Spectral pattern recognition in under-sampled functions

Fourier optics and an optical bench model are used to construct an ensemble of candidate functions representing variational patterns in an undersampled two dimensional functiong(x,y). The known sample functions(x,y) is the product ofg(x,y) and a set of unit impulses on the sample point patternp(x,y) which, from the optical point of view, is an aperture imposing strict mathematical limits on what the sample can tell aboutg(x,y). The laws of optics enforce much needed—and often lacking—conceptual discipline in reconstructing candidate variational patterns ing(x,y). The Fourier transform (FT) ofs(x,y) is the convolution of the FT's ofg(x,y) andp(x,y). If the convolution shows aliasing or confounding of frequencies undersampling is surely present and all reconstructions are indeterminate. Then information from outsides(x,y) is required and it is easily expressed in frequency terms so that the principles of optical filtering and image reconstruction can be applied. In the application described and pictured the FT ofs(x,y) was filtered to eliminate unlikely or uninteresting high frequency amplitude maxima. A menu of the 100 strongest remaining terms was taken as indicating the principle variational patterns ing(x,y). Subsets of 10 terms from the menu were chosen using stepwise regression. By so restricting the subset size both the variance and the span of their inverse transforms were made consistent with those of the data. The amplitudes of the patterns being overdetermined, it was possible to estimate the phases also. The inverse transforms of 9 patterns so selected are regarded as ensembles of reconstructions, that is as stochastic process models, from which estimates of the mean and other moments can be calculated.This paper was presented at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.