the information encoded in conscious experience, an approach which has served psychology well in the past.

3.1 The Cartesian Theatre and the Homunculus Problem

This "picture-in-the-head" or "Cartesian theatre" concept of visual representation has been criticized on the grounds that there would have to be a miniature observer to view this miniature internal scene, resulting in an infinite regress of observers within observers (Dennett 1991, 1992, O'Regan 1992, Pessoa et al. 1998). In fact there is no need for an internal observer of the scene, since the internal representation is simply a data structure like any other data in a computer, except that this data is expressed in spatial form (Earle 1998, Singh & Hoffman 1998). For if a picture in the head required a homunculus to view it, then the same argument would hold for any other form of information in the brain, which would also require a homunculus to read or interpret that information. In fact any information encoded in the brain needs only to be available to other internal processes rather than to a miniature copy of the whole brain. The fact that the brain does go to the trouble of constructing a full spatial analog of the external environment merely suggests that it has ways to make use of this spatial data. For example field theories of navigation have been proposed (Koffka 1935 pp 42-46, Gibson & Crooks, 1938) in which perceived objects in the perceived environment exert spatial field-like forces of attraction and repulsion, drawing the body towards attractive percepts, and repelling it from aversive percepts, as a spatial computation taking place in a spatial medium. If this idea seems to "fly in the face of what we know about the neural substrates of space perception" (Pessoa et al. 1998 author's response R3.2 p. 789), it is our theories of spatial representation that are in urgent need of revision. For to deny the spatial nature of the perceptual representation in the brain is to deny the spatial nature so clearly evident in the world we perceive around us. To paraphrase Descartes, it is not only the existence of myself that is verified by the fact that I think, but when I experience the vivid spatial presence of objects in the phenomenal world, those objects are certain to exist, at least in the form of a subjective experience, with properties as I experience them to have, i.e. location, spatial extension, color, and shape. I think them, therefore they exist. All that remains uncertain is whether those percepts exist also as objective external objects as well as internal perceptual ones, and whether their perceived properties correspond to objective properties. But their existence and fully spatial nature in my internal perceptual world is beyond question if I experience them so, even if only as a hallucination.

3.2 Bounded Nature of the Perceptual World

The idea of perception as a literal volumetric replica of the world inside your head immediately raises the question of boundedness, i.e. how an explicit spatial representation can encode the infinity of external space in a finite volumetric system. The solution to this problem can be found by inspection. For phenomenological examination reveals that perceived space is not infinite, but is bounded. This can be seen most clearly in the night sky, where the distant stars produce a dome-like percept that presents the stars at equal distance from the observer, and that distance is perceived to be less than infinite. The lower half of perceptual space is usually filled with a percept of the ground underfoot, but it too becomes hemispherical when viewed from far enough above the surface, for example from an airplane or a hot air balloon. The dome of the sky above, and the bowl of the earth below therefore define a finite approximately spherical space (Heelan 1983) that encodes distances out to infinity within a representational structure that is both finite and bounded. While the properties of perceived space are approximately Euclidean near the body, there are peculiar global distortions evident in perceived space that provide clear evidence of the phenomenal world being an internal rather than external entity.

3.3 The Phenomenon of Perspective

Consider the phenomenon of perspective, as seen for example when standing on a long straight road that stretches to the horizon in a straight line in opposite directions. The sides of the road appear to converge to a point both up ahead and back behind, but while converging, they are also perceived to pass to either side of the percipient, and at the same time, the road is perceived to be straight and parallel throughout its entire length. This property of perceived space is so familiar in everyday experience as to seem totally unremarkable. And yet this most prominent violation of Euclidean geometry offers clear evidence for the non-Euclidean nature of perceived space. For the two sides of the road must therefore in some sense be perceived as being bowed, and yet while bowed, they are also perceived as being straight. This can only mean that the space within which we perceive the road to be embedded, must itself be curved. In fact, the observed warping of perceived space is exactly the property that allows the finite representational space to encode an infinite external space. This property is achieved by using a variable representational scale, i.e. the ratio of the physical distance in the perceptual representation relative to the distance in external space that it represents. This scale is observed to vary as a function of distance from the center of our perceived world, such that objects close to the body are encoded at a larger representational scale than objects in the distance, and beyond a certain limiting distance the representational scale, at least in the depth dimension, falls to zero, i.e. objects beyond a certain distance lose all perceptual depth. This is seen for example where the sun and moon and distant mountains appear as if cut out of paper and pasted against the dome of the sky.

Figure 1

The perceptual representation of a man walking down a long straight road. The sides of the road are perceived to be parallel and equidistant throughout their length, and yet at the same time they are perceived to converge to a point both up ahead and behind, and that point is perceived at a distance that is less than infinite. This peculiar violation of Euclidean geometry is perhaps the best evidence for the internal nature of the perceived world, for it shows evidence for the perspective projection due to the optics of the eye, out in the world around us.

The distortion of perceived space is suggested in figure 1 which depicts the perceptual representation for a man walking down a road. The phenomenon of perspective is by definition a transformation defined from a three-dimensional world through a focal point to a two-dimensional surface. The appearence of perspective on the retinal surface therefore is no mystery, and is similar in principle to the image formed by the lens in a camera. What is remarkable in perception is the perspective that is observed not on a two- dimensional surface, but somehow embedded in the three-dimensional space of our perceptual world. Nowhere in the objective world of external reality is there anything that is remotely similar to the phenomenon of perspective as we experience it phenomenologically, where a perspective foreshortening is observed not on a two-dimensional image, but in three dimensions on a solid volumetric object. The appearence of perspective in the three-dimensional world we perceive around us is perhaps the strongest evidence for the internal nature of the world of experience, for it shows that the world that appears to be the source of the light that enters our eye, must actually be downstream of the retina, for it exhibits the traces of perspective distortion imposed by the lens of the eye, although in a completely different form.

This view of perspective offers an explanation for another otherwise paradoxical but familiar property of perceived space whereby more distant objects are perceived to be both smaller, and yet at the same time to be perceived as undiminished in size. This corresponds to the difference in subject's reports depending on whether they are given objective v.s. projective instruction (Coren et al., 1979. p. 500) in how to report their observations, showing that both types of information are available perceptually. This duality in size perception is often described as a cognitive compensation for the foreshortening of perspective, as if the perceptual representation of more distant objects is indeed smaller, but is somehow labeled with the correct size as some kind of symbolic tag representing objective size attached to each object in perception. However this kind of explanation is misleading, for the objective measure of size is not a discrete quantity attached to individual objects, but is more of a continuum, or gradient of difference between objective and projective size, that varies monotonically as a function of distance from the percipient. In other words, this phenomenon is best described as a warping of the space itself within which the objects are represented, so that objects that are warped coherently along with the space in which they are embedded appear undistorted perceptually. The mathematical form of this warping will be discussed in more detail below.

3.4 The Embodied Percipient

This model of spatial representation emphasizes another aspect of perception that is often ignored in models of vision, that our percept of the world includes a percept of our own body within that world, and our body is located at a very special location at the center of that world, and it remains at the center of perceived space even as we move about in the external world. Perception is embodied by its very nature, for the percept of our body is the only thing that gives an objective measure of scale in the world. The little man at the center of the spherical world of perception therefore is not a miniature observer of the internal scene, but is itself a spatial percept, constructed of the same perceptual material as the rest of the spatial scene, for that scene would be incomplete without a replica of the percipient's own body in his perceived world.

4 The Gestalt Properties of Perception

One of the most formidable obstacles facing computational models of the perceptual process is that perception exhibits certain global Gestalt properties such as emergence, reification, multistability, and invariance that are difficult to account for either neurophysiologically, or even in computational terms such as computer algorithms. The ubiquity of these properties in all aspects of perception, as well as their preattentive nature suggests that Gestalt phenomena are fundamental to the nature of the perceptual mechanism. I propose that no useful progress can possibly be made in our understanding of neural processing until the computational principles behind Gestalt theory have been identified.

4.1 Emergence

Figure 2 shows a picture that is familiar in vision circles, for it reveals the principle of emergence in a most compelling form. For those who have never seen this picture before, it appears initially as a random pattern of irregular shapes. A remarkable transformation is observed in this percept as soon as one recognizes the subject of the picture as a dalmation dog in patchy sunlight in the shade of overhanging trees. What is remarkable about this percept is that the dog is perceived so vividly despite the fact that much of its perimeter is missing. Furthermore, visual edges which form a part of the perimeter of the dog are locally indistinguishable from other less significant edges. Therefore any local portion of this image does not contain the information necessary to distinguish significant from insignificant edges.

Figure 2

The dog picture is familiar in vision circles for it demonstrates the principle of emergence in perception. The local regions of this image do not contain sufficient information to distinguish significant form contours from insignificant noisy edges. As soon as the picture is recognized as that of a dog in the dappled sunshine under overhanging trees, the contours of the dog pop out perceptually, filling in visual edges in regions where no edges are present in the input.

