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Image denoising by generalized total variation regularization and least squares fidelity

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Abstract

Inspired by the ability of \(\ell _p\)-regularized algorithms and the close connection of total variation (TV) to the \(\ell _1\) norm, a \(p\)th-power type TV denoted as TV\(_p\) is proposed for \(0\le p \le 1\). The TV\(_p\)-regularized problem for image denoising is nonconvex thus difficult to tackle directly. Instead, we deal with the problem by proposing a weighted TV (WTV) minimization where the weights are updated iteratively to locally approximate the TV\(_p\)-regularized problem. The difficulty of WTV minimization is dealt with in a modified split Bregman framework. Numerical results are presented to demonstrate improved denoising performance of the new algorithm with \(p<1\) relative to that obtained by the standard TV minimization and several recent denoising methods from the literature on a variety of images.

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References

  • Allard, W. K. (2008). Total variation regularization for image denoising, ii. examples. SIAM Journal on Imaging Sciences, 1(4), 400–417.

    Article  MATH  MathSciNet  Google Scholar 

  • Allard, W. K. (2009). Total variation regularization for image denoising, iii. examples. SIAM Journal on Imaging Sciences, 2(2), 532–568.

    Article  MATH  MathSciNet  Google Scholar 

  • Beck, A., & Teboulle, M. (2009). Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. Image Processing, IEEE Transactions on, 18(11), 2419–2434.

    Article  MathSciNet  Google Scholar 

  • Candès, E., Romberg, J., & Tao, T. (2006). Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 52(2), 489–509.

    Article  MATH  Google Scholar 

  • Candes, E., Wakin, M., & Boyd, S. (2008). Enhancing sparsity by reweighted \(\ell _1\) minimization. Journal of Fourier Analysis and Applications, 14(5), 877–905.

    Article  MATH  MathSciNet  Google Scholar 

  • Chambolle, A. (2004). An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20(1), 89–97.

    MathSciNet  Google Scholar 

  • Chan, T., Esedoglu, S., Park, F., & Yip, A. (2006). Total variation image restoration: Overview and recent developments. In N. Paragios, Y. Chen, & O. Faugeras (Eds.), Handbook of mathematical models in computer vision (pp. 17–31). New York: Springer.

  • Chan, T. F., & Esedoglu, S. (2005). Aspects of total variation regularized l1 function approximation. SIAM Journal on Applied Mathematics, 65(5), 1817–1837.

    Article  MATH  MathSciNet  Google Scholar 

  • Chartrand, R. (2007). Exact reconstruction of sparse signals via nonconvex minimization. Signal Processing Letters, IEEE, 14(10), 707–710.

    Article  Google Scholar 

  • Chartrand, R., & Yin, W. (2008). Iteratively reweighted algorithms for compressive sensing. In Acoustics, speech and signal processing, 2008. ICASSP 2008. IEEE international conference on, pp. 3869–3872. IEEE.

  • Chen, Q., Montesinos, P., Sun, Q. S., Heng, P. A., & Xia, D. S. (2010). Adaptive total variation denoising based on difference curvature. Image and Vision Computing, 28(3), 298–306.

    Article  Google Scholar 

  • Chen, S., Donoho, D., & Saunders, M. (2001). Atomic decomposition by basis pursuit. SIAM Review, 43(1), 129–159.

    Article  MATH  MathSciNet  Google Scholar 

  • Chopra, A., & Lian, H. (2010). Total variation, adaptive total variation and nonconvex smoothly clipped absolute deviation penalty for denoising blocky images. Pattern Recognition, 43(8), 2609–2619.

    Article  MATH  Google Scholar 

  • Dabov, K., Foi, A., Katkovnik, V., & Egiazarian, K. (2007). Image denoising by sparse 3-d transform-domain collaborative filtering. Image Processing, IEEE Transactions on, 16(8), 2080–2095.

    Article  MathSciNet  Google Scholar 

  • Foucart, S., & Lai, M. (2009). Sparsest solutions of underdetermined linear systems via lq-minimization for \(0< q<1\). Applied and Computational Harmonic Analysis, 26(3), 395–407.

    Article  MATH  MathSciNet  Google Scholar 

  • Fu, H., Ng, M. K., Nikolova, M., & Barlow, J. L. (2006). Efficient minimization methods of mixed l2-l1 and l1-l1 norms for image restoration. SIAM Journal on Scientific computing, 27(6), 1881–1902.

