Image Algebra and Automatic Shape Recognition

Shape recognition can be carried out with set theory operations. In addition to the usual set theory operations of union, complement, etc., the operation of erosion between two sets (or images) is incorporated. Then it is shown that these operations between sets serve as a basis for representation theorems for all, generally nonlinear, translation invariant transformations, (i.e., transformations which commute with translations). By treating images and shapes as point sets in n-dimensional Cartesian space (n = 2 for binary images, n = 3 for gray scale images, and larger n for images that incorporate color, polarization, and the like), the problem of shape or pattern recognition is converted to the problem of detecting the occurrences of specific sets within an image. This problem is closely related to the operation of erosion. By introducing complement images and complement shapes, a generic computer for automatic shape recognition is found which provides a constructive proof of the representation theorem and related results.