Although Gestalt theory did not offer any specific computational mechanism to explain emergence in visual perception, Koffka (1935) suggested a physical analogy of the soap bubble to demonstrate the operational principle behind emergence. The spherical shape of a soap bubble is not encoded in the form of a spherical template or abstract mathematical code, but rather that form emerges from the parallel action of innumerable local forces of surface tension acting in unison. The characteristic feature of emergence is that the final global form is not computed in a single pass, but continuously, like a relaxation to equilibrium in a dynamic system model. In other words the forces acting on the system induce a change in the system configuration, and that change in turn modifies the forces acting on the system. The system configuration and the forces that drive it therefore are changing continuously in time until equilibrium is attained, at which point the system remains in a state of dynamic equilibrium, i.e. its static state belies a dynamic balance of forces ready to spring back into motion as soon as the balance is upset.

4.2 Reification

The Kanizsa figure (Kanzsa 1979) shown in figure 3 A, is one of the most familiar illusions introduced by Gestalt theory. In this figure the triangular configuration is not only recognized as being present in the image, but that triangle is filled-in perceptually, producing visual edges in places where no edges are present in the input, and those edges in turn are observed to bound a uniform triangular region that is brighter than the white background of the figure. Idesawa (1991) and Tse (1999a, 1999b) have extended this concept with a set of even more sophisticated illusions such as those shown in Figure 3 B through D, in which the illusory percept takes the form of a three-dimensional volume. These figures demonstrate that the visual system performs a perceptual reification, i.e. a filling-in of a more complete and explicit perceptual entity based on a less complete visual input. Reification is a general principle of perceptual processing, of which boundary completion and surface filling-in are more specific computational components. The identification of this generative aspect of perception was one of the most significant contributions of Gestalt theory.

Figure 3

A: the Kanizsa triangle. B: Tse's volumetric worm. C: Idesawa's spiky sphere. D: Tse's "sea monster".

4.3 Multistability

A familiar example of multistability in perception is seen in the Necker cube, shown in Figure 4 A. Prolonged viewing of this stimulus results in spontaneous reversals, in which the entire percept is observed to invert in depth. Figure 4 B shows how large regions of the percept invert coherently in bistable fashion. Even more compelling examples of multistability are seen in surrealistic paintings by Salvator Dali, and etchings by Escher, in which large and complex regions of the image are seen to invert perceptually, losing all resemblance to their former appearence (Attneave 1971). The significance for theories of visual processing is that perception cannot be considered as simply a feed-forward processing performed on the visual input to produce a perceptual output, as it is most often characterized in computational models of vision, but rather perception must involve some kind of dynamic process whose stable states represent the final percept.

Figure 4

A: The Necker cube demonstrates multistability in perception. B: This figure shows how large regions of the percept flip coherently between perceptual states.

4.4 Invariance

A central focus of Gestalt theory was the issue of invariance, i.e. how an object, like a square or a triangle, can be recognized regardless of its rotation, translation, or scale, or whatever its contrast polarity against the background, or whether it is depicted solid or in outline form, or whether it is defined in terms of texture, motion, or binocular disparity. This invariance is not restricted to the two-dimensional plane, but is also observed through rotation in depth, and even in invariance to perspective transformation. For example the rectangular shape of a table top is recognized even when its retinal projection is in the form of a trapezoid due to perspective, and yet when viewed from any particular perspective we can still identify the exact contours in the visual field that correspond to the boundaries of the perceived table, to the highest resolution of the visual system. The ease with which these invariances are handled in biological vision suggests that invariance is fundamental to the visual representation.

Our failure to find a neurophysiological explanation for Gestalt phenomena does not suggest that no such explanation exists, only that we must be looking for it in the wrong places. The enigmatic nature of Gestalt phenomena only highlights the importance of the search for a computational mechanism that exhibits these same properties. In the next section I present a model that demonstrates how these Gestalt principles can be expressed in a computational model that is isomorphic with the subjective experience of vision.

5 The Computational Mechanism of Perception

The basic function of visual perception can be described as the transformation from a two-dimensional retinal image, or a pair of images in the binocular case, to a solid three-dimensional percept. Figure 5 A depicts a two-dimensional stimulus that produces a three-dimensional percept of a solid cube complete in three dimensions. For simplicity, a simple line drawing is depicted in the figure, but the argument applies more appropriately to a view of a real cube observed in the world. Every point on every visible surface of the percept is experienced at a specific location in depth, and each of those surfaces is experienced as a planar continuum, with a specific three-dimensional slope in depth. The information in this perceptual experience can therefore be expressed as a three-dimensional model, as suggested in figure 5 B, constructed on the basis of the input image in figure 5 A.

Figure 5

A: A line drawing stimulates B: a volumetric spatial percept with an explicit depth value at every point on every visible surface, and an amodal percept of hidden rear surfaces. C: The central "Y"-vertex from A, which tends to be perceived as a corner in depth. D: A dynamic rod-and-rail model of the emergence of the depth percept in C by relaxation of local constraints.

The transformation from a two-dimensional image space to a three-dimensional perceptual space is known as the inverse optics problem, since the intent is to reverse the optical projection in the eye, in which three-dimensional information from the world is collapsed into a two-dimensional image. However the inverse optics problem is underconstrained, for there are an infinite number of possible three-dimensional configurations that can give rise to the same two-dimensional projection. How does the visual system select from this infinite range of possible percepts to produce the single perceptual interpretation observed phenomenally? The answer to this question is of central significance to understanding the principles behind perception, for it reveals a computational strategy quite unlike anything devised by man, and certainly unlike the algorithmic decision sequences embodied in the paradigm of digital computation. The transformation observed in visual perception gives us the clearest insight into the nature of this unique computational strategy. I propose that the principles of emergence, reification, and multistability are intimately involved in this reconstruction, and that in fact these Gestalt properties are exactly the properties needed for the visual system to address the fundamental ambiguities inherent in reflected light imagery.

The principle behind the perceptual transformation can be expressed in general terms as follows: for any given visual input there is an infinite range of possible configurations of objects in the external world which could have given rise to that same stimulus. The configuration of the stimulus constrains the range of those possible perceptual interpretations to those that line up with the stimulus in the two dimensions of the retinal image. Now although each individual interpretation within that range is equally likely with respect to the stimulus, some of those perceptual alternatives are intrinsically more likely than others, in the sense that they are more typical of objects commonly found in the world. I propose that the perceptual representation has the property that the more likely structural configurations are also more stable in the perceptual representation, and therefore the procedure used by the visual system is to essentially construct or reify all possible interpretations of a visual stimulus in parallel, as constrained by the configuration of the input, and then to select from that range of possible percepts the most stable perceptual configuration by a process of emergence. In other words, perception can be viewed as the computation of the intersection of two sets of constraints, which might be called extrinsic v.s. intrinsic constraints. The extrinsic constraints are those defined by the visual stimulus, whereas the intrinsic constraints are those defined by the structure of the percept. The configuration of the input encodes the extrinsic constraints, while the stability of the perceptual representation encodes the intrinsic constraints.

Arnheim (1969) presents an insightful analysis of this concept, which can be reformulated as follows. Consider (for simplicity) just the central "Y" vertex of figure 5 A depicted in figure 5 C. Arnheim proposes that the extrinsic constraints of inverse optics can be expressed for this stimulus using a rod- and-rail analogy as shown in figure 5 D. The three rods, representing the three edges in the visual input, are constrained in two dimensions to the configuration seen in the input, but are free to slide in depth along four rails. The rods must be elastic between their end-points, so that they can expand and contract in length. By sliding along the rails, the rods can take on any of the infinite three-dimensional configurations corresponding to the two-dimensional input of figure 5 C. For example the final percept could theoretically range from a percept of a convex vertex protruding from the depth of the page, to a concave vertex intruding into the depth of the page, with a continuum of intermediate perceptual states between these limits. There are other possibilities beyond these, for example percepts where each of the three rods is at a different depth and therefore they do not meet in the middle of the stimulus. However these alternative perceptual states are not all equally likely to be experienced. Hochberg & Brooks (1960) showed that the final percept is the one that exhibits the greatest simplicity, or prägnanz. In the case of the vertex of figure 5 C the percept tends to appear as three rods whose ends coincide in depth at the center, and meet at a mutual right angle, defining either a concave or convex corner. This reduces the infinite range of possible configurations to two discrete perceptual states. This constraint can be expressed emergently in the rod and rail model by joining the three rods flexibly at the central vertex, and installing spring forces that tend to hold the three rods at mutual right angles at the vertex. With this mechanism in place to define the intrinsic or structural constraints, the rod-and-rail model becomes a dynamic system that slides in depth along the rails, and this system is bistable between a concave and convex right angled percept, as observed phenomenally in figure 5 C. Although this model reveals the dynamic interaction between intrinsic and extrinsic constraints, this particular analogy is hard-wired to modeling the percept of the triangular vertex of figure 5 C. I will now develop a more general model that operates on this same dynamic principle, but is designed to handle arbitrary input patterns.

