PATTERN-ANALYSIS WITH 2-DIMENSIONAL SPECTRAL LOCALIZATION - APPLICATIONS OF 2-DIMENSIONAL S-TRANSFORMS

An image is a function, f(x,y), of the independent space variables x a nd y. The global Fourier spectrum of the image is a complex function F (k(x),k(y)) of the wave numbers k(x) and k(y). The global spectrum may be viewed as a construct of the spectra of an arbitrary number of seg ments of f(x,y), leading to the concept of a local spectrum at every p oint of f(x, y). The two-dimensional S transform is introduced here as a method of computation of the local spectrum at every point of an im age. In addition to the variables x and y, the 2-D S transform retains the variables k(x) and k(y), being a complex function of four variabl es. Visualisation of a function of four variables is difficult. We ski rt around this by removing one degree of freedom, through examination of 'slices'. Each slice of the 3-D S transform would then be a complex function of three variables, with separate amplitude and phase compon ents. By ranging through judiciously chosen slice locations the entire S transform can be examined. Images with strictly periodic patterns a re best analysed with a global Fourier spectrum. On the other hand, th e 2-D S transform would be more useful in spectral characterisation of aperiodic or random patterns.