Learning doubly sparse transforms for images

IEEE Transactions on Image Processing

Volumn 22, Issue 12, 2013, Pages 4598-4612

  Ravishankar, Saiprasad ^a   Bresler, Yoram ^a  

^a University of Illinois at Urbana Champaign   (United States)

Author keywords

compressed sensing; dictionary learning; image denoising; image representation; sparse representation; Sparsifying transforms; structured transforms

Indexed keywords

COMPRESSION STANDARDS; DICTIONARY LEARNING; DISCRETE COSINE TRANSFORM(DCT); IMAGE REPRESENTATIONS; K-SVD ALGORITHMS; PROBLEM FORMULATION; RECENT RESEARCHES; SPARSE REPRESENTATION;

COMPRESSED SENSING; IMAGE DENOISING; IMAGE PROCESSING;

DISCRETE COSINE TRANSFORMS;

EID: 84884838010     PISSN: 10577149     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIP.2013.2274384     Document Type: Article

References (59)

1. M. Elad, P. Milanfar, and R. Rubinstein, "Analysis versus synthesis in signal priors," Inverse Problems, vol. 23, no. 3, pp. 947-968, 2007. (Pubitemid 46838835)
2. A. M. Bruckstein, D. L. Donoho, and M. Elad, "From sparse solutions of systems of equations to sparse modeling of signals and images," SIAM Rev., vol. 51, no. 1, pp. 34-81, 2009.
3. D. Donoho, "Compressed sensing," IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289-1306, Apr. 2006.
4. B. K. Natarajan, "Sparse approximate solutions to linear systems," SIAM J. Comput., vol. 24, no. 2, pp. 227-234, Apr. 1995.
5. G. Davis, S. Mallat, and M. Avellaneda, "Adaptive greedy approximations," J. Construct. Approx., vol. 13, no. 1, pp. 57-98, 1997.
6. Y. Pati, R. Rezaiifar, and P. Krishnaprasad, "Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition," in Proc. Asilomar Conf. Signals, Syst. Comput., vol. 1. 1993, pp. 40-44.
7. S. G. Mallat and Z. Zhang, "Matching pursuits with timefrequency dictionaries," IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3397-3415, Dec. 1993.
8. W. Dai and O. Milenkovic, "Subspace pursuit for compressive sensing signal reconstruction," IEEE Trans. Inf. Theory, vol. 55, no. 5, pp. 2230-2249, May 2009.
9. S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM J. Sci. Comput., vol. 20, no. 1, pp. 33-61, 1998. (Pubitemid 128689501)
10. I. F. Gorodnitsky, J. George, and B. D. Rao, "Neuromagnetic source imaging with FOCUSS: A recursive weighted minimum norm algorithm," Electrocephalogr. Clin. Neurophysiol., vol. 95, no. 4, pp. 231-251, 1995.
11. B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, "Least angle regression," Ann. Stat., vol. 32, no. 2, pp. 407-499, 2004. (Pubitemid 41250302)
12. S. Nam, M. E. Davies, M. Elad, and R. Gribonval, "Cosparse analysis modeling - Uniqueness and algorithms," in Proc. IEEE ICASSP, May 2011, pp. 5804-5807.
13. R. Rubinstein, T. Faktor, and M. Elad, "K-SVD dictionary-learning for the analysis sparse model," in Proc. IEEE ICASSP, Mar. 2012, pp. 5405-5408.
14. R. Rubinstein and M. Elad, "K-SVD dictionary-learning for the analysis sparse model," in Proc. SPARS, Jun. 2011, p. 73.
15. A. Chambolle, "An algorithm for total variation minimization and applications," J. Math. Imag. Vis., vol. 20, nos. 1-2, pp. 89-97, 2004.
16. M. Yaghoobi, S. Nam, R. Gribonval, and M. E. Davies, "Noise aware analysis operator learning for approximately cosparse signals," in Proc. IEEE ICASSP, Mar. 2012, pp. 5409-5412.
17. S. Ravishankar and Y. Bresler, "Learning sparsifying transforms," IEEE Trans. Signal Process., vol. 61, no. 5, pp. 1072-1086, Mar. 2013.