5.1 A Gestalt Bubble Model

For the perceptual representation I propose a volumetric block or matrix of dynamic computational elements, as suggested in figure 6 A, each of which can exist in one of two states, transparent or opaque, with opaque state units being active at all points in the volume of perceptual space where a colored surface is experienced. In other words upon viewing a stimulus like figure 5 A, the perceptual representation of this stimulus is modeled as a three-dimensional pattern of opaque state units embedded in the volume of the perceptual matrix in exactly the configuration observed in the subjective percept when viewing figure 5 A, i.e. with opaque-state elements at all points in the volumetric space that are within a perceived surface in three dimensions, as suggested in figure 5 B. All other elements in the block are in the transparent state to represent the experience of the spatial void within which perceived objects are perceived to be embedded. More generally opaque state elements should also encode the subjective dimensions of color, i.e. hue, intensity, and saturation, and intermediate states between transparent and opaque would be required to account for the perception of semi-transparent surfaces, although for now, the discussion will be limited to two states and the monochromatic case. The transformation of perception can now be defined as the turning on of the appropriate pattern of elements in this volumetric representation in response to the visual input, in order to replicate the three-dimensional configuration of surfaces experienced in the subjective percept.

Figure 6

A: The Gestalt Bubble model consisting of a block of dynamic local elements which can be in one of several states. B: The transparent state, no neighborhood interactions. C: The opaque coplanarity state which tends to complete smooth surfaces. D: The opaque orthogonality stsate which tends to complete perceptual corners. E: The opaque occlusion state which tends to complete surface edges.

The perceived surfaces due to a stimulus like 5 A appear to span the structure of the percept defined by the edges in the stimulus, somewhat like a milky bubble surface clinging to a cubical wire frame. Although the featureless portions of the stimulus between the visual edges offer no explicit visual information, a continuous surface is perceived within those regions, as well as across the white background behind the block figure, with a specific depth and surface orientation value encoded explicitly at each point in the percept. This three-dimensional surface interpolation function can be expressed in the perceptual model by assigning every element in the opaque state a surface orientation value in three dimensions, and by defining a dynamic interaction between opaque state units to fill in the region between them with a continuous surface percept. In order to express this process as an emergent one, the dynamics of this surface interpolation function must be defined in terms of local field-like forces analogous to the local forces of surface tension active at any point in a soap bubble. Figure 6 C depicts an opaque state unit representing a local portion of a perceived surface at a specific three-dimensional location and with a specific surface orientation. The planar field of this element, depicted somewhat like a planetary ring in figure 6 C, represents both the perceived surface represented by this element, as well as a field-like influence propagated by that element to adjacent units. This planar field fades smoothly with distance from the center with a Gaussian function. The effect of this field is to recruit adjacent elements within that field of influence to take on a similar state, i.e. to induce transparent state units to switch to the opaque state, and opaque state units to rotate towards a similar surface orientation value. The final state and orientation taken on by any element is computed as a spatial average or weighted sum of the states of neighboring units as communicated through their planar fields of influence, i.e. with the greatest influence from nearby opaque elements in the matrix. The influence is reciprocal between neighboring elements, thereby defining a circular relation as suggested by the principle of emergence. In order to prevent runaway positive feedback and uncontrolled propagation of surface signal, an inhibitory dynamic is also incorporated in order to suppress surface formation out of the plane of the emergent surface, by endowing the local field of each unit with an inhibitory field in order to suppress the opaque state in neighboring elements in all directions outside of the plane of its local field. The mathematical specification of the local field of influence between opaque state units is outlined in greater detail in the appendix. However the intent of the model is expressed more naturally in the global properties as described here, so the details of the local field influences are presented as only one possible implementation of the concept, provided in order to ground this somewhat nebulous idea in more concrete terms.

The global properties of the system should be such that if the elements in the matrix were initially assigned randomly to either the transparent or opaque state, with random surface orientations for opaque- state units, the mutual field-like influences would tend to amplify any group of opaque-state elements whose planar fields happened to be aligned in an approximate plane, and as that plane of active units feeds back on its own activation, the orientations of its elements would conform ever closer to that of the plane, while elements outside of the plane would be suppressed to the transparent state. This would result in the emergence of a single plane of opaque-state units as a dynamic global pattern of activation embedded in the volume of the matrix, and that surface would be able to flex and stretch much like a bubble surface, although unlike a real bubble, this surface is defined not as a physical membrane, but as a dynamic sheet of active elements embedded in the matrix. This volumetric surface interpolation function will now serve as the backdrop for an emergent reconstruction of the spatial percept around a three- dimensional skeleton or framework constructed on the basis of the visual edges in the scene.

5.3 Local Effects of a Visual Edge

A visual edge can be perceived as an object in its own right, like a thin rod or wire surrounded by empty space. More often however an edge is seen as a discontinuity in a surface, either as a corner or fold, or perhaps as an occlusion edge like the outer perimeter of a flat figure viewed against a more distant background. The interaction between a visual edge and a perceived surface can therefore be modeled as follows. The two-dimensional edge from the retinal stimulus projects a different kind of field of influence into the depth dimension of the volumetric matrix, as suggested by the gray shading in figure 6 A, to represent the three-dimensional locus of all possible edges that project to the two-dimensional edge in the image. In other words, this field expresses the inverse optics probability field or extrinsic constraint due to a single visual edge. Wherever this field intersects opaque-state elements in the volume of the matrix, it changes the shape of their local fields of influence from a coplanar interaction to an orthogonal, or corner interaction as suggested by the local force field in figure 6 D. The corner of this field should align parallel to the visual edge, but otherwise remain unconstrained in orientation except by interactions with adjacent opaque units. Visual edges can also denote occlusion, and so opaque-state elements can also exist in an occlusion state, with a coplanarity interaction in one direction only, as suggested by the occlusion field in figure 6 E. Therefore, in the presence of a single visual edge, a local element in the opaque state should have an equal probability of changing into the orthogonality or occlusion state, with the orthogonal or occlusion edge aligned parallel to the inducing visual edge. Elements in the orthogonal state tend to promote orthogonality in adjacent elements along the perceived corner, while elements in the occlusion state promote occlusion along that edge. In other words, an edge will tend to be perceived as a corner or occlusion percept along its entire length, although the whole edge may change state back and forth as a unit in a multistable manner. The appendix presents a more detailed mathematical description of how these orthogonality and occlusion fields might be defined. The presence of the visual edge in figure 6 A therefore tends to crease or break the perceived surface into one of the different possible configurations shown in figure 7 A through D. The final configuration selected by the system would depend not only on the local image region depicted in figure 7, but also on forces from adjacent regions of the image, in order to fuse the orthogonal or occlusion state elements seamlessly into nearby coplanar surface percepts.

Figure 7

A through D: Several possible stable states of the Gestalt Bubble model in response to a single visual edge.

5.4 Global Effects of Configurations of Edges

Visual illusions like the Kanizsa figure shown in figure 3 A suggest that edges in a stimulus that are in a collinear configuration tend to link up in perceptual space to define a larger global edge connecting the local edges. This kind of collinear boundary completion is expressed in this model as a physical process analogous to the propagation of a crack or fold in a physical medium. A visual edge which fades gradually produces a crease in the perceptual medium that tends to propagate outward beyond the edge as suggested in figure 8 A. If two such edges are found in a collinear configuration, the perceptual surface will tend to crease or fold between them as suggested in figure 8 B. This tendency is accentuated if additional evidence from adjacent regions support this configuration. This can be seen in figure 8 C where fading horizontal lines are seen to link up across the figure to create a percept of a folded surface in depth, which would otherwise appear as a regular hexagon, as seen in figure 8 D.

Figure 8

A: Boundary completion in the Gestalt bubble model: A single line ending creates a crease in the perceptual surface. B: Two line endings generate a crease joining them. C. A regular hexagon figure transforms into D: a percept of a folded surface in depth, with the addition of suggestive lines, with the assistance of a global gestalt that is consistent with that perceptual interpretation.

Gestalt theory emphasized the significance of closure as a prominant factor in perceptual segmentation, since an enclosed contour is seen to promote a figure / ground segregation (Koffka 1935 p. 178). For example an outline square tends to be seen as a square surface in front of a background surface that is complete and continuous behind the square, as suggested in the perceptual model depicted in figure 9 A. The problem is that closure is a "gestaltqualität", a quality defined by a global configuration that is difficult to specify in terms of any local featural requirements, especially in the case of irregular or fragmented contours as seen in figure 9 B. In this model an enclosed contour breaks away a piece of the perceptual surface, completing the background amodally behind the occluding foreground figure. In the presence of irregular or fragmented edges the influence of the individual edge fragments act collectively to break the perceptual surface along that contour as suggested in figure 9 C, like the breaking of a physical surface that is weakened along an irregular line of cracks or holes. The final scission of figure from ground is therefore driven not so much by the exact path of the individual irregular edges, as it is by the global configuration of the emergent gestalt.

Figure 9

A: The perception of closure and figure / ground segregation are explained in the Gestalt bubble model exactly as perceived, in this case as a foreground square in front of a background surface that completes behind the square. B: Even irregular and fragmented surfaces produce a figure / ground segregation. C: The perceived boundary of the fragmented figure follows the global emergent gestalt rather than the exact path of individual edges.

