Recent Developments in Image Algebra

Abstract

Image algebra is a mathematical theory concerned with the transformation and analysis of images. This chapter focuses on the analysis and transformation of images by computers, the main goal is the establishment of a comprehensive and unifying mathematical theory of image transformations, image analysis, and image understanding in the discrete as well as the continuous domain. This chapter defines “image algebra” as a heterogeneous or many-valued algebra with multiple sets of operands. The manipulation of images for purposes of image enhancement, analysis, and understanding involves operations not only on images, but also on different types of values and quantities associated with these images. Thus, the basic operands of image algebra are images and certain values or quantities associated with images. An image consists of two things, a collection of points and values associated with these points. Images are, therefore, endowed with two types of information—namely, the spatial relationships of the points and also some type of numeric or other descriptive information associated with these points. To make these notions mathematically precise, the chapter formally defines the concepts of value set, point set, and image.