Automatic kernel design procedure for Cohen's bilinear class of representations as applied to in-line Fresnel holograms

Abstract

Cohen's bilinear class of shift-invariant space–frequency representations provides an automated means for extracting three-dimensional particle locations from in-line Fresnel holograms without any focusing. Choosing kernel parameters of a fixed-kernel representation in order to achieve the trade-off between auto-term sharpness and cross-term suppression while processing a multiple-particle hologram is a tedious task, especially if the hologram considered is crowded. Hence, this paper proposes an automatic kernel design procedure in order to eliminate this parameter selection task altogether and obtain a signal adaptive representation that matches the particular hologram analyzed. An ambiguity function (AF) domain analysis of a two-dimensional (2-D), multiple-particle hologram reveals AF slices of it that carry the auto-term information. By applying the Radon transform (RT) and the inverse RT to moduli of these slices successively, a 2-D discrete AF domain kernel that matches the hologram is obtained in separable form. This procedure is used in our fixed-frequency slice technique recently proposed for 2-D holograms, and also in computing complete space–frequency patterns for one-dimensional holograms, for particle-location analysis of them.