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Models for Multiplicative Noise Removal

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Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

Abstract

Image denoising is the most important step in image processing for further image analysis. It is an important topic in many applications, such as object recognition, digital entertainment, etc. The digital image can be corrupted with noise during acquisition, storage, and transmission. Noise can be classified as additive noise, multiplicative noise, and non-additive non-multiplicative noise (such as salt and pepper noise, Poisson noise). The main properties of a good image denoising model are that it will remove noise while preserving details of the image.

This chapter aims to present a review of multiplicative denoising models, especially for the multiplicative Gamma noise. Similar to denoising for additive Gaussian noise, these denoising approaches can be categorized as variational methods, non-local methods, and deep neural network-based methods. Due to space constraints, this chapter only discusses some of them. The rest of this chapter is organized as follows. Section “Introduction” is an introduction and section “Variational Methods with Different Data Fidelity Terms” describes variational methods with different data fidelity terms. Section “Variational Methods with Different Regularizers” introduces variational methods with different regularizers. Sections “Multitasks” to “DNN Method” describe multitasks, nonlocal, and deep neural network (DNN) methods. Finally, section “Conclusion” presents our conclusions.

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References

  • Abolhassani, M., Rostami, Y.: Speckle noise reduction by division and digital processing of a hologram. Optik 123(10), 937–939 (2012)

    Article  Google Scholar 

  • Aubert, G., Aujol, J.-F.: A variational approach to removing multiplicative noise. SIAM J. Appl. Math. 68(4), 925–946 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Box, G.E.P., Cox, D.R.: An analysis of transformations. J. R. Stat. Soc.: Ser. B (Methodol.) 26(2), 211–243 (1964)

    Google Scholar 

  • Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Buades, A., Coll, B., Morel, J.-M.: A non-local algorithm for image denoising. In: 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), vol 2, pp. 60–65. IEEE (2005)

    Google Scholar 

  • Burger, H.C., Schuler, C.J., Harmeling, S.: Image denoising: can plain neural networks compete with BM3D? In: 2012 IEEE Conference on Computer Vision and Pattern Recognition, pp. 2392–2399. IEEE (2012)

    Google Scholar 

  • Candes, E.J., Wakin, M.B., Boyd, S.P.: Enhancing sparsity by reweighted l1 minimization. J. Fourier Anal. Appl. 14(5–6), 877–905 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Chambolle, A., Pock, T.: An introduction to continuous optimization for imaging. Acta Numer. 25, 161–319 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Chatterjee, P., Milanfar, P.: Is denoising dead? IEEE Trans. Image Process. 19(4), 895–911 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, D.-Q., Cheng, L.-Z.: Spatially adapted total variation model to remove multiplicative noise. IEEE Trans. Image Process. 21(4), 1650–1662 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, Y., Pock, T.: Trainable nonlinear reaction diffusion: a flexible framework for fast and effective image restoration. IEEE Trans. Pattern Anal. Mach Intell. 39(6), 1256–1272 (2016)

    Article  Google Scholar 

  • Chesneau, C., Fadili, J., Starck, J.-L.: Stein block thresholding for image denoising. Appl. Comput. Harmon. Anal. 28(1), 67–88 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)

    Article  MathSciNet  Google Scholar 

  • Daubechies, I., DeVore, R., Fornasier, M., Güntürk, C.S.: Iteratively reweighted least squares minimization for sparse recovery. Commun. Pure Appl. Math.: J. Issued Courant Inst. Math. Sci. 63(1), 1–38 (2010)

    Google Scholar 

  • Deledalle, C.-A., Denis, L., Tupin, F.: Iterative weighted maximum likelihood denoising with probabilistic patch-based weights. IEEE Trans. Image Process. 18(12), 2661–2672 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Denis, L., Tupin, F., Darbon, J., Sigelle, M.: SAR image regularization with fast approximate discrete minimization. IEEE Trans. Image Process. 18(7), 1588–1600 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Dong, Y., Zeng, T.: A convex variational model for restoring blurred images with multiplicative noise. SIAM J. Imaging Sci. 6(3), 1598–1625 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Durand, S., Fadili, J., Nikolova, M.: Multiplicative noise removal using L1 fidelity on frame coefficients. J. Math. Imaging Vis. 36(3), 201–226 (2010)

    Article  Google Scholar 

  • Elad, M., Aharon, M.: Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15(12), 3736–3745 (2006)

    Article  MathSciNet  Google Scholar 

  • Gilboa, G., Darbon, J., Osher, S., Chan, T.: Nonlocal convex functionals for image regularization. UCLA CAM-report, pp. 06–57 (2006)

    Google Scholar 

  • Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Han, Y., Feng, X.-C., Baciu, G., Wang, W.-W.: Nonconvex sparse regularizer based speckle noise removal. Pattern Recogn. 46(3), 989–1001 (2013)

    Article  Google Scholar 

  • Hao, Y., Feng, X., Xu, J.: Multiplicative noise removal via sparse and redundant representations over learned dictionaries and total variation. Signal Process. 92(6), 1536–1549 (2012)