18. S. Mallat, A Wavelet Tour of Signal Processing. San Diego, CA, USA: Academic, 1999.
19. E. J. Candès and D. L. Donoho, "Ridgelets: A key to higher-dimensional intermittency?" Phil. Trans. R. Soc. London A, vol. 357, no. 1760, pp. 2495-2509, 1999.
20. M. N. Do and M. Vetterli, "The contourlet transform: An efficient directional multiresolution image representation," IEEE Trans. Image Process., vol. 14, no. 12, pp. 2091-2106, Dec. 2005. (Pubitemid 41781516)
21. E. J. Candès and D. L. Donoho, "Curvelets - A surprisingly effective nonadaptive representation for objects with edges," in Curves and Surfaces. Nashville, TN, USA: Vanderbilt Univ. Press, 1999, pp. 105-120.
22. W. K. Pratt, J. Kane, and H. C. Andrews, "Hadamard transform image coding," Proc. IEEE, vol. 57, no. 1, pp. 58-68, Jan. 1969.
23. J. B. Allen and L. R. Rabiner, "A unified approach to short-time Fourier analysis and synthesis," Proc. IEEE, vol. 65, no. 11, pp. 1558-1564, Nov. 1977.
24. C. Lemaréchal and C. Sagastizábal, "Practical aspects of the Moreau- Yosida regularization: Theoretical preliminaries," SIAM J. Optim., vol. 7, no. 2, pp. 367-385, 1997.
25. B. A. Olshausen and D. J. Field, "Emergence of simple-cell receptive field properties by learning a sparse code for natural images," Nature, vol. 381, no. 6583, pp. 607-609, 1996. (Pubitemid 26177476)
26. K. Engan, S. Aase, and J. Hakon-Husoy, "Method of optimal directions for frame design," in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Mar. 1999, pp. 2443-2446.
27. M. Aharon, M. Elad, and A. Bruckstein, "K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation," IEEE Trans. Signal Process., vol. 54, no. 11, pp. 4311-4322, Nov. 2006. (Pubitemid 44637761)
28. M. Yaghoobi, T. Blumensath, and M. Davies, "Dictionary learning for sparse approximations with the majorization method," IEEE Trans. Signal Process., vol. 57, no. 6, pp. 2178-2191, Jun. 2009.
29. R. Gribonval and K. Schnass, "Dictionary identification - Sparse matrixfactorization via l1-minimization," IEEE Trans. Inf. Theory, vol. 56, no. 7, pp. 3523-3539, Jul. 2010.
30. J. Mairal, F. Bach, J. Ponce, and G. Sapiro, "Online learning for matrix factorization and sparse coding," J. Mach. Learn. Res., vol. 11, pp. 19-60, Mar. 2010.
31. K. Skretting and K. Engan, "Recursive least squares dictionary learning algorithm," IEEE Trans. Signal Process., vol. 58, no. 4, pp. 2121-2130, Apr. 2010.
32. R. Rubinstein, A. M. Bruckstein, and M. Elad, "Dictionaries for sparse representation modeling," Proc. IEEE, vol. 98, no. 6, pp. 1045-1057, Jun. 2010.
33. Q. Geng, H. Wang, and J. Wright, "On the local correctness of 1-minimization for dictionary learning," 2011 [Online]. Available: http://arxiv.org/abs/1101.5672
34. D. A. Spielman, H. Wang, and J. Wright, "Exact recovery of sparselyused dictionaries," J. Mach. Learn. Res., Proc. Track, vol. 23, pp. 37.1- 37.18, 2012.
35. M. Elad and M. Aharon, "Image denoising via sparse and redundant representations over learned dictionaries," IEEE Trans. Image Process., vol. 15, no. 12, pp. 3736-3745, Dec. 2006. (Pubitemid 44811686)
36. J. Mairal, M. Elad, and G. Sapiro, "Sparse representation for color image restoration," IEEE Trans. Image Process., vol. 17, no. 1, pp. 53-69, Jan. 2008.
37. M. Aharon and M. Elad, "Sparse and redundant modeling of image content using an image-signature-dictionary," SIAM J. Imag. Sci., vol. 1, no. 3, pp. 228-247, 2008.