5.5 Vertices and Intersections

In the case of vertices or intersections between visual edges, the different edges interact with one another favoring the percept of a single vertex at that point. For example the three edges defining the three-way "Y" vertex shown in figure 5 C promote the percept of a single three-dimensional corner, whose depth profile depends on whether the corner is perceived as convex or concave. In the case of figure 5 A, the cubical percept constrains the central "Y" vertex as a convex rather than a concave trihedral percept. I propose that this dynamic behavior can be implemented using the same kinds of local field-forces described in the appendix to promote mutually orthogonal completion in three dimensions, wherever visual edges meet at an angle in two dimensions. Figure 10 A depicts the three-dimensional influence of the two-dimensional Y-vertex when projected on the front face of the volumetric matrix. Each plane of this three-planed structure promotes the emergence of a corner or occlusion percept at some depth within that plane. But the effects due to these individual edges are not independent. Consider for example, first the vertical edge projecting from the bottom of the vertex. By itself, this edge might produce a folded percept as suggested in figure 10 B, which could occur through a range of depths, and a variety of orientations in depth, and in concave or convex form. But the two angled planes of this percept each intersect the other two fields of influence due to the other two edges of the stimulus, as suggested in figure 10 B, thus favoring the emergence of those edges' perceptual folds at that same depth, resulting in a single trihedral percept at some depth in the volumetric matrix, as suggested in figure 10 C. Any dimension of this percept that is not explicitly specified or constrained by the visual input, remains unconstrained. In other words, the trihedral percept is embedded in the volumetric matrix in such a way that its three component corner percepts are free to slide inward or outward in depth, to rotate through a small range of angles, and to flip in bistable manner between a convex and concave trihedral configuration. The model now expresses the multistability of the rod-and-rail analogy shown in figure 5 D, but in a more generalized form that is no longer hard-wired to the Y-vertex input shown in figure 5 C, but can accommodate any arbitrary configuration of lines in the input image. A local visual feature like an isolated Y-vertex generally exhibits a larger number of stable states, whereas in the context of adjacent features the number of stable solutions is often diminished. This explains why the cubical percept of figure 5 A is stable, while its central Y-vertex alone as shown in figure 5 C is bistable. The fundamental multistability of figure 5 A can be revealed by the addition of a different spatial context, as depicted in figure 10 D.

Figure 10

A: The three-dimensional field of influence due to a two-dimensional Y-vertex projected into the depth dimension of the volumetric matrix. B: Each field of influence, for example the one due to the vertical edge, stimulates a folded surface percept. The folded surface intersects the other fields of influence due to the other two edges, thereby tending to produce a percept of a single corner percept. C: One of many possible emergent surface percepts in response to that stimulus, in the form of a convex trihedral surface percept. D: The percept can also be of a concave trihedral corner, as seen sometimes at the center in this bistable figure.

5.6 Perspective Cues

Perspective cues offer another example of a computation that is inordinately complicated in most models. However in a fully reified spatial model perspective can be computed relatively easily with only a small change in the geometry of the model. Figure 11 A shows a trapezoid stimulus, which has a tendency to be perceived in depth, i.e. the shorter top side tends to be perceived as being the same length as the longer base, but apparently diminished by perspective. Arnheim (1969) suggests a simple distortion to the volumetric model to account for this phenomenon, which can be reformulated as follows. The height and width of the volumetric matrix are diminished as a function of depth, as suggested in figure 11 B, transforming the block shape into a truncated pyramid that tapers in depth. The vertical and horizontal dimensions represented by that space however are not diminished, in other words, the larger front face and the smaller rear face of the volumetric structure represent equal areas in perceived space, by unequal areas in representational space, as suggested by the converging grid lines in the figure. All of the spatial interactions described above, for example the collinear propagation of corner and occlusion percepts, would be similarly distorted in this space. Even the angular measure of orthogonality is distorted somewhat by this transformation. For example the perceived cube depicted in the solid volume of figure 11 B is metrically shrunken in height and width as a function of depth, but since this shrinking is in the same proportion as the shrinking of the space itself, the depicted irregular cube represents a percept of a regular cube with equal sides and orthogonal faces. The propagation of the field of influence in depth due to a two-dimensional visual input on the other hand does not shrink with depth. A projection of the trapezoid of figure 11 A would occur in this model as depicted in figure 11 C, projecting the trapezoidal form backward in parallel, independent of the convergence of the space around it. The shaded surfaces in figure 11 C therefore represent the locus of all possible spatial interpretations of the two-dimensional trapezoid stimulus of figure 11 A, or the extrinsic constraints for the spatial percept due to this stimulus. For example one possible perceptual interpretation is of a trapezoid parallel to the plane of the page, which can be perceived to be either nearer or farther in depth, but since the size scale shrinks as a function of depth, the percept will be experienced as larger in absolute size (as measured against the shrunken spatial scale) when perceived as farther away, and as smaller in absolute size (as measured against the expanded scale) when perceived to be closer in depth. This corresponds to the phenomenon known as Emmert's Law (Coren et al. 1979), whereby a retinal after-image appears larger when viewed against a distant background than when viewed against a nearer background. Now there are also an infinite number of alternative perceptual interpretations of the trapezoidal stimulus, some of which are depicted by the dark shaded lines of figure 11 D. Most of these alternative percepts are geometrically irregular, representing figures with unequal sides and odd angles. But of all these possibilities, there is one special case, depicted in black lines in figure 11 D, in which the convergence of the sides of the perceived form happens to coincide exactly with the convergence of the space itself. In other words, this particular percept represents a regular rectangle viewed in perspective, with parallel sides and right angled corners, whose nearer (bottom) and farther (top) horizontal edges are the same length in the distorted perceptual space. While this rectangular percept represents the most stable interpretation, other possible interpretations might be suggested by different contexts. The most significant feature of this concept of perceptual processing is that the result of the computation is expressed not in the form of abstract variables encoding the depth and slope of the perceived rectangle, but in the form of an explicit three- dimensional replica of the surface as it is perceived to exist in the world.

Figure 11

A: A trapezoidal stimulus that tends to be perceived as a rectangle viewed in perspective. B: The perspective modified spatial representation whose dimensions are shrunken in height and bredth as a function of depth. C: The parallel projection of a field of influence into depth of the two-dimensional trapezoidal stimulus. D: Several possible perceptual interpretations of the trapezoidal stimulus, one of which (depicted in black outline) represents a regular rectangle viewed in perspective, because the convergence of its sides exactly matches the convergence of the space itself.

5.7 Bounding the Representation

An explicit volumetric representation of perceived space as proposed here must necessarily be bounded in some way in order to allow a finite representational space to map to the infinity of external space, as suggested in figure 2. The nonlinear compression of the depth dimension observed in phenomenal space can be modeled mathematically with a vergence measure, which maps the infinity of Euclidean distance into a finite bounded range, as suggested in figure 12 A.This produces a representation reminiscent of museum diaramas, like the one depicted in figure 12 B, where objects in the foreground are represented in full depth, but the depth dimension gets increasingly compressed with distance from the viewer, eventually collapsing into a flat plane corresponding to the background. This vergence measure is presented here merely as a nonlinear compression of depth in a monocular spatial representation, as opposed to a real vergence value measured in a binocular system, although this system could of course serve both purposes in biological vision. Assuming unit separation between the eyes in a binocular system, this compression is defined by the equation

n = 2 atan(1/2r)

where n is the vergence measure of depth, and r is the Euclidean range, or distance in depth. Actually, since vergence is large at short range and smaller at long range, it is actually the "p-compliment" vergence measure r that is used in the representation, where r = (p-n), and r ranges from 0 at r = 0, to p at r = infinity.

Figure 12

A: A vergence representation maps infinite distance into a finite range. B: This produces a mapping reminiscent of a museum diarama. C: The compressed reference grid in this compressed space defines intervals that are perceived to be of uniform size.

What does this kind of compression mean in an isomorphic representation? If the perceptual frame of reference is compressed along with the objects in that space, then the compression need not be perceptually apparent. Figure 12 C depicts this kind of compressed reference grid. The unequal intervals between adjacent grid lines in depth define intervals that are perceived to be of equal length, so the flattened cubes defined by the distorted grid would appear perceptually as regular cubes, of equal height, breadth, and depth. This compression of the reference grid to match the compression of space would, in a mathematical system with infinite resolution, completely conceal the compression from the percipient. In a real physical implementation there are two effects of this compression that would remain apparent perceptually, due to the fact that the spatial matrix itself would have to have a finite perceptual resolution. The resolution of depth within this space is reduced as a function of depth, and beyond a certain limiting depth, all objects are perceived to be flattened into two dimensions, with zero extent in depth. This phenomenon is observed perceptually, where the sun, moon, and distant mountains appear as if they are pasted against the flat dome of the sky.

