Analysis of image detection based on fourier plane nonlinear filtering in a joint transform correlator

Signal and image and detection systems based on nonlinear operations of Fourier-transformed data often exhibit greater selectivity than standard matched-filtering techniques. One such system is the joint transform correlator. We analyze the performance of the nonlinear joint transform correlator in terms of the output signal-to-noise ratio; this signal-to-noise ratio is evaluated in terms of both output contrast (peak-to-noise floor) and output variability (peak-to-peak standard deviation). The main assumption used is that the signal energy is small relative to that of the additive noise; this assumption is defensible in practice owing to the generally small spatial extent of target images relative to scenes. With respect to the first performance measure, this study is an extension of that in an earlier paper [Appl. Opt. 34, 5218 (1995)]. The previous analysis was carried out under a restriction that the signal and noise spectra were to be similar (actually multiples of one another). In the current study there is no such constraint, and all analysis of the second measure is new. The analysis is supported by simulation. A benefit of analytical rather than simulational study is that conclusions can be drawn with greater confidence. One of the most interesting of these is that the smooth square-root Fourier plane nonlinearity, more usually known as the k-law processor with k = 0.5, offers extremely robust performance with respect to relative noise bandwidth.