An uncertainty analysis of some real functions for image processing applications

There are many benefits to be gained in image processing and compression by the use of analyzing functions which are local in both space and spatial frequency. It is often assumed that these benefits are in some way proportional to the degree of joint locality of the functions being used. Within the limits imposed by the uncertainty principle, there can be great variation in this joint locality across different local function families. While there is no generally accepted joint locality metric appropriate for visual applications, Gabor's joint uncertainty is often cited to justify the use of a set of functions. It has been shown that complex Gabor functions optimize this metric. There is some debate however regarding which, of the restricted class of real functions, has the lowest joint uncertainty. In this paper we examine three families of real functions and directly evaluate the Gabor metric for joint uncertainty. In contrast to previous attempts to prove the optimality of any one function, this analysis provides an explicit numerical basis for comparison of these real functions.