The other two dimensions of space can also be bounded by converting the x and y of Euclidean space into azimuth and elevation angles, a and b, producing an angle / angle / vergence representation, as shown in figure 13 A. Mathematically this transformation converts the point P(a,b,r) in polar coordinates to point Q(a,b,r) in this bounded spherical representation. In other words, azimuth and elevation angles are preserved by this transformation while the radial distance in depth r is compressed to the vergence representation r as described above. This spherical coordinate system has the ecological advantage that the space near the body is represented at the highest spatial resolution, whereas the less important more distant parts of space are represented at lower resolution. All depths beyond a certain radial distance are mapped to the surface of the representation which corresponds to perceptual infinity.

Figure 13

A: An azimuth / elevation / vergence representation maps the infinity of three-dimensional Euclidean space into a finite perceptual space. B: The deformation of the infinite Cartesian grid caused by the perspective transformation of the azimuth / elevation / vergence representation. C: A view of a man walking down a road represented in the perspective distorted space. D: A section of the spherical space depicted in the same format as the perspective space shown in figure 11.

The mathematical form of this distortion is depicted in figure 13 B, where the distorted grid depicts the perceptual representation of an infinite Cartesian grid with horizontal and vertical grid lines spaced at equal intervals. This geometrical transformation from the infinite Cartesian grid actually represents a unique kind of perspective transformation on the Cartesian grid. In other words, the transformed space looks like a perspective view of a Cartesian grid when viewed from inside, with all parallel lines converging to a point in opposite directions. The significance of this observation is that by mapping space into a perspective-distorted grid, the distortion of perspective is removed, in the same way that plotting log data on a log plot removes the logarithmic component of the data. Figure 13 C shows how this space would represent the perceptual experience of a man walking down a road. If the distorted reference grid of figure 13 B is used to measure lines and distances in figure 13 C, the bowed line of the road on which the man is walking is aligned with the bowed reference grid and therefore is perceived to be straight. Therefore the distortion of straight lines into curves in the perceptual representation is not immediately apparent to the percipient, because they are perceived to be straight. However in a global sense there are peculiar distortions that are apparent to the percipient caused by this deformation of Euclidean space. For while the sides of the road are perceived to be parallel, they are also perceived to meet at a point on the horizon. The fact that two lines can be perceived to be both straight and parallel and yet to converge to a point both in front and behind the percipient indicates that our internal representation itself must be curved. The proposed representation of space has exactly this property. Parallel lines do not extend to infinity but meet at a point beyond which they are no longer represented. Likewise the vertical walls of the houses in figure 13 C bow outwards away from the observer, but in doing so they follow the curvature of the reference lines in the grid of figure 13 B, and are therefore perceived as being both straight, and vertical. Since curved lines in this spherical representation represent straight lines in external space, all of the spatial interactions discussed in the previous section, including the coplanar interactions, and collinear creasing of perceived surfaces, must follow the grain or curvature of collinearity defined within this distorted coordinate system. The distance scale encoded in the grid of figure 13 B replaces the regularly spaced Cartesian grid by a nonlinear collapsing grid whose intervals are spaced ever closer as they approach perceptual infinity but nevertheless represent equal intervals in external space. This nonlinear collapsing scale thereby provides an objective measure of distance in the perspective-distorted perceptual world. For example the houses in figure 13 C would be perceived to be approximately the same size and depth, although the farther house is experienced at a lower perceptual resolution.

Figure 13 D depicts how a slice of Euclidean space of fixed height and width would appear in the perceptual sphere, extending to perceptual infinity in one direction, like a slice cut from the spherical representation of figure 13 C. This slice is similar to the truncated pyramid shape shown in figure 11 B, with the difference that the horizontal and vertical scale of representational space diminishes in a nonlinear fashion as a function of distance in depth. In other words, the sides of the pyramid in figure 13 B converge in curves rather than in straight lines, and the pyramid is no longer truncated, but extends in depth all the way to the vanishing point at representational infinity. An input image is projected into this spherical space using the same principles as before.

5.8 Brain Anchoring

One of the most disturbing properties of the phenomenal world for models of the perceptual mechanism involves the subjective impression that the phenomenal world rotates relative to our perceived head as our head turns relative to the world, and that objects in perception are observed to translate and rotate while maintaining their perceived structural integrity and recognized identity. This suggests that the internal representation of external objects and surfaces is not anchored to the tissue of the brain, as suggested by current concepts of neural representation, but that perceptual structures are free to rotate and translate coherently relative to the neural substrate, as suggested in Köhler's field theory (Köhler & Held 1947). This issue of brain anchoring is so troublesome that it is often cited as a counter-argument for an isomorphic representation, since it is difficult to conceive of the solid spatial percept of the surrounding world having to be reconstructed anew in all its rich spatial detail with every turn of the head (Gibson 1979, O'Regan 1992). However an argument can be made for the adaptive value of a neural representation of the external world that could break free of the tissue of the sensory or cortical surface in order to lock on to the more meaningful coordinates of the external world, if only a plausible mechanism could be conceived to achieve this useful property.

Even in the absence of a neural model with the required properties, the invariance property can be encoded in a perceptual model. In the case of rotation invariance, this property can be quantified by proposing that the spatial structure of a perceived object and its orientation are encoded as separable variables. This would allow the structural representation to be updated progressively from successive views of an object that is rotating through a range of orientations. However the rotation invariance property does not mean that the encoded form has no defined orientation, but rather that the perceived form is presented to consciousness at the orientation and rate of rotation that the external object is currently perceived to possess. In other words, when viewing a rotating object, like a person doing a cartwheel, or a skater spinning about her vertical axis, every part of that visual stimulus is used to update the corresponding part of the internal percept even as that percept rotates within the perceptual manifold to remain in synchrony with the rotation of the external object. The perceptual model need not explain how this invariance is achieved neurophysiologically, it must merely express the invariance property computationally, regardless of the "neural plausibility" or computational efficiency of that calculation. For the perceptual model is more a quantitative description of the phenomenon rather than a theory of neurocomputation. The property of translation invariance can be similarly quantified in the representation by proposing that the structural representation can be calculated from a stimulus that is translating across the sensory surface, to update a perceptual effigy that translates with respect to the representational manifold, while maintaining its structural integrity. This accounts for the structural constancy of the perceived world as it scrolls past a percipient walking through a scene, with each element of that scene following the proper curved perspective lines as depicted in figure 1, expanding outwards from a point up ahead, and collapsing back to a point behind, as would be seen in a cartoon movie rendition of figure 1.

The fundamental invariance of such a representation offers an explanation for another property of visual perception, i.e. the way that the individual impressions left by each visual saccade are observed to appear phenomenally at the appropriate location within the global framework of visual space depending on the direction of gaze. This property can be quantified in the perceptual model as follows. The two- dimensional image from the spherical surface of the retina is copied onto a spherical surface in front of the eyeball of the perceptual effigy, from whence the image is projected radially outwards in an expanding cone into the depth dimension of the internal perceptual world as suggested in figure 14, as an inverse analog of the cone of light received from the world by the eye. Eye, head, and body orientation relative to the external world are taken into account in order to direct the visual projection of the retinal image into the appropriate sector of perceived space, as determined from proprioceptive and kinesthetic sensations in order to update the image of the body configuration relative to external space. The percept of the surrounding environment therefore serves as a kind of three-dimensional frame buffer expressed in global coordinates, that accumulates the information gathered in successive visual saccades and maintains an image of that external environment in the proper orientation relative to a spatial model of the body, compensating for body rotations or translations through the world. Portions of the environment that have not been updated recently gradually fade from perceptual memory, which is why it is easy to bump one's head after bending for some time under an overhanging shelf, or why it is possible to advance only a few steps safely after closing one's eyes while walking.

Figure 14

The image from the retina is projected into the perceptual sphere from the center outward in the direction of gaze, as an inverse analog of the cone of light that enters the eye in the external world, taking into account eye, head, and body orientation in order to update the appropriate portion of perceptual space.

6 Discussion

The picture of visual processing revealed by the phenomenological approach is radically different from the picture revealed by neurophysiological studies. In fact, the computational transformations observed phenomenologically are implausible in terms of contemporary concepts of neurocomputation and even in terms of computer algorithms. However the history of psychology is replete with examples of plausibility arguments based on the limited technology of the time which were later invalidated by the emergence of new technologies. The outstanding achievements of modern technology, especially in the field of information processing systems, might seem to justify our confidence to judge the plausibility of proposed processing algorithms. And yet, despite the remarkable capabilities of modern computers, there remain certain classes of problems that appear to be fundamentally beyond the capacity of the digital computer. In fact the very problems that are most difficult for computers to address, such as extraction of spatial structure from a visual scene especially in the presence of attached shadows, cast shadows, specular reflections, occlusions, perspective distortions, as well as the problems of navigation in a natural environment, etc. are problems that are routinely handled by biological vision systems, even those of simpler animals. On the other hand, the kinds of problems that are easily solved by computers, such as perfect recall of vast quantities of meaningless data, perfect memory over indefinite periods, detection of the tiniest variation in otherwise identical data, exact repeatability of even the most complex computations, are the kinds of problems that are inordinately difficult for biological intelligence, even that of the most complex of animals. It is therefore safe to assume that the computational principles of biological vision are fundamentally different from those of digital computation, and therefore plausibility arguments predicated on contemporary concepts of what is computable are not applicable to biological vision. If we begin with the assumption that our most basic concepts of neurocomputation are incomplete, the evidence for a Gestalt Bubble model of perceptual processing becomes overwhelming.

