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Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 4314527, 10 pages
https://doi.org/10.1155/2018/4314527
Research Article

Error in the Reconstruction of Nonsparse Images

1Faculty of Electrical Engineering, University of Montenegro, 81000 Podgorica, Montenegro
2GIPSA Lab, INP, University Grenoble Alpes, 38400 Saint-Martin-d’Hères, France

Correspondence should be addressed to Miloš Daković; em.ca@solim

Received 31 August 2017; Revised 9 December 2017; Accepted 28 December 2017; Published 12 February 2018

Academic Editor: Joan Serra-Sagrista

Copyright © 2018 Miloš Brajović et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.