Abstract
Many algorithms have been proposed to find sparse representations over redundant dictionaries or transforms. This paper gives an overview of these algorithms by classifying them into three categories: greedy pursuit algorithms, l p norm regularization based algorithms, and iterative shrinkage algorithms. We summarize their pros and cons as well as their connections. Based on recent evidence, we conclude that the algorithms of the three categories share the same root: l p norm regularized inverse problem. Finally, several topics that deserve further investigation are also discussed.
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References
Mallat S. A Wavelet Tour of Signal Processing. San Diego: Academic Press, 1998
Mallat S, Zhang Z. Matching pursuits with time-frequency dictionaries. IEEE Trans Signal Process, 1993 41(3): 3397–3415
Davis G, Mallat S, Zhang Z. Adaptive time-frequency decompositions. Opt Engin, 1994 33(7): 2183–2191
Chen S, Donoho D, Saunders M. Atomic decomposition by basis pursuit. SIAM J Sci Comput, 1999 20: 33–61
Elad M. Why simple shrinkage is still relevant for redundant representations?. IEEE Trans Inf Theory, 2006 52(12): 5559–5569
Tibshirani R. Regression shrinkage and selection via the lasso. J Royal Statist Soc Ser B (Methodological), 1996 58(1): 267–288
Osborne M R, Presnell B, Turlach B A. A new approach to variable selection in least squares problems. IMA J Num Anal, 2000, 20(3): 389–403
Efron B, Hastie T, Johnstone I, et al.Least angle regression. Annals Statist, 2004 32: 407–499
Starck J L, Nguyen M K, Murtagh F. Wavelets and curvelets for image deconvolution: A combined approach. Signal Process, 2003 83: 2279–2283
Figueiredo M A T, Nowak R D. An EM algorithm for waveletbased image restoration. IEEE Trans Image Process, 2003 12(8): 906–916
Fischer S, Crist’obal G, Redondo R. Sparse overcomplete Gabor wavelet representation based on local competitions. IEEE Trans Signal Process, 2006 15(2): 265–272
Wang B, Wang Y, Selesnick I, et al. Video coding using 3-D dual-tree wavelet transform. EURASIP J Image Video Process, Special issue on “Wavelets in Source Coding, Communications, and Networks”, Volume 2007 (2007), Article ID. 42761, 15 pages, doi: 10.1155/2007/42761
Yang J Y, Xu W L, Dai Q H, et al. Image compression using 2-D dual-tree discrete wavelet transform (DDWT). In: Proc. IEEE International Symposium on Circuits and Systems, New Orleans, 2007. 297–300
Figueiredo M A T, Nowak R D, Wright S J. Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse Problems. Technical Report, University of Wisconsin, 2007
DeVore R A. Nonlinear Approximation, Acta Numerica. Cambridge: Cambridge Univ. Press, 1998. 51–150
Pati Y C, Rezaiifar R, Krishnaprasad P S. Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. In: Proc. 27th Annu. Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, 1993. 40–44
Cotter S, Rao B. Application of tree-based searches to matching pursuit. In: Proc. International Conference on Acoustics, Speech, and Signal Processing, Salt Lake City, UT, USA, 2001. 3933–3936
Jost P, Vandergheynst P, Frossard P. Tree-based pursuit: Algorithm and properties. IEEE Trans Signal Process, 2006 54(12): 4685–4697
Gorodnitsky I F, Rao B D. Sparse signal reconstructions from limited data using FOCUSS: A re-weighted minimum norm algorithm. IEEE Trans Signal Process, 1997 45(3): 600–616
Rao B D, Kreutz-Delgado K. An affine scaling methodology for best basis selection. IEEE Trans Signal Process, 1999 47(1): 187–200
Huggins P S, Zucker S W. Greedy basis pursuit. IEEE Trans Signal Process, 2007 55(7): 3760–3772
Reeves T H, Kingsbury N G. Overcomplete image coding using iterative projection-based noise shaping. In: Proc. International Conference on Image Processing, Rochester, NY, 2002. 597–600
Fischer S, Cristóbal G. Minimum entropy transform using Gabor wavelets for image compression. In: Proc. International Conference on Image Analysis and Processing, Palermo, Italy, 2001. 428–433
Fadili M J, Starck J L, Murtagh F. Inpainting and zooming using sparse representations. Computer J, 2007, doi:10.1093/comjnl/bxm055
Pece A E C. The problem of sparse coding. J Math Imag Vision, 2002 17: 89–108