The phenomena of hallucinations and dreams demonstrate that the mind is capable of generating complete spatial percepts of the world, including a percept of the body and the space around it (Revonsuo 1995). It is unlikely that this remarkable capacity is used only to create such illusory percepts. More likely, dreams and hallucinations reveal the capabilities of an imaging system that is normally driven by the sensory input, generating perceptual constructs that are coupled to external reality.

Studies of mental imagery (Kosslyn 1980, 1994) have characterized the properties of this imaging capacity, and confirmed the three-dimensional nature of the encoding and processing of mental imagery. Pinker (1980) shows that the scanning time between objects in a remembered three-dimensional scene increases linearly with increasing distance between objects in three dimensions. Shepard & Metzler. (1971) show that the time for rotation of mental images is proportional to the angle through which they are rotated. Kosslyn shows that it takes time to expand the size of mental images, and that smaller mental images are more difficult to scrutinize (Kosslyn 1975). As unexpected as these findings may seem for theorists of neural representation, they are perfectly consistent with the subjective experience of mental imagery. On the basis of these findings, Pinker (1988) derived a volumetric spatial medium to account for the observed properties of mental image manipulation which is very similar to the model proposed here, i.e. with a volumetric azimuth/elevation coordinate system that is addressable both in subjective viewer-centered, and objective viewer-independent coordinates, and with a compressive depth scale.

The phenomenon of hemi-neglect (Kolb & Whishaw 1996) reveals the effects of damage to the spatial representation, destroying the capacity to represent spatial percepts in one half of phenomenal space. Such patients are not simply blind to objects to one side, but are blind to the very existence of a space in that direction as a potential holder of objects. For example, neglect patients will typically eat food only from the right half of their plate, and express surprise at the unexpected appearance of more food when their plate is rotated 180 degrees. This condition even persists when the patient is cognitively aware of their deficit (Sacks 1985). Bisiach et al. (1978,1981) show how this condition can also impair mental imaging ability. They describe a neglect patient who, when instructed to recall a familiar scene viewed from a certain direction, can recall only objects from the right half of his remembered space. When instructed to mentally turn around and face in the opposite direction, the patient now recalls only objects from the other side of the scene, that now fall in the right half of his mental image space. The condition of hemi-neglect therefore suggests damage to the left half of a three-dimensional imaging mechanism that is used both for perception and for the generation of mental imagery. Note that hemi-neglect also includes a neglect of the left side of the body, which is consistent with the fact that the body percept is included as an integral part of the perceptual representation. Whatever the physiological reality behind the phenomenon of hemi-neglect, the Gestalt Bubble model offers at least a concrete description of this otherwise paradoxical phenomenon.

The idea that this spatial imaging system employs an explicit volumetric spatial representation is suggested by the fact that disparity tuned cells have been found in the cortex (Barlow et al. 1967), as predicted by the Projection Field Theory of binocular vision (Kaufman 1974, Boring 1933, Charnwood 1951, Marr & Poggio. 1976, Julesz 1971), which is itself a volumetric model. Psychophysical evidence for a volumetric representation comes from the fact that perceived objects in depth exhibit attraction and repulsion in depth (Westheimer & Levi. 1987, Mitchison 1993) in a manner that is suggestive of a short- range attraction and longer-range repulsion in depth, analogous to the center-surround processing in the retina. Brookes & Stevens. (1989) discuss the analogy between brightness and depth perception, and show that a number of brightness illusions that have been attributed to such center-surround processing have corresponding illusions in depth. Similarly, Anstis & Howard. (1978) have demonstrated a Craik- O'Brien-Cornsweet illusion in depth by cutting the near surface of a block of wood with a depth profile matching the brightness cusp of the brightness illusion, resulting in an illusory percept of a difference in depth of the surfaces on either side of the cusp. As in the brightness illusion, therefore, the depth difference at the cusp appears to propagate a perceptual influence out to the ends of the block, suggesting a spatial diffusion of depth percept between depth edges.

The many manifestations of constancy in perception have always posed a serious challenge for theories of perception because they reveal that the percept exhibits properties of the distal object rather than the proximal stimulus, or pattern of stimulation on the sensory surface. The Gestalt Bubble model explains this by the fact that the information encoded in the internal perceptual representation itself reflects the properties of the distal object rather than the proximal stimulus. Size constancy is explained by the fact that objects perceived to be more distant are represented closer to the outer surface of the perceptual sphere, where the collapsing reference grid corrects for the shrinkage of the retinal image due to perspective. An object perceived to be receding in depth therefore is expected perceptually to shrink in retinal size along with the shrinking of the grid in depth, and conversely, shrinking objects tend to be perceived as receding. Rock & Brosgole. (1964), show that perceptual grouping by proximity is determined not by proximity in the two-dimensional retinal projection of the figure, but rather by the three-dimensional perceptual interpretation. A similar finding is shown by Green & Odum (1986). Shape constancy is exemplified by the fact that a rectangle seen in perspective is not perceived as a trapezoid, as its retinal image would suggest. The Müller-Lyer and Ponzo illusions are explained in similar fashion (Tausch 1954, Gregory 1963, Gillam 1971, 1980), the converging lines in those figures suggesting a surface sloping in depth, so that features near the converging ends are measured against a more compressed reference grid than the corresponding feature near the diverging ends of those lines.

Several researchers have presented psychophysical evidence for a spatial interpolation in depth, which is difficult to account for except with a volumetric representation in which the interpolation is computed explicitly in depth (Attneave 1982). Kellman et al. (1996) have demonstrated a coplanar completion of perceived surfaces in depth in a manner analogous to the collinear completion in the Kanizsa figure. Barrow & Tenenbaum. (1981, p. 94 and Figure 6.1) show how a two-dimensional wire-frame outline held in front of a dynamic random noise pattern stimulates a three-dimensional surface percept spanning the outline like a soap film, and that perceived surface undergoes a Necker reversal together with the reversal of the perimeter wire. Ware & Kennedy. (1978) have shown that a three-dimensional rendition of the Ehrenstein illusion constructed of a set of rods converging on a circular hole, creates a three-dimensional version of the illusion that is perceived as a spatial structure in depth, even when rotated out of the fronto- parallel plane, complete with a perception of brightness at the center of the figure. This illusory percept appears to hang in space like a faintly glowing disk in depth, reminiscent of the neon color spreading phenomenon. A similar effect can be achieved with a three-dimensional rendition of the Kanizsa figure. If the Ehrenstein and Kanizsa figures are explained by spatial interpolation, then the corresponding three- dimensional versions of these illusions must involve a volumetric computational matrix to perform the interpolation in depth.

Collett (1985) has investigated the interaction between monocular and binocular perception using stereoscopically presented line drawings in which some features are presented only monocularly, i.e. their depth information is unspecified. Collett shows that such features tend to appear perceptually at the same depth as adjacent binocularly specified features, as if under the influence of an attractive force in depth generated by the binocular feature. In ambiguous cases the percept is often multi-stable, jumping back and forth in depth, especially when monocular perspective cues conflict with the binocular disparity information. The perceived depth of the monocularly specified surfaces is measured psychophysically using a three-dimensional disparity-specified cursor, whose depth is adjusted by the subject to match the depth of the perceived surface at that point. Subjects report a curious interaction between the cursor and the perceived surface, which is observed to flex in depth towards the cursor at small disparity differences, in the manner of the attraction and repulsion in depth reported by Westheimer & Levi. (1987). This dynamic influence is suggestive of a grouping by proximity mechanism, expressed as a field-like attraction between perceived features in depth, and the flexing of the perceived surface near the 3-D cursor, as well as the multistability in the presence of conflicting perspective and disparity cues, are suggestive of a Gestalt Bubble model.

Carman & Welch. (1992) employ a similar cursor to measure the perceived depth of three-dimensional illusory surfaces seen in Kanizsa figure stereograms, whose inducing edges are tilted in depth in a variety of configurations, as shown in Figure 15 A. Note how the illusory surface completes in depth by coplanar interpolation defining a smooth curving surface. The subjects in this experiment also reported a flexing of the perceived surface in depth near the disparity-defined cursor. Equally interesting is the "port hole" illusion seen in the reverse-disparity version of this figure, where the circular completion of the port holes generates an ambiguous unstable semi-transparent percept at the center of the figure that is characteristic of the Gestalt Bubble model. Kellman & Shipley (1991) and Idesawa (1991) report the emergence of more complex illusory surfaces in depth, using similar illusory stereogram stimuli as shown in Figure 15 B and C. It is difficult to deny the reality of a precise high-resolution spatial interpolation mechanism in the face of these compelling illusory percepts. Whatever the neurophysiological basis of these phenomena, the Gestalt Bubble model offers a mathematical framework for a precise description of the information encoded in these elaborate spatial percepts, independent of the confounding factor of neurophysiological considerations.

