Statistical image algebra: a Bayesian approach

A mathematical structure used to express image processing transforms, the AFATL image algebra has proved itself useful in a wide variety of applications. The theoretical foundation for the image algebra includes many important constructs for handling a wide variety of image processing problems: questions relating to linear and nonlinear transforms, including decomposition techniques [9], [5]; mapping of transformations to computer architectures [8], [4]; neural networks [1 1], [6]; recursive transforms [10]; and data manipulation on hexagonal arrays. However, statistical notions have been included only on a very elementary level in [12], and on a more sophisticated level in [2]. In this paper we present an extension of the current image algebra that includes a Bayesian statistical approach that is similar in spirit to parts of [2]. Here we show how images are modeled as random vectors, probability functions or mass functions are modeled as images, and conditional probability functions are modeled are templates. The remainder of the paper gives a brief discussion of the current image algebra, an example of the use of image algebra to express high-level image processing transforms, and the presentation of the statistical development of the image algebra.