Direct construction of phase-only filters

TLDR: A new direct construc­ tion of phase-only filters which might be useful for threshold optical correlation detectors is given and leads to a significant improvement in signal-to-noise ratio (SNR) over previous methods.

Abstract: Robert R. Kallman North Texas State University, Mathematics Department, Denton, Texas 76203. Received 18 July 1987. 0003-6935/87/245200-02$02.00/0. © 1987 Optical Society of America. The purpose of this Letter is to give a new direct construc­ tion of phase-only filters which might be useful for threshold optical correlation detectors. This construction is a descen­ dant of the constructions introduced in Kallman and leads to a significant improvement in signal-to-noise ratio (SNR) over previous methods. Simulations suggest that the result­ ing filters and their optimized binarizations can be designed to contain a great deal of information, to be stable under perturbations in the training set, and to have a very low false alarm rate. An intrinsic numerical measure of the performance or SNR of a discriminant function h against a training set will first be formulated. This formulation takes into account that a phase-only filter (POF) or binary phase-only filter (BPOF), unlike a synthetic discriminant function (SDF), cannot control the actual size of the recognition spike in the output correlation plane when a valid target is centered in the filter input plane. Start with 2-D images ƒ1,... ,fn,..., ƒm, the training set. Think of ƒ 1 t . . . , ƒn as objects one is seeking and ƒ n + 1 , . . . , ƒm as objects one is not seeking and definitely does not want to confuse with ƒ 1 , . . . , ƒn. n may well be equal to m. For any x = (x1,x2) in the plane and any function g, let gx(y) = g(y x) for any y = (y1,y1) in the plane. If one initially thinks of g as centered over the origin, think of gx as being g translated so as to be centered over x. One wants the measured optical intensities |+hx,ƒi,| (1 ≤ i ≤ n) in the output plane all to be large for x the origin [note that h(0.0) = h] and that |+hx,ƒi,| (1 ≤ i ≤ m) be as small as possible for all x values outside of some a priori chosen box (or any other region) Bi about the origin. Here Bi is empty for n + 1 ≤ i ≤ m, and +p,q, denotes the usual inner product between the pair of functions p and q or the pair of vectors p and q. The SNR of h, a number intrinsically associated with h, is defined to be