Guided Signal Reconstruction Theory

TL;DR

This paper introduces a new axiomatic framework for signal reconstruction using a guiding set and sample consistency, proposing novel frame-less methods, analyzing their properties, and demonstrating practical algorithms with stability bounds.

Contribution

It formulates a new axiomatic approach and introduces frame-less reconstruction methods based on shortest pathways between sets, with theoretical analysis and iterative algorithms.

Findings

Existence and uniqueness conditions are established.

New stability and error bounds are derived.

Algorithms are demonstrated for image magnification.

Abstract

An axiomatic approach to signal reconstruction is formulated, involving a sample consistent set and a guiding set, describing desired reconstructions. New frame-less reconstruction methods are proposed, based on a novel concept of a reconstruction set, defined as a shortest pathway between the sample consistent set and the guiding set. Existence and uniqueness of the reconstruction set are investigated in a Hilbert space, where the guiding set is a closed subspace and the sample consistent set is a closed plane, formed by a sampling subspace. Connections to earlier known consistent, generalized, and regularized reconstructions are clarified. New stability and reconstruction error bounds are derived, using the largest nontrivial angle between the sampling and guiding subspaces. Conjugate gradient iterative reconstruction algorithms are proposed and illustrated numerically for image magnification.