38. J. Mairal, G. Sapiro, and M. Elad, "Learning multiscale sparse representations for image and video restoration," SIAM Multiscale Model. Simul., vol. 7, no. 1, pp. 214-241, 2008.
39. G. Yu, G. Sapiro, and S. Mallat, "Image modeling and enhancement via structured sparse model selection," in Proc. 17th IEEE ICIP, Sep. 2010, pp. 1641-1644.
40. H. Y. Liao and G. Sapiro, "Sparse representations for limited data tomography," in Proc. IEEE ISBI, May 2008, pp. 1375-1378.
41. S. Ravishankar and Y. Bresler, "MR image reconstruction from highly undersampled k-space data by dictionary learning," IEEE Trans. Med. Imag., vol. 30, no. 5, pp. 1028-1041, May 2011.
42. S. Ravishankar and Y. Bresler, "Multiscale dictionary learning for MRI," in Proc. ISMRM, 2011, p. 2830.
43. G. Peyré and J. Fadili, Learning analysis sparsity priors, in Proc. Int. Conf. SampTA, May 2011.
44. M. Yaghoobi, S. Nam, R. Gribonval, and M. Davies, "Analysis operator learning for overcomplete cosparse representations," in Proc. EUSIPCO, 2011, pp. 1470-1474.
45. B. Ophir, M. Elad, N. Bertin, and M. Plumbley, "Sequential minimal eigenvalues - An approach to analysis dictionary learning," in Proc. EUSIPCO, 2011, pp. 1465-1469.
46. M. Yaghoobi, S. Nam, R. Gribonval, and M. E. Davies, "Constrained overcomplete analysis operator learning for cosparse signal modelling," IEEE Trans. Signal Process., vol. 61, no. 9, pp. 2341-2355, May 2013.
47. M. W. Marcellin, M. J. Gormish, A. Bilgin, and M. P. Boliek, "An overview of JPEG-2000," in Proc. Data Compress. Conf., 2000, pp. 523-541.
48. M. Lustig, D. Donoho, and J. Pauly, "Sparse MRI: The application of compressed sensing for rapid MR imaging," Magn. Resonance Med., vol. 58, no. 6, pp. 1182-1195, 2007. (Pubitemid 350234205)
49. M. Lustig, J. M. Santos, D. L. Donoho, and J. M. Pauly, "k-t SPARSE: High frame rate dynamic MRI exploiting spatio-temporal sparsity," in Proc. ISMRM, 2006, p. 2420.
50. R. Rubinstein, M. Zibulevsky, and M. Elad, "Double sparsity: Learning sparse dictionaries for sparse signal approximation," IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1553-1564, Mar. 2010.
51. B. Ophir, M. Lustig, and M. Elad, "Multi-scale dictionary learning using wavelets," IEEE J. Sel. Topics Signal Process., vol. 5, no. 5, pp. 1014-1024, Sep. 2011.
52. M. Zibulevsky and B. A. Pearlmutter, "Blind source separation by sparse decomposition in a signal dictionary," Neural Comput., vol. 13, no. 4, pp. 863-882, 2001. (Pubitemid 33594308)
53. R. Pytlak, Conjugate Gradient Algorithms in Nonconvex Optimization. Berlin Heidelberg, Germany: Springer-Verlag, 2009.
54. J. A. Fill and S. Janson, "Quicksort asymptotics," J. Algorithms, vol. 44, no. 1, pp. 4-28, 2002. (Pubitemid 35167314)
55. J. Portilla, "Image restoration through L0 analysis-based sparse optimization in tight frames," in IEEE ICIP, Nov. 2009, pp. 3909-3912.
56. S. Ravishankar and Y. Bresler, "Learning doubly sparse transforms for image representation," in Proc. IEEE Int. Conf. Image Process., Sep./Oct. 2012, pp. 685-688.
57. M. Elad. (2009). Michael Elad Personal Page [Online]. Available: http://www.cs.technion.ac.il/~elad/Various/KSVD-Matlab-ToolBox.zip
58. R. Rubinstein. (accessed 2012, Oct. 9) [Online]. Available: http://www.cs.technion.ac.il/~ronrubin/Software/ksvdsbox11.zip
59. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, "Image denoising by sparse 3-D transform-domain collaborative filtering," IEEE Trans. Image Process., vol. 16, no. 8, pp. 2080-2095, Aug. 2007. (Pubitemid 47225460)