Pattern analysis with two-dimensional spectral localisation: Applications of two-dimensional S transforms

An image is a function, f(x,y), of the independent space variables x and 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 segments of f(x,y), leading to the concept of a local spectrum at every point 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 image. 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 variables. Visualisation of a function of four variables is difficult. We skirt 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 components. By ranging through judiciously chosen slice locations the entire S transform can be examined. Images with strictly periodic patterns are best analysed with a global Fourier spectrum. On the other hand, the 2-D S transform would be more useful in spectral characterisation of aperiodic or random patterns.