    Article  MATH  MathSciNet  Google Scholar 

  • Goldstein, T., & Osher, S. (2009). The split bregman method for l1 regularized problems. SIAM Journal on Imaging Sciences, 2(2), 323–343.

    Article  MATH  MathSciNet  Google Scholar 

  • Grant, M., & Boyd, S. (2008). Graph implementations for nonsmooth convex programs. In V. Blondel, S. Boyd, & H. Kimura (Eds.), Recent advances in learning and control. Lecture Notes in Control and Information Sciences, Springer, pp. 95–110. http://stanford.edu/~boyd/graph_dcp.html.

  • Grant, M., & Boyd, S. (2012). CVX: Matlab software for disciplined convex programming, version 2.0 beta. http://cvxr.com/cvx.

  • Hu, Y., & Jacob, M. (2012). Higher degree total variation (hdtv) regularization for image recovery. Image Processing, IEEE Transactions on, 21(5), 2559–2571.

    Article  MathSciNet  Google Scholar 

  • Jaya Manmadha Rao, M., Anuradha, S., & Reddy, K. (2010). Impulse noise removal in digital images. International Journal of Computer Applications, 10(8), 39–42.

    Google Scholar 

  • Larson, R. (2013). Elementary linear algebra (7th ed.). Belmont: Brooks/Cole, Cengage Learning.

    Google Scholar 

  • Lee, Y. J., Lee, S., & Yoon, J. (2013). A framework for moving least squares method with total variation minimizing regularization. Journal of Mathematical Imaging and Vision. doi:10.1007/s10851-013-0428-5.

  • Louchet, C., & Moisan, L. (2011). Total variation as a local filter. SIAM Journal on Imaging Sciences, 4(2), 651–694.

    Article  MATH  MathSciNet  Google Scholar 

  • Mallat, S. (2009). A wavelet tour of signal processing: The sparse way (3rd ed.). Amsterdam: Elsevier.

    Google Scholar 

  • Nikolova, M. (2002). Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers. SIAM Journal on Numerical Analysis, 40(3), 965–994.

    Article  MATH  MathSciNet  Google Scholar 

  • Rudin, L., Osher, S., & Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1–4), 259–268.

    Article  MATH  Google Scholar 

  • Wang, Y., Yang, J., Yin, W., & Zhang, Y. (2008). A new alternating minimization algorithm for total variation image reconstruction. SIAM Journal on Imaging Sciences, 1(3), 248–272.

    Article  MATH  MathSciNet  Google Scholar 

  • Wohlberg, B., & Rodriguez, P. (2007). An iteratively reweighted norm algorithm for minimization of total variation functionals. Signal Processing Letters, IEEE, 14(12), 948–951.

    Article  Google Scholar 

  • Yan, J., & Lu, W.-S. (2011). New algorithms for sparse representation of discrete signals based on \(\ell _p\)-\(\ell _2\) optimization. In Proc. IEEE PacRim Conf., 2011, pp. 73–78.

  • Yan, J., & Lu, W.-S. (2011). Power-iterative strategy for \(\ell _p\)-\(\ell _2\) optimization for compressive sensing: Towards global solution. In Proc. IEEE Asilomar Conf., 2011, pp. 1153–1157.

  • Yan, J., & Lu, W.-S. (2012). Smoothed \(\ell _p\)-\(\ell _2\) solvers for signal denoising. In Proc. IEEE ICASSP Conf., 2012, pp. 3801–3804.

  • Zibulevsky, M., & Elad, M. (2010). L1-L2 optimization in signal and image processing. Signal Processing Magazine, IEEE, 27(3), 76–88.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Victoria, 3800 Finnerty Rd., Victoria, BC , V8P 5C2, Canada

    Jie Yan & Wu-Sheng Lu

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  1. Jie Yan
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  2. Wu-Sheng Lu
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Correspondence to Jie Yan.

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Yan, J., Lu, WS. Image denoising by generalized total variation regularization and least squares fidelity. Multidim Syst Sign Process 26, 243–266 (2015). https://doi.org/10.1007/s11045-013-0255-2

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  • DOI: https://doi.org/10.1007/s11045-013-0255-2

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