Zhang C, Yin Z, Chen X, et al.Signal overcomplete representation and sparse decomposition based on redundant dictionaries. Chin Sci Bull, 2005 50(23): 2672–2677
Donoho D, Huo X. Uncertainty principles and ideal atomic decompositions. IEEE Trans Information Theory, 2001 47(7): 2845–2862
Donoho D, Tsaig Y, Drori I, et al. Sparse solution of underdetermined linear equations of stagewise orthogonal matching pursuit. Technical Report, Department of Statistics, Stanford University, 2006
Neff R, Zakhor A. Matching pursuit video coding: I. dictionary approximation. IEEE Trans Circuits Syst Video Tech, 2002 12(1): 13–26
Ventura R F, Vandergheynst P, Frossard P. Low-rate and flexible image coding with redundant representations. IEEE Trans Image Process, 2006 15(3): 726–739
Gribonval R. Fast matching pursuit with a multiscale dictionary of Gaussian chirps. IEEE Trans Signal Process, 2001 49(5): 994–1001
Donoho D L. For most large underdetermined systems of linear equations, the minimal ell-1 norm solution is also the sparsest solution. Commun Pure Appl Math, 2006 59(7): 797–829
Chartrand R. Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process Lett, 2007 14(1): 707–710
Donoho D. De-noising by soft-thresholding. IEEE Trans Inf Theory, 1995 41(3): 613–627
Daubechies I, Defrise M, De Mol C. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun Pure Appl Math, 2004 57(11): 1413–1457
Daubechies I, Teschke G., Vese L. Iteratively solving linear inverse problems under general convex constraints. Inverse Problem Imag, 2007 1(1): 29–46
Yang J Y, Peng Y G, Xu W L, et al.Way to sparse representation: a comparative study. Tsinghua Sci Tech, 2009 14(4): in press
Donoho D, Tsaig Y. Fast solution of l 1-norm minimization problems when the solution may be sparse. Technical Report, Stanford University, 2006
Blumensath T, Davies M E. Iterative thresholding for sparse approximations. Technical Report, The University of Edinburgh, 2004
Olshausen B A, Field D J. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 1996 381: 607–609
Candes E J, Donoho D. Curvelets-a surprisingly effective nonadaptive representation for objects with edges. In: Laurent P J, Le Mehaute A, Schumaker L L, eds. Curves and Surfaces. Nashville: Vanderbilt University Press, TN, 2000. 105–120
Pennec E L, Mallat S. Sparse geometric image representations with bandelets. IEEE Trans Image Process, 2005 14: 423–438
Do M N, Vetterli M. The contourlet transform: an efficient directional multiresolution image representation. IEEE Trans Image Process, 2005 14(12): 2091–2106
Gribonval R, Nielsen R. Sparse decompositions in unions of bases. IEEE Trans Inf Theory, 2003 49: 3320–3325
Tropp J. Greed is good: Algorithmic results for sparse approximation. IEEE Trans Inf Theory, 2004 50(10): 2231–2242
Donoho D, Elad M. Maximal sparsity representation via l 1 minimization. In: Proc. the National Academy of Sciences, 2003 100: 2197–2202
Elad M, Bruckstein A M. A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans Inf Theory, 2002 48: 2558–2567
Herrity K K, Gilbert A C, Tropp J A. Sparse approximation via iterative thresholding. In: Proc. International Conference on Acoustics, Speech and Signal Processing, 2006, 3: 624–627
Sharon Y, Wright J, Ma Y. Computation and relaxation of conditions for equivalence between l1 and l0 minimization. Technical Report, University of Illinois, 2007
Donoho D, Elad M, Temlyakov V. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans Inf Theory, 2006 52: 6–18
Fadili M J, Starck J L. Sparse representation-based image deconvolution by iterative thresholding. Astronomical Data Analysis, France: Marseille, 2006
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Supported by the Joint Research Fund for Overseas Chinese Young Scholars of the National Natural Science Foundation of China (Grant No. 60528004) and the Key Project of the National Natural Science Foundation of China (Grant No. 60528004)
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Yang, J., Peng, Y., Xu, W. et al. Ways to sparse representation: An overview. Sci. China Ser. F-Inf. Sci. 52, 695–703 (2009). https://doi.org/10.1007/s11432-009-0045-5
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DOI: https://doi.org/10.1007/s11432-009-0045-5