Figure 15

Perceptual interpolation in depth in illusory figure stereograms, adapted from A: Carman et al. (1992), B: Kellman et al. (1991), and C: Idesawa (1991). Opposite disparity percepts are achieved by binocular fusion of either the first and second, or the second and third columns of the figure.

The sophistication of the perceptual reification capacity is revealed by the apparent motion phenomenon (Coren et al. 1979) which, in its simplest form consists of a pair of alternately flashing lights, that generates a percept of a single light moving back and forth between the flashing stimuli. With more complex variations of the stimulus, the illusory percept is observed to change color or shape in mid-flight, to carry illusory contours, or to carry a texture region bounded by an illusory contour between the alternately flashing stimuli (Coren et al. 1979). Most pertinent to the discussion of a spatial representation is the fact that the illusory percept is observed to make excursions into the third dimension when that produces a simpler percept. For example if an obstacle is placed between the flashing stimuli so as to block the path between them, the percept is observed to pass either in front of, or behind the obstacle in depth. Similarly, if the two flashing stimuli are in the shape of angular features like a "<" and ">" shape, this angle is observed to rotate in depth between the flashing stimuli, preserving a percept of a rigid rotation in depth, in preference to a morphological deformation in two dimensions. The fact that the percept transitions so readily into depth suggests the fundamental nature of the depth dimension for perception.

While the apparent motion effects reify whole perceptual gestalts, the elements of this reification, such as the field-like diffusion of perceived surface properties, are seen in such diverse phenomena as the perceptual filling-in of the Kanizsa figure (Takeichi et al. 1992), the Craik-O'Brien-Cornsweet effect (Cornsweet 1970), the neon color spreading effect (Bressan 1993), the filling-in of the blind spot (Ramachandran 1992), color bleeding due to retinal stabilization (Heckenmuller 1965, Yarbus 1967), the motion capture effect (Ramachandran & Anstis. 1986), and the aperture problem in motion perception (Movshon et al. 1986). In all of these phenomena, a perceived surface property (brightness, transparency, color, motion, etc.) is observed to spread from a localized origin, not into a fuzzy ill-defined region, but rather, into a sharply bounded region containing a homogeneous perceptual quality, and this filling-in occurs as readily in depth in a perspective view as in the frontoparallel plane. The time has come to recognize that these phenomena do not represent exceptional or special cases, nor are they illusory in the sense of lacking a neurophysiological counterpart. Rather, these phenomena reveal a general principle of neurocomputation that is ubiquitous in biological vision.

Evidence for the spherical nature of perceived space dates back to observations by Helmholtz (1925). A subject in a dark room is presented with a horizontal line of point-lights at eye level in the frontoparallel plane, and instructed to adjust their displacement in depth, one by one, until they are perceived to lie in a straight line in depth. The result is a line of lights that curves inwards towards the observer, the amount of curvature being a function of the distance of the line of lights from the observer. Helmholtz recognized this phenomenon as evidence of the non-Euclidean nature of perceived space. The Hillebrand- Blumenfeld alley experiments (Hillebrand 1902, Blumenfeld 1913) extended this work with different configurations of lights, and mathematical analysis of the results (Luneburg 1950, Blank 1958) characterized the nature of perceived space as Riemannian with constant Gaussian curvature (see Graham 1965 and Foley 1978 for a review). In other words, perceived space bows outward from the observer, with the greatest distortion observed proximal to the body, as suggested by the Gestalt Bubble model. Heelan (1983) presents a more modern formulation of the hyperbolic model of perceived space, and provides further supporting evidence from art and illusion.

It is perhaps too early to say definitively whether the model presented here can be formulated to address all of the phenomena outlined above. What is becoming increasingly clear however is the inadequacy of the conventional feed-forward abstraction approach to account for these phenomena, and that therefore novel and unconventional approaches to the problem should be given serious consideration. The general solution offered by the Gestalt Bubble model to all of these problems in perception is that the internal perceptual representation encodes properties of the distal object rather than of the proximal stimulus, that the computations of spatial perception are most easily performed in a fully spatial matrix, in a manner consistent with the subjective experience of perception.

7 Conclusion

I have presented an elaborate model of perception that incorporates many of the concepts and principles introduced by the original Gestalt movement. While the actual mechanisms of the proposed model remain somewhat vague and poorly specified, this is not a model that makes no predictions. Indeed this model, even in its present general form makes the following very specific predictions:

These "predictions" are so immediately manifest in the subjective experience of perception that they need hardly be tested psychophysically. And yet curiously, these most obvious properties of perception have been systematically ignored by neural modelers, even though the central significance of these phenomena was highlighted decades ago by the Gestaltists. There are two reasons why these prominent aspects of perception have been consistently ignored. The first results from the outstanding success of the single-cell recording technique, which has shifted our theoretical emphasis from field-like theories of whole aspects of perception, to point-like theories of the elements of neural computation. Like the classical Introspectionists, who refused to acknowledge perceptual experiences that were inconsistent with their preconceived notions of sensory representation, the Neuroreductionists of today refuse to consider aspects of perception that are inconsistent with current theories of neural computation, and some of them are even prepared to deny consciousness itself in a heroic attempt to save the sinking paradigm.

There is another factor that has made it possible to ignore these most salient aspects of perception, which is that perceptual entities, such as the solid volumes and empty spaces we perceive around us, are easily confused with real objects and spaces in the objective external world. The illusion of perception is so compelling that we mistake the percept of the world for the real world itself. And yet this Naïve Realist view that we can somehow perceive the world directly, is inconsistent with the physics of perception. If perception is a consequence of neural processing of the sensory input, a percept cannot in principle escape the confines of our head to appear in the world around us, any more than a computation in a digital computer can escape the confines of the computer. We cannot therefore in principle have direct experience of objects in the world itself, but only of the internal effigies of those objects generated by mental processes. The world we see around us therefore can only be an elaborate, though very compelling illusion, which must in reality correspond to perceptual data structures and processes occurring actually within our own head. As soon as we examine the world we see around us, not as a physical scientist observing the physical world, but as a perceptual scientist observing a rich and complex internal percept, only then does the rich spatial nature of perceptual processing become immediately apparent. It was this central insight into the illusion of consciousness that formed the key inspiration of the Gestalt movement, from which all of their other ideas were developed. The central message of Gestalt theory therefore is that the primary function of perceptual processing is the generation of a miniature, virtual-reality replica of the external world inside our head, and that the world we see around us is not the real external world, but is exactly that miniature internal replica. It is only in this context that the elaborate model presented here begins to seem plausible.


The Coplanarity Field

The mathematical form of the coplanarity interaction field can be described as follows. Consider the field strength F due to an element in the opaque state at some point in the volume of the spatial matrix, with a certain surface orientation, depicted in figure 16 A as a vector, representing the normal to the surface encoded by that element. The strength of the field F should peak within the plane at right angles to this normal vector (depicted as a circle in figure 16 A) as defined in polar coordinates by the function Fa = sin(a), where a is the angle between the surface normal and some point in the field, that ranges from zero, parallel to the normal vector, to p, in the opposite direction. The sine function peaks at a = p/2, as shown in Figure 16 B, producing an equatorial belt around the normal vector as suggested schematically in cross-section in Figure 16 C, where the gray shading represents the strength of the field. The strength of the field should actually decay with distance from the element, for example with an exponential decay function, as defined by the equation Far = e-r2 sin(a) as shown in Figure 16 D, where r is the radial distance from the element. This produces a fading equatorial band, as suggested schematically in cross- section in Figure 16 E. The equatorial belt of the function described so far would be rather fat, resulting in a lax or fuzzy coplanarity constraint, but the constraint can be stiffened by raising the sine to some positive power P, producing the equation Far = e-r2 sin(a)P which will produce a sharper peak in the function as shown in Figure 16 F, producing a sharper in-plane field depicted schematically in cross-section in Figure 16 G. In order to control runaway positive feedback and suppress the uncontrolled proliferation of surfaces, the field function should be normalized, in order to project inhibition in directions outside the equatorial plane. This can be achieved with the equation Far = e-r2 2 sin(a)P - 1 which has the effect of shifting the equatorial function half way into the negative region as shown in Figure 16 H, producing the field suggested in cross section in Figure 16 I.

Figure 16

Progressive construction of the equation for the coplanarity field from one element to another, as described in the text.

