Plato's Cave: The Bubble World

The Bubble World

The Bounded Percept

The parable of Plato's Cave suggests that the world you see around you is a percept inside your head. The model I have presented thus far is a fully spatial model, i.e. its surfaces and volumes are represented as surfaces and volumes, somewhat like an architect's model, standing in the lobby of the building of which it is a miniature copy. But an architect's model must of necessity be of finite size, which is generally achieved by establishing abrupt limits beyond which the model is no longer defined. Such abrupt limits are not seen in the spatial percept of surrounding space, and yet at the same time the perceptual space appears bounded. The following figure illustrates this point.

While the space near the body appears to be approximately Euclidean, such as the building in the foreground, objects in the distance appear strangely foreshortened, so that the sun and moon and distant mountains all appear to be at about the same perceptual distance from the observer, although geometrically the mountains are orders of magnitude closer than the moon, which in turn is orders of magnitude closer than the sun. The sun and moon appear as flattened disks, and the mountains appear in silhouette almost as if cut out of cardboard and pasted against the dome of the sky.

Objects between the observer and the mountains gradually become more flattened with distance from the observer, as seen in the building in the background, whose depth is seen with less resolution than are its height and width. This phenomenon is reminiscent of dioramas, illustrated below, as seen in museum exhibits, where a three-dimensional foreground model is displayed in front of a two-dimensional background painting, with a smooth transition in between the two representations, creating the impression of a full three-dimensional scene. In order to create the smooth transition, the depth dimension of the spatial models is progressively reduced with proximity to the backplane painting. The fact that these displays create such a vivid sensation of depth suggests that this representation is meaningful to human perception.

Bounding in Depth

This kind of mapping can be defined mathematically with a vergence representation which, when applied to the bubble model discussed above, results in a representation like the one shown below, where surfaces and volumes are represented in a fully spatial manner, while the spatial resolution of depth approaches zero as distances approach infinity. Boundary completion and surface filling-in can be computed spatially in this representation, which nevertheless is bounded in the depth dimension.

Bounding in All Three Dimensions

In order to bound the representation in the other two dimensions, the X and Y of Cartesian coordinates can be replaced by azimuth and elevation angles in polar coordinates, which are also bounded dimensions. All three dimensions of this spherical representation are closed, or bounded, which means that the boundless infinity of Euclidean space can be remapped within a finite sphere in an angle / angle / vergence representation, with the ecological advantage that the most important region of space, that closest to the body, is mapped at the highest resolution, while the least significant region is mapped at the lowest resolution.

The figure below illustrates how such a representation would reflect the percept of a man walking down the road.

Boundary and Surface Completion in 3-D

In order to perform boundary completion and surface filling-in in the manner described in the Bubble Model above, the geometrical distortions of the perceptual representation must be applied to the boundary and surface completion operations. Consider again the two-dimensional case of the Kanizsa figure depicted below (left) whose illusory square emerges by collinearity interactions between the visible edges of the inducing pac-man figures. In the Directed Diffusion model this completion by collinearity is mediated by a matrix of oriented edge sensitive cells, represented by the short black lines below (center), whose activation diffuses along lines of collinearity, represented by the long gray lines below (center). These lines therefore represent a grid or reference frame of collinearity. For example the figure below (right) represents the response of the matrix below (center) to the Kanizsa stimulus below (left).

In order to extend this idea to three dimensions, the matrix of cells must be expanded to solid volume of cells, as shown below, where the neural connectivity defines a three-dimensional web of colinearity in all directions through every point in the volume. The percept of a cube in this representation for example would be defined by the activation of oriented cells at the eight corners of the cube representing vertices with three mutually orthogonal edges that meet at the corner, as well as by lines of activation connecting the corners along the sides of the cube, and planar activation connecting the sides along the faces of the cube.

Applying the angle / angle / vergence transformation to this infinite cartesian grid of collinearity connections produces the bounded spherical volume of tissue shown below, where the central point represents the center of the head, and the spherical surface represents perceptual infinity. This distorted grid is like a perspective view of the infinite Cartesian grid above, and, by the laws of perspective, all lines that are locally parallel meet at a point at perceptual infinity in both directions. This distorted grid performs an inverse perspective transformation on the visual input, allowing veridical judgements of colinearity to be made on an image containing perspective distortions. Notice that this representation is approximately Euclidean near the center, for instance within the central bulging square below, and becomes progressively more distorted near the surface of the sphere.

The idea of this distorted reference grid is that points of neural activation in this system will appear to be perceptually colinear when they fall along one of the curved lines in the figure. In other words, this system represents perceptual collinearity by a set of curves which become progressively more distorted as they approach infinity, at the surface of the sphere. Notice also that the perception of orthogonality also experiences a similar distortion. Edges which are perceived as orthogonal are also geometrically orthogonal at the center of the representation, but are squashed into greater or lesser angles near the edge of the space, as shown below. The "90 degrees" caption in the figure refers to the perception of orthogonality for edges in the representation that meet at the angles shown.

The figure below shows how such a system would represent a cubical room to an observer at the center of that room. The lines which define the boundaries of each wall bulge outwards from the observer, and yet they follow lines of colinearity as shown above, and thus produce a percept of straightness. The corners of the room are similarly distorted, but again they conform to perceptual orthogonality as defined by the lattice of connectivity, and therefore are perceived as orthogonal corners.

Consider again now the percept of a man walking down a road. This figure shows how the infinite dimensions of the outside world can be captured inside a perceptual bubble of finite dimensions. Notice how the perceptual image of the road appears locally parallel near the man's feet, but converges to a point up ahead and behind. Because the edges of the road generally follow the grain or texture of the matrix of connectivity in the perceptual mechanism as shown above, the man judges the sides of the road to be parallel, even though paradoxically they appear to meet at a point both ahead and behind. Likewise, the vertical walls of the house seen below seem to bend away from the man in the perceptual representation, but again they generally follow the grain of collinearity representing vertical edges, which converge to a point vertically above and below, and therefore they appear to the man to be parallel to one another and vertical, just as in the example of the percept of a cubical room shown above. The angles of intersection between the vertical walls and the horizontal ground are similarly distorted in the perceptual representation, but conform to orthogonality in that representation and are therefore perceived as orthogonal.

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