An Evaluation of the Sparsity Degree for Sparse Recovery with Deterministic Measurement Matrices
- 541 Downloads
The paper deals with the estimation of the maximal sparsity degree for which a given measurement matrix allows sparse reconstruction through ℓ 1-minimization. This problem is a key issue in different applications featuring particular types of measurement matrices, as for instance in the framework of tomography with low number of views. In this framework, while the exact bound is NP hard to compute, most classical criteria guarantee lower bounds that are numerically too pessimistic. In order to achieve an accurate estimation, we propose an efficient greedy algorithm that provides an upper bound for this maximal sparsity. Based on polytope theory, the algorithm consists in finding sparse vectors that cannot be recovered by ℓ 1-minimization. Moreover, in order to deal with noisy measurements, theoretical conditions leading to a more restrictive but reasonable bounds are investigated. Numerical results are presented for discrete versions of tomography measurement matrices, which are stacked Radon transforms corresponding to different tomograph views.
KeywordsCompressed sensing Deterministic matrix Sparsity Greedy algorithm Sparcity degree
The authors wish to thank the reviewers for their careful reading and relevants remarks that improve significatively the readability of this work.
- 7.Candès, E.J., Romberg, J.: Practical signal recovery from random projections. In: Wavelet Applications in Signal and Image Processing XI. Proc. SPIE, vol. 5914 (2004) Google Scholar
- 9.Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, R., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer, New York (2010) Google Scholar
- 11.Davenport, M., Duarte, M.F., Eldar, Y.C., Kutyniok, G.: Introduction to compressed sensing. In: Compressed Sensing: Theory and Applications. Cambridge University Press, Cambridge (2011) Google Scholar
- 13.Donoho, D.L.: Neighborly polytopes and sparse solutions of underdetermined linear equations. Tech. rep., Department of Statistics, Stanford University (2004) Google Scholar
- 23.Krzakala, F., Mézard, M., Sausset, F., Sun, Y.F., Zdeborová, L.: Statistical physics-based reconstruction in compressed sensing. Phys. Rev. X. 2(021005), x+18 (2012) Google Scholar
- 24.Liang, D., Zhang, H.F., Ying, L.: Compressed-sensing photoacoustic imaging based on random optical illumination. Int. J. Funct. Inform. Pers. Med. 2(4), 394–406 (2009) Google Scholar
- 27.Pustelnik, N., Dossal, Ch., Turcu, F., Berthoumieu, Y., Ricoux, P.: A greedy algorithm to extract sparsity degree for ℓ 1/ℓ 0-equivalence in a deterministic context. In: Proc. Eur. Sig. and Image Proc. Conference, Bucharest, Romania pp. 859–863 (2012) Google Scholar