Detection of Differences in Real Distributions

Abstract

If two coherently illuminated distributions are spatially separated at the front focal plane of a spherical lens and the resulting diffraction pattern is square-law recorded in the back focal plane, the Fourier transform of the record contains comparative data about the two input distributions. The transform of the record presents data in a reference frame that simplifies the problem of identifying and locating differences between the two input distributions. The process of transforming, detecting, and retransforming is a method of convolving the two input distributions. If the inputs are identical, the autocorrelation function is derived. If only real functions are evaluated, the autocorrelation function is symmetrical about the correlation peak. An asymmetry indicates a difference between the inputs, i.e., a cross-correlation function. The value of the technique is that it reduces the procedure for comparing spatial distributions to a search for asymmetry about an easily defined point. The same process may be used to identify spatial distributions by heterodyning a reference distribution with the field of view to be searched. The cross correlation function derived contains a sharp correlation peak for each target within the field of view.