Abstract
An artificial retina is a plane circuit, consisting of a matrix of photocaptors; each has its own memory, consisting in a small number of cells (3 to 5), arranged in parallel planes. The treatment consists in logical operations between planes, plus translations of any plane: they are called “elementary operations” (EO). A retina operator (RO) is a transformation of the image, defined by a specific representation of a Boolean function of n variables (n is the number of neighboring cells taken into account). What is the best way to represent an RO by means of EO, considering the strong limitation of memory? For most retina operators, the complexity (i.e., the number of EO needed) is exponential, no matter what representation is used, but, for specific classes, threshold functions and more generally symmetric functions, we obtain a result several orders of magnitude better than previously known ones. It uses a new representation, called “Block Addition of Variables.” For instance, the threshold function T 25,12 (find if at least 12 pixels are at 1 in a square of 5 × 5) required 62 403 599 EO to be performed. With our method, it requires only 38 084 operations, using three memory cells.
Highlights
An artificial retina is a plane circuit, consisting of a matrix of photocaptors; each has its own memory, consisting in a small number of cells (3 to 5), arranged in parallel planes
We establish some general results about the complexity of the retina operator (RO), that is, we evaluate the number of elementary operations (EO) needed to decompose any RO
The computation of complexity, is done assuming that there are enough memory planes, so that any sequence of elementary operations” (EO) can be performed. Despite this assumption, which is quite strong, we will see that most retina operators of size n have a measure of complexity which is exponential in n
Summary
The “image” is the set of excited photocaptors (the ones which are in the 1 state); it can be viewed as a subset of Z2. If there are three memory bits for each captor, we see that as three planes, parallel to the image plane; each memory bit is represented by the point in the corresponding parallel plane, just below the image. This representation is quite convenient for parallel computation. (iv) to realize the AND functions between two planes This means that if the pixel in plane 1 is at 1 and the corresponding pixel (same coordinates) of plane 2 is at 1, the first pixel remains at 1; it is set to 0 in all other cases;. (v) to realize the OR function between two planes; (vi) to realize the exclusive OR (XOR) between two planes: the first pixel is set to 1 if and only if only one of the pixels is at 1
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