Hierarchical CoSaMP for compressively sampled sparse signals with nested structure

Stefania Colonnese,Roberto Cusani,Stefano Rinauro,Gaetano Scarano,Katia Mangone,Mauro Biagi

doi:10.1186/1687-6180-2014-80

Abstract

This paper presents a novel procedure, named Hierarchical Compressive Sampling Matching Pursuit (CoSaMP), for reconstruction of compressively sampled sparse signals whose coefficients are organized according to a nested structure. The Hierarchical CoSaMP is inspired by the CoSaMP algorithm, and it is based on a suitable hierarchical extension of the support over which the compressively sampled signal is reconstructed. We analytically demonstrate the convergence of the Hierarchical CoSaMP and show by numerical simulations that the Hierarchical CoSaMP outperforms state-of-the-art algorithms in terms of accuracy for a given number of measurements at a restrained computational complexity.

Highlights

The burgeoning field of compressive sampling (CS) addresses the recovery of signals which are sparse either in the original domain or in a different representation domain achieved by a suitable invertible transform
5 Numerical simulation We present numerical results assessing the performance of Hierarchical CoSaMP (HCoSaMP) in reconstructing signals characterized by the nested approximation property (NAP); we both investigate the case of signals compressible in the discrete wavelet transform (DWT) domain and the interesting case of signals compressible in the so-called graph-based transform domain, among which the texture images stand as an example of paramount relevance
We investigate the case of a texture image, where typical assumptions found in dealing with natural images do not hold, and we show that the HCoSaMP outperforms selected state-of-the-art

Summary

Introduction

The burgeoning field of compressive sampling (CS) addresses the recovery of signals which are sparse either in the original domain or in a different representation domain achieved by a suitable invertible transform. Thereby, it is argued that in specific applications, the number of measurements may be in principle sufficient to recover the signal under concern, exactly or in an approximate form, while still not being enough for the CoSaMP to converge. We provide application examples on images obtained by oceanographic monitoring [6,7] and natural images [8], as well as on texture images [9] For the former cases, we select the well-known discrete wavelet transform as a sparsifying transform, whereas for the latter, we resort to the graphbased transform, originally established for depth map encoding, as a sparsity-achieving representation within the reconstruction procedure.

Compressive sampling basics

Restrict such a support to L0

Findings

Conclusion