Error in the Reconstruction of Nonsparse Images

Miloš Brajović,Miloš Daković,Cornel Ioana,Ljubiša Stanković,Isidora Stanković

doi:10.1155/2018/4314527

Abstract

Sparse signals, assuming a small number of nonzero coefficients in a transformation domain, can be reconstructed from a reduced set of measurements. In practical applications, signals are only approximately sparse. Images are a representative example of such approximately sparse signals in the two-dimensional (2D) discrete cosine transform (DCT) domain. Although a significant amount of image energy is well concentrated in a small number of transform coefficients, other nonzero coefficients appearing in the 2D-DCT domain make the images be only approximately sparse or nonsparse. In the compressive sensing theory, strict sparsity should be assumed. It means that the reconstruction algorithms will not be able to recover small valued coefficients (above the assumed sparsity) of nonsparse signals. In the literature, this kind of reconstruction error is described by appropriate error bound relations. In this paper, an exact relation for the expected reconstruction error is derived and presented in the form of a theorem. In addition to the theoretical proof, the presented theory is validated through numerical simulations.

Highlights

Signals that can be characterized by a small number of nonzero coefficients are referred to as sparse signals [1,2,3,4,5,6,7,8,9,10,11]
We present an exact relation for the expected squared error in approximately sparse or nonsparse signals in the 2D-discrete cosine transform (DCT) domain
Representing the mathematical model for the compressive sampling procedure, where A is an NA × MN measurement matrix. It is defined as the partial inverse 2D-DCT matrix, containing rows of Ψ that correspond to the available pixel positions

Summary

Introduction

Signals that can be characterized by a small number of nonzero coefficients are referred to as sparse signals [1,2,3,4,5,6,7,8,9,10,11]. Signal samples can be considered as measurements (observations) in the case when a linear signal transform is the sparsity domain. It is important to note that, in practice, digital images are usually only approximately sparse or nonsparse in the 2D-DCT domain [21,22,23,24] It means that besides the coefficients with significant values, carrying most of the signal energy, small valued coefficients may appear instead of zero-valued ones.

Basic Definitions

Reconstruction Error Energy

The Reconstruction Process and the Proof

Numerical Results

Conclusions