Ambiguity function magnitude inversion and applications of morphological dilation in POCS.

Abstract

This dissertation examines morphological dilation for applications in Projection onto Convex Sets (POCS) as well as the inversion of ambiguity function magnitude. In general, POCS solvers implement Least-Squares (LS) algorithms which intuitively minimize the Euclidean distance or L^2-norm of a proposed solution. However, there are situations where other error metrics can be advantageous. One such metric is the weighted minimized-maximum error, or minimax which minimizes the L^∞-norm. Multiple methods for evaluating the weighted, minimax error are investigated, and this dissertation will introduced a modified alternating projections algorithm utilizing morphological dilation on context sets to solve for the minimax. This is shown to have notable improvements over standard POCS solvers for selective signal synthesis applications, including Fresnel diffraction synthesis and Computed Tomography (CT) and associative memory image reconstruction. When multiple, conflicting objective functions are present, minimax solvers can be demonstrated to be an unbiased solver among conflicting constraints, avoiding the Least-Squares tendency to shift a solution towards the centroid.

In addition, the ambiguity function magnitude inversion is shown to be possible and a regularized method for quickly inverting a given function to a valid family of source signals is detailed. The ambiguity function is a fundamental aspect of radar signal processing that is frequently described as non-invertible from its magnitude as the transform is not one-to-one. In the past, an inversion to constant phase shift is possible with the full magnitude and phase of the function, but the phase information is frequently stripped as extraneous for analysis. Unfortunately this practice prevents a clear inversion. However, this paper demonstrates that an inversion to a valid spawning signal is possible, and outlines a regularized method for achieving the desired magnitude response. This will give radar designers direct control over crafting ambiguity functions with mission-critical characteristics.