Plato's Cave: The Lighted Bubble World

The Lighted Bubble World

The architecture of connectivity for the perceptual sphere proposed by the Bubble World model has some interesting geometrical properties that suggest a solution to the problem of lightness, brightness, and form perception. Every point on the surface of the sphere is connected by the texture of colinearity through every point in the volume of the sphere, and the angle at which those lines of collinearity pass through a point within the sphere depends only on the location of the surface point on the sphere. This property follows from the fact that a point on the spherical surface represents a specific direction in space, and all parallel lines pointing in that direction in Euclidean space will pass through that point on the spherical surface in perceptual space.

Imagine therefore that this same texture of connectivity is designed to carry brightness information. A "light source" at a point on the spherical surface will thereby create a percept of illumination from the corresponding direction in space. Likewise reciprocally, illumination on the "sunny side" of all perceived objects in the space will be communicated in the opposite direction to create the percept of an illuminant in the opposite direction. In such a system a perception of any object will automatically produce a percept of an illuminant, which in turn will produce an expectation of illumination for other objects in the space.

Specifically, every point on a perceived surface must represent or encode two values- the perceived brightness of that point, i.e. how much light appears to be reflected from that surface, and the perceived reflectance or "lightness" of the surface, i.e. the "color" it is perceived to be "painted". An important function of the visual system is to factor these two quantities in order to distinguish lightness edges from illuminance edges. In the Lighted Bubble Model this is achieved by giving these two variables different properties of spatial interaction- the "lightness" value tends to diffuse across surfaces in space creating a uniform percept even around corners, while the reflected light is expected to be greater on surfaces facing the illumination source.

Consider the ambiguous figure below. If it is perceived as a flat figure of two diamond shapes, both shapes are facing the same direction in space, and therefore they project contradictory information about the illumination from that direction, one suggesting a lighter, and one a darker light source. The light source value thus takes on an intermediate value. The difference in brightness must therefore be accounted for by a difference in surface reflectance, therefore the lightness variable takes on different values across the central edge.

Alternatively, the figure can be perceived as a three-dimensional figure of a single folded surface, which in the absence of contradictory evidence might be supposed to have uniform lightness. The "lightness value" would thus tend to propagate throughout the surface even crossing the fold in the middle. The difference in brightness seen on the two faces therefore would suggest a difference in illumination due to the geometrical fold, which can be perceived as either a concave or a convex corner. A concave interpretation suggests a light source from the right, whereas a convex interpretation suggests a light source from the left.

The Lighted Bubble model suggests that as this percept pops in a bistable manner between the two alternative interpretations, it also alternately "turns on" a supposed light source on the appropriate side. That light source in turn "illuminates" all other perceived objects in the space, changing their expected brightness value in a geometry-dependent manner, i.e. they are expected to appear brighter on the sunny side, and less bright on the shady side. This effect is seen in the figure below, where the single block which lacks shading on the left side is perceived to have a "lighter paint" on that side than the other blocks, and this white side even appears lighter than the white top of the block, which is actually the same color.

The interaction between the brightness and form perception goes in both directions, as can be seen in the example below, where the presence of a single unambiguous feature in a field of ambiguous ones is sufficient to establish with certainty the direction of illumination which, in turn, establishes with certainty which of the two alternative spatial interpretations is valid.

The important principle embodied in this model is that the complex geometrical interactions between perceived form and illuminant are modeled by a system which replicates those same geometrical properties, and thus need not calculate geometrical abstractions, but simply performs a parallel "ray tracing" to render the required percept in a three-dimensional context.

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