Reconstruction of Randomly Sampled Sparse Signals Using an Adaptive Gradient Algorithm

TL;DR

This paper introduces an adaptive gradient algorithm for reconstructing sparse signals from randomly sampled data, focusing on missing sample recovery in the time domain with improved efficiency and a uniqueness theorem.

Contribution

It proposes a novel adaptive gradient-based method with a new parameter adaptation criterion for sparse signal reconstruction from random samples.

Findings

Improved computational efficiency through gradient direction angle criterion.

Successful reconstruction demonstrated on statistical examples.

Reconstruction of nonuniformly sampled signals using recalculation procedure.

Abstract

Sparse signals can be recovered from a reduced set of samples by using compressive sensing algorithms. In common methods the signal is recovered in the sparse domain. A method for the reconstruction of sparse signal which reconstructs the remaining missing samples/measurements is recently proposed. The available samples are fixed, while the missing samples are considered as minimization variables. Recovery of missing samples/measurements is done using an adaptive gradient-based algorithm in the time domain. A new criterion for the parameter adaptation in this algorithm, based on the gradient direction angles, is proposed. It improves the algorithm computational efficiency. A theorem for the uniqueness of the recovered signal for given set of missing samples (reconstruction variables) is presented. The case when available samples are a random subset of a uniformly or nonuniformly sampled signal is considered in this paper. A recalculation procedure is used to reconstruct the nonuniformly sampled signal. The methods are illustrated on statistical examples.