Approximate performance of the nonlinear joint transform correlator in signallike noise

Abstract

We present a statistical-analysis technique for a nonlinear joint transform correlator (JTC) based on two assumptions: the noise and the signal spectra are identical, and the signal energy is small relative to the noise energy. The first assumption, while admittedly convenient, is also defensible in that it is a worst case and in that image and scene noise can be similar in texture. The second is also reasonable, given that even a clearly visible signal may have small energy compared with the scene noise if it is of limited extent; in any case, the results appear moderately faithful even for the case that signal and noise energies are equal. We discover that the optimal Fourier-plane transformation is spatially variant and tends to remove the Fourier amplitudes of the input image, and indeed functions in a way very similar to the spatially variant binary JTC. We also see that the classic (or spatially invariant linear) JTC is a very inferior technique for signallike noise, that the best spatially variant binary JTC uses a threshold proportional to the noise power spectrum, and that, if a spatially invariant binary-thresholded JTC is desired, then the median Fourier-plane value is an excellent choice of threshold. The performance predictions are verified by simulation and appear to be reasonable even for the highly nonlinear binary schemes.