The field described so far is un-oriented, i.e. it has a magnitude, but no direction at any sample point (r,a). What is actually required is a field with a direction, that would have maximal influence on adjacent elements that are oriented parallel to it, i.e. elements that are coplanar with it in both position and orientation. We can describe this orientation of the field with the parameter q, that represents the orientation at which the field F is sampled, expressed as an angle relative to the normal vector; in other words, the strength of the influence F exerted on an adjacent element located at a point (r,a) varies with the deviation q of that element from the direction parallel to the normal vector, as shown in Figure 17, such that the maximal influence is felt when the two elements are parallel, i.e. when q = 0, as in Figure 17 A, and falls off smoothly as the other element's orientation deviates from that orientation as in Figure 17 B and C. This can be expressed with a cosine function, such that the influence F of an element on another element in a direction a and separation r from the first element, and with a relative orientation q would be defined by

Farq = e-r2 [2 sin(a)P - 1] | cos(q)Q | (EQ 1)

This cosine function allows the coplanar influence to propagate to near-coplanar orientations, thereby allowing surface completion to occur around smoothly curving surfaces. The tolerance to such curvature can also be varied parametrically by raising the cosine function to a positive power Q, as shown in Equation 1. So the in-plane stiffness of the coplanarity constraint is adjusted by parameter P, while the angular stiffness is adjusted by parameter Q. The absolute value on the cosine function in Equation 1 allows interaction between elements when q is between p/2 and p.

Figure 17

Orientation of the field of influence between one element and another. For an element located at polar coordinates (r,q), the influence varies as a cosine function of q, the angle between the normal vectors of the two interacting elements.

The Occlusion Field

The orthogonality and occlusion fields have one less dimension of symmetry than the coplanarity field, and therefore they are defined with reference to two vectors through each element at right angles to each other, as shown in Figure 18 A. For the orthogonality field, these vectors represent the surface normals to the two orthogonal planes of the corner, while for the occlusion field one vector is a surface normal, and the other vector points within that plane in a direction orthogonal to the occlusion edge. The occlusion field G around the local element is defined in polar coordinates from these two vector directions, using the angles a and b respectively, as shown in Figure 18 A. The plane of the first surface is defined as for the coplanarity field, with the equation Gabr = e-r2 sin(a)P. For the occlusion field this planar function should be split in two, as shown in Figure 18 B to produce a positive and a negative half, so that this field will promote surface completion in one direction only, and will actually suppress surface completion in the negative half of the field. This can be achieved by multiplying the above equation by the sign (plus or minus, designated by the function sgn()) of a cosine on the orthogonal vector, i.e. Gabr = e-r2 sin(a)P sgn(cos(b)). Because of the negative half-field in this function, there is no need to normalize the equation. However the oriented component of the field can be added as before, resulting in the equation

Gabrq = e-r2 [sin(a)P sgn(cos(b))] | cos(q)Q | (EQ 2)

Again, the maximal influence will be experienced when the two elements are parallel in orientation, i.e. when q = 0. As before, the orientation cosine function is raised to the positive power Q, to allow parametric adjustment of the stiffness of the coplanarity constraint.

The Orthogonality Field

The orthogonality field H can be developed in a similar manner, beginning with the planar function divided into positive and negative half-fields, i.e. with the equation Habr = e-r2 sin(a)P sgn(cos(b)) but then adding another similar plane from the orthogonal surface normal, producing the equation Habr = e-r2 [sin(a)P sgn(cos(b)) + sin(b)P sgn(cos(a))]. This produces two orthogonal planes, each with a negative half-field, as shown schematically in Figure 18 C. Finally, this equation must be modified to add the oriented component to the field, represented by the vector q, such that the maximal influence on an adjacent element will be experienced when that element is either within one positive half-plane and at one orientation, or is within the other positive half-plane and at the orthogonal orientation. The final equation for the orthogonality field therefore is defined by

Habrq = e-r2 [sin(a)P sgn(cos(b)) | cos(q)Q | + [sin(b)P sgn(cos(a)) | cos(q)Q |] (EQ 3)

Edge Consistency and Inconsistency Constraints

There is another aspect of the field-like interaction between elements that remains to be defined. Both the orthogonal and the occlusion states are promoted by appropriately aligned neighboring elements in the coplanar state. Orthogonal and occlusion elements should also feel the influence of neighboring elements in the orthogonal and occlusion states, because a single edge should have a tendency to become either an orthogonal corner percept, or an occlusion edge percept along its entire length. Therefore orthogonal or occlusion elements should promote like-states, and inhibit unlike-states in adjacent elements along the same corner or edge. The interaction between like-state elements along the edge will be called the edge-consistency constraint, and the corresponding field of influence will be designated E, while the complementary interaction between unlike-state elements along the edge is called the edge-inconsistency constraint, whose corresponding edge-inconsistency field will be designated I. These interactions are depicted schematically in Figure 19

Figure 19

A and B: Edge consistency constraint as an excitatory influence between like-state elements along a corner or edge percept. C and D: Edge inconsistency constraint as an inhibitory influence between unlike-state elements along a corner or edge percept. E: The direction along the edge expressed as the intersection of the orthogonal planes defined by the sine functions on the two orthogonal vectors.

The spatial direction along the edge can be defined by the product of the two sine functions sin(a) sin(b) defining the orthogonal planes, denoting the zone of intersection of those two orthogonal planes, as suggested in Figure 19 E. Again, this field can be sharpened by raising these sine functions to a positive power P, and localized by applying the exponential decay function. The edge consistency constraint E therefore has the form Eabr = e-r2 [sin(a)P sin(b)P]. As for the orientation of the edge-consistency field, this will depend now on two angles,q and f, representing the orientations of the two orthogonal vectors of the adjacent orthogonal or occlusion elements relative to the two normal vectors respectively. Both the edge- consistency and the edge-inconsistency fields, whether excitatory between like-state elements, or inhibitory between unlike-state elements, should peak when both pairs of reference vectors are parallel to the normal vectors of the central element, i.e. when q and f are both equal to zero. The full equation for the edge-consistency field E would therefore be

Eabrqf = e-r2 [sin(a)P sin(b)P] cos(q)Q cos(f)Q (EQ 4)

where this equation is applied only to like-state edge or corner elements, while the edge-inconsistency field I would be given by

Iabrqf = e-r2 [sin(a)P sin(b)P] cos(q)Q cos(f)Q (EQ 5)

applied only to unlike-state elements. The total influence R on an occlusion element therefore is calculated as the sum of the influence of neighboring coplanar, orthogonal, and occlusion state elements as defined by

Rabrqf = Gabrqf + Eabrqf - Iabrqf (EQ 6)

and the total influence S on an orthogonal state element is defined by

Sabrqf = Habrqf + Eabrqf - Iabrqf (EQ 7)

Influence of the Visual Input

A two-dimensional visual edge has an influence on the three-dimensional interpretation of a scene, since an edge is suggestive of either a corner or an occlusion at some orientation in three dimensions whose two-dimensional projection coincides with that visual edge. This influence however is quite different from the local field-like influences described above, because the influence of a visual edge should penetrate the volumetric matrix with a planar field of influence to all depths, and should activate all local elements within the plane of influence that are consistent with that edge. Subsequent local interactions between those activated elements serves to select which subset of them should finally represent the three- dimensional percept corresponding to the two-dimensional image. For example, a vertical edge as shown in Figure 20 A would project a vertical plane of influence, as suggested by the light shading in Figure 20 A, into the depth dimension of the volumetric matrix, where it stimulates the orthogonal and occlusion states which are consistent with that visual edge. For example it would stimulate corner and occlusion states at all angles about a vertical axis, as shown in Figure 20 A, where the circular disks represent different orientations of the positive half-fields of either corner or occlusion fields. However a vertical edge would also be consistent with corners or occlusions about axes tilted relative to the image plane but within the plane of influence, for example about the axes depicted in Figure 20 B. The same kind of stimulation would occur at every point within the plane of influence of the edge, although only one point is depicted in the figure. When all elements consistent with this vertical edge have been stimulated, the local field-like interactions between adjacent stimulated elements will tend to select one edge or corner at some depth and at some tilt, thereby suppressing alternative edge percepts at that two-dimensional location at different depths and at different tilts. At equilibrium, some arbitrary edge or corner percept will emerge within the plane of influence as suggested in Figure 20 C, which depicts only one such possible percept, while edge consistency interactions will promote like-state elements along that edge, producing a single emergent percept consistent with the visual edge. In the absence of additional influences, for example in the isolated local case depicted in Figure 20 C, the actual edge that emerges will be unstable, i.e. it could appear anywhere within the plane of influence of the visual edge through a range of tilt angles, and could appear as either an occlusion or a corner edge. However when it does appear, it propagates its own field-like influence into the volumetric matrix, in this example the corner percept would propagate a planar percept of two orthogonal surfaces that will expand into the volume of the matrix, as suggested by the arrows in Figure 20 C. The final percept therefore will be influenced by the global pattern of activity, i.e. the final percept will construct a self-consistent perceptual whole, whose individual parts reinforce each other by mutual activation by way of the local interaction fields, although that percept would remain unstable in all unconstrained dimensions. For example the corner percept depicted in Figure 20 C would snake back and forth unstably within the plane of influence, rotate back and forth along its axis through a small angle, and flip alternately between the corner and occlusion states, unless the percept is stabilized by other features at more remote locations in the matrix.

Figure 20

The influence of a visual edge, in this case a vertical edge, is to A: stimulate local elements in the occlusion or corner percept states at orientations about a vertical axis, or B: about a tilted axis within the plane of influence of the edge. At equilibrium C: a single unified percept emerges, in this case of a perceived corner at some depth and tilt in the volume of the matrix.


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