Plato's Cave: Combinatorial Explosion
## Combinatorial Explosion in Computing the Gestalt Laws

Consider the Gestalt law of proximity. A conventional algorithm might
compute this law with an algorithm something like...

At first sight this appears perfectly feasable. In practice however
there are multiple problems. First of all, the objects must be
identified and cataloged before this algorithm can work on them. But
it is this similarity grouping which generates the low-level
segmentation in the first place. In other words, the objects which
must be grouped initially are the lowest level visual primitives, such
as edges, lines, dots, etc. A natural image can contain thousands of
such primitives, many of which may be illusory, such as attatched
shadows, cast shadows, reflectance edges, textures, etc. so an
algorithm which explicitly measures the distance between such features
might have to generate a proximity table which is larger than the
original image on which it is based.
Furthermore, the proximity relations are not defined by an absolute
distance, but rather by relative groupings or clusterings, which may
occur at a multitude of spatial scales, as shown below.

A full encoding of the rules of this algorithm would be even more
complex, and would lead to yet more combinatorial problems.
Furthermore, the Gestalt grouping laws do not act independently but
appear to influence each other, so that the final pre-attentive
percept is some analog combination of all of the Gestalt grouping laws
acting in unison (note the influence of "good continuation", "symmetry
(five-fold), and "closure" above). A full algorithmic coding of these
rules in symbolic form would be practically impossible.
An analog Gestalt mechanism to compute the law of proximity in the
above figure might be defined something like this: for every dot in
the image, glue a marble at the corresponding location on a thin
rubber sheet stretched over a rectangular frame. When all the marbles
are glued in place, the gradient of the rubber sheet sagging under the
weight of the marbles represents the strength and direction of the
grouping force. While this mechanical analog has problems of its own,
the *principle* behind this mechanism allows it to compute the
grouping between any number of points with equal ease, and the analog
nature of the system allows it to be coupled to other similar analog
mechanisms to compute the other Gestalt grouping laws, so that the
final grouping can be an analog combination of all of the analog laws
acting simultaneously.

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