TABLE OF CONTENTS The goal of image modeling or representation is to find proper ways to mathematically describe and analyze images. It is therefore the most fundamental step in image processing. One, however, must realize that there is no absolutely the best representation, since optimality inevitably depends upon specific processing tasks, as in the case of different representations of natural numbers: the decimal system is more convenient than the dyadic one in daily life, but the latter is more natural for digital or quantum computers. The current chapter introduces five general and useful approaches to image representation, based upon which many successful image processors are to be developed in later chapters.
Digital images are most commonly presented as a matrix of scalars for gray-scale images or vectors for color images, since they are often captured by charge coupled device (CCD) arrays (as in digital cameras) or displayed by liquid crystal arrays (as for laptop or pocket computers). Pixel-matrix representation is, however, by no means the most efficient from the information-theoretic point of view. In what follows, we shall call such direct matrix representation u = ( u i,j) or its analog idealization u = u( x, y) with ( x, y) 
? = ( a, b) ( c, d), the physical image.
A representation of a given class U of physical images refers to a transform 
