Circuits, Systems, and Signal Processing

, Volume 33, Issue 7, pp 2151–2171 | Cite as

Image Denoising Via Sparse Dictionaries Constructed by Subspace Learning

  • Yin Kuang
  • Lei Zhang
  • Zhang Yi


In this paper, we propose a combinational algorithm for the removal of zero-mean white and homogeneous Gaussian additive noise from a given image. Image denoising is formulated as an optimization problem. This is iteratively solved by a weighted basis pursuit (BP) in the closed affine subspace. The patches extracted from a given noisy image can be sparsely and approximately represented by adaptively choosing a few nearest neighbors. The approximate reconstruction of these denoised patches is performed by the sparse representation on two dictionaries, which are built by a discrete cosine transform and the noisy patches, respectively. Experiments show that the proposed algorithm outperforms both BP denoising and Sparse K-SVD. This is because the underlying structure of natural images is better captured and preserved. The results are comparable to those of the block-matching 3D filtering algorithm.


Image denoising Optimization problem Weighted BPDN K-SVD Sparse K-SVD Closed affine subspace learning 



This work was supported by National Basic Research Program of China (973 Program) under Grant No. 2011CB302201, National Nature Science Foundation of China under Grants Nos. 61322203 and 61332002, and Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20120181130007.


  1. 1.
    M. Aharon, M. Elad, A. Bruckstein, K-SVD and its non-negative variant for dictionary design, in Proceedings of the SPIE Conference Wavelets (2005), pp. 327–339Google Scholar
  2. 2.
    M. Aharon, M. Elad, A. Bruckstein, K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54(11), 4311–4322 (2006)CrossRefGoogle Scholar
  3. 3.
    M. Bahoura, H. Ezzaidi, FPGA-implementation of discrete wavelet transform with application to signal denoising. Circuits Syst. Signal Process. 31(3), 987–1015 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    O. Bryt, M. Elad, Compression of facial images using the K-SVD algorithm. J. Vis. Commun. Image Represent. 19(4), 270–282 (2008)CrossRefGoogle Scholar
  5. 5.
    A. Buades, B. Coll, J.M. Morel, A review of image denoising methods, with a new one. SIAM Multiscale Model. Simul. 4(2), 490–530 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    J. Cao, Z. Lin, The detection bound of the probability of error in compressed sensing using Bayesian approach, in Proceedings of the 2012 IEEE International Symposium on Circuits and Systems (ISCAS) (2012), pp. 2577–2580Google Scholar
  7. 7.
    P. Chatterjee, P. Milanfar, Clustering-based denoising with locally learned dictionaries. IEEE Trans. Image Process. 18(7), 1438–1451 (2009)CrossRefMathSciNetGoogle Scholar
  8. 8.
    S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129–159 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Image denoising with block-matching and 3 D filtering, in Proceedings of SPIE (2006), pp. 354–365Google Scholar
  10. 10.
    K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)CrossRefMathSciNetGoogle Scholar
  11. 11.
    M. Elad, M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15(12), 3736–3745 (2006)CrossRefMathSciNetGoogle Scholar
  12. 12.
    T. Faktor, Y.C. Eldar, M. Elad, Denoising of image patches via sparse representations with learned statistical dependencies, in Proceedings of the International Conference on Acoustics Speech and Signal Processing (2011), pp. 5820–5823Google Scholar
  13. 13.
    M. Grant, S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21, Accessed 26 Oct 2010
  14. 14.
    L. Jacques, D.K. Hammond, M.J. Fadili, Dequantizing compressed sensing: when oversampling and non-gaussian constraints combine. IEEE Trans. Inf. Theory 57(1), 559–571 (2011)CrossRefMathSciNetGoogle Scholar
  15. 15.
    C. Kervrann, J. Boulanger, Optimal spatial adaptation for patchbased image denoising. IEEE Trans. Image Process. 15(10), 2866–2878 (2006)CrossRefGoogle Scholar
  16. 16.
    A.B. Lee, K.S. Pedersen, D. Mumford, The nonlinear statistics of high-contrast patches in natural images. Int. J. Comput. Vis. 54(1–3), 83–103 (2003)CrossRefzbMATHGoogle Scholar
  17. 17.
    C. Liu, Y.V. Zakharov, T. Chen, Broadband underwater localization of multiple sources using basis pursuit de-noising. IEEE Trans. Image Process. 60(4), 1708–1717 (2012)CrossRefMathSciNetGoogle Scholar
  18. 18.
