Dilated POCS: Minimax Convex Optimization

TL;DR

This paper introduces dilated POCS, a novel convex optimization method that employs morphological dilation to find minimax solutions in signal reconstruction, expanding the capabilities of traditional POCS.

Contribution

The paper presents dilated POCS, a new approach that uses dilation of convex sets to achieve minimax solutions, offering an alternative to MMSE in nonintersecting constraints.

Findings

Dilated POCS effectively finds minimax solutions in image reconstruction.

The method provides a new imaging modality for image synthesis.

Morphological erosion can be used to refine overlapping sets.

Abstract

Alternating projection onto convex sets (POCS) provides an iterative procedure to find a signal that satisfies two or more convex constraints when the sets intersect. For nonintersecting constraints, the method of simultaneous projections produces a minimum mean square error (MMSE) solution. In certain cases, a minimax solution is more desirable. Generating a minimax solution is possible using dilated POCS. The minimax solution uses morphological dilation of nonintersecting signal convex constraints. The sets are progressively dilated to the point where there is intersection at a minimax solution. Examples are given contrasting the MMSE and minimax solutions in problems of tomographic reconstruction of images. Dilated POCS adds a new imaging modality for image synthesis. Lastly, morphological erosion of signal sets is suggested as a method to shrink the overlap when sets intersect at more than one point.