    G. Luo, Wavelet notch filter design of spread-spectrum communication systems for high-precision wireless positioning. Circuits Syst. Signal Process. 31(2), 651–668 (2012)CrossRefGoogle Scholar
  19. 19.
    J. Mairal, F. Bach, J. Ponce, G. Sapiro, A. Zisserman, Non-local sparse models for image restoration, in Proceedings of the IEEE International Conference on Computer Vision (2009), pp. 2272–2279Google Scholar
  20. 20.
    J. Mairal, M. Elad, G. Sapiro, Sparse representation for color image restoration. IEEE Trans. Image Process. 17(1), 53–69 (2008)CrossRefMathSciNetGoogle Scholar
  21. 21.
    J. Mairal, G. Sapiro, M. Elad, Multiscale sparse image representation with learned dictionaries, in Proceedings of the IEEE International Conference on Image Processing (2007), pp. III-105–III-108Google Scholar
  22. 22.
    J. Mairal, G. Sapiro, M. Elad, Learning multiscale sparse representations for image and video restoration. SIAM Multiscale Model. Simul. 7(1), 214–241 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    S.G. Mallat, Z. Zhang, Matching pursuits with time–frequency dictionaries. IEEE Trans. Image Process. 41(12), 3397–3415 (1993)CrossRefzbMATHGoogle Scholar
  24. 24.
    A. Müller, D. Sejdinovic, R. Piechocki, Approximate message passing under finite alphabet constraints, in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) (2012), pp. 3177–3180Google Scholar
  25. 25.
    B.A. Olshausen, D.J. Field, Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381(6583), 607–609 (1996)CrossRefGoogle Scholar
  26. 26.
    Y.C. Pati, R. Rezaiifar, P.S. Krishnaprasad, Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition, in Proceedings of the 27th Anual Asilomar Conference on Signals, Systems and Computers (1993), pp. 40–44Google Scholar
  27. 27.
    T. Peleg, Y.C. Eldar, M. Elad, Exploiting statistical dependencies in sparse representations for signal recovery. IEEE Trans. Image Process. 60(5), 2286–2303 (2012)CrossRefMathSciNetGoogle Scholar
  28. 28.
    G. Pope, C. Studer, M. Baes, Coherence-based recovery guarantees for generalized basis-pursuit de-quantizing, in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) (2012), pp. 3669–3672Google Scholar
  29. 29.
    J. Portilla, V. Strela, M.J. Wainwright, E.P. Simoncelli, Image denoising using a scale mixture of Gaussians in the wavelet domain. IEEE Trans. Image Process. 12(11), 1338–1351 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    M. Protter, M. Elad, Image sequence denoising via sparse and redundant representations. IEEE Trans. Image Process. 18(1), 27–35 (2009)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Z.M. Ramadan, Efficient restoration method for images corrupted with impulse noise. Circuits Syst. Signal Process. 31(4), 1397–1406 (2012)CrossRefMathSciNetGoogle Scholar
  32. 32.
    D.H. Rao, P.P. Panduranga, A survey on image enhancement techniques: classical spatial filter, neural network, cellular neural network, and fuzzy filter, in Proceedings of the IEEE International Conference on Information Technology (2006), pp. 2821–2826Google Scholar
  33. 33.
    R. Rubinstein, M. Zibulevsky, M. Elad, Double sparsity: learning sparse dictionaries for sparse signal approximation. IEEE Trans. Image Process. 58(3), 1553–1564 (2010)CrossRefMathSciNetGoogle Scholar
  34. 34.
    H.S. Seung, D.D. Lee, The manifold ways of perception. Science 290(5500), 2268–2269 (2000)CrossRefGoogle Scholar
  35. 35.
    J.N. Tehrani, A. McEwan, C. Jin, A.V. Schaik, L1 regularization method in electrical impedance tomography by using the l1-curve (Pareto frontier curve). Appl. Math. Model. 36(3), 1095–1105 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Y. Yang, Y. Wei, Neighboring coefficients preservation for signal denoising. Circuits Syst. Signal Process. 31(2), 827–832 (2012)CrossRefMathSciNetGoogle Scholar
  37. 37.
    G. Yu, G. Sapiro, S. Mallat, Image modeling and enhancement via structured sparse model selection, in 2010 Proceedings of the 17th IEEE International Conference on Image Processing (2009), pp. 1641–1644Google Scholar
  38. 38.
    A. Zahedi, M.H. Kahaei, Spectrum estimation by sparse representation of autocorrelation function. IEICE Trans Fundam Electron Commun Comput Sci E 95A(7), 1185–1186 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Machine Intelligence Laboratory, College of Computer ScienceSichuan UniversityChengdu People’s Republic of China
  2. 2.College of Computer ScienceChengdu Normal UniversityChengdu People’s Republic of China

Personalised recommendations