    Article  Google Scholar 

  • Hoekman, D.H.: Speckle ensemble statistics of logarithmically scaled data (radar). IEEE Trans. Geosci. Remote Sens. 29(1), 180–182 (1991)

    Article  Google Scholar 

  • Hu, X., Wu, Y.H., Li, L.: Analysis of a new variational model for image multiplicative denoising. J. Inequal. Appl. 2013(1), 568 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Huang, Y.-M., Moisan, L., Ng, M.K., Zeng, T.: Multiplicative noise removal via a learned dictionary. IEEE Trans. Image Process. 21(11), 4534–4543 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Huang, Y.-M., Yan, H.-Y., Zeng, T.: Multiplicative noise removal based on unbiased box-cox transformation. Commun. Comput. Phys. 22(3), 803–828 (2017)

    Article  MathSciNet  Google Scholar 

  • Jin, Z., Yang, X.: Analysis of a new variational model for multiplicative noise removal. J. Math. Anal. Appl. 362(2), 415–426 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Kang, M., Yun, S., Woo, H.: Two-level convex relaxed variational model for multiplicative denoising. SIAM J. Imaging Sci. 6(2), 875–903 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Krishnan, D., Fergus, R.: Fast image deconvolution using hyper-Laplacian priors. In: Advances in Neural Information Processing Systems, pp. 1033–1041 (2009)

    Google Scholar 

  • Laus, F., Steidl, G.: Multivariate myriad filters based on parameter estimation of Student-t distributions. SIAM J. Imaging Sci. 12(4), 1864–1904 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  • Le, T., Vese, L.: Additive and multiplicative piecewise-smooth segmentation models in a variational level set approach. UCLA CAM Report 03-52, University of California at Los Angeles, Los Angeles (2003)

    Google Scholar 

  • Lebrun, M., Colom, M., Buades, A., Morel, J.-M.: Secrets of image denoising cuisine. Acta Numer. 21, 475 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Li, F., Shen, C., Fan, J., Shen, C.: Image restoration combining a total variational filter and a fourth-order filter. J. Vis. Commun. Image Represent. 18(4), 322–330 (2007)

    Article  Google Scholar 

  • Lu, J., Shen, L., Xu, C., Xu, Y.: Multiplicative noise removal in imaging: an exp-model and its fixed-point proximity algorithm. Appl. Comput. Harmon. Anal. 41(2), 518–539 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Makitalo, M., Foi, A.: Optimal inversion of the Anscombe transformation in low-count Poisson image denoising. IEEE Trans. Image Process. 20(1), 99–109 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Mäkitalo, M., Foi, A.: Noise parameter mismatch in variance stabilization, with an application to Poisson–Gaussian noise estimation. IEEE Trans. Image Process. 23(12), 5348–5359 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Mei, J.-J., Dong, Y., Huang, T.-Z., Yin, W.: Cauchy noise removal by nonconvex ADMM with convergence guarantees. J. Sci. Comput. 74(2), 743–766 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Na, H., Kang, M., Jung, M., Kang, M.: Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters. Inverse Probl. Imaging 13(1), 117 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Na, H., Kang, M., Jung, M., Kang, M.: An exp model with spatially adaptive regularization parameters for multiplicative noise removal. J. Sci. Comput. 75(1), 478–509 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Nikolova, M., Ng, M.K., Tam, C.-P.: Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Trans. Image Process. 19(12), 3073–3088 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Ochs, P., Dosovitskiy, A., Brox, T., Pock, T.: On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision. SIAM J. Imaging Sci. 8(1), 331–372 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Rudin, L., Lions, P.-L., Osher, S.: Multiplicative denoising and deblurring: theory and algorithms. In: Geometric Level Set Methods in Imaging, Vision, and Graphics, pp. 103–119. Springer, New York (2003)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Setzer, S., Steidl, G., Teuber, T.: Deblurring Poissonian images by split Bregman techniques. J. Vis. Commun. Image Represent. 21(3), 193–199 (2010)

    Article  Google Scholar 

  • Shama, M.-G., Huang, T.-Z., Liu, J., Wang, S.: A convex total generalized variation regularized model for multiplicative noise and blur removal. Appl. Math. Comput. 276, 109–121 (2016)

    MATH  Google Scholar 

  • Shao, L., Yan, R., Li, X., Liu, Y.: From heuristic optimization to dictionary learning: a review and comprehensive comparison of image denoising algorithms. IEEE Trans. Cybern. 44(7), 1001–1013 (2013)

    Article  Google Scholar 

  • Shi, J., Osher, S.: A nonlinear inverse scale space method for a convex multiplicative noise model. SIAM J. Imaging Sci. 1(3), 294–321 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Singh, P., Jain, L.: A review on denoising of images under multiplicative noise. Int. Res. J. Eng. Technol. (IRJET) 03(04), 574–579 (2016)

    Google Scholar 

  • Steidl, G., Teuber, T.: Removing multiplicative noise by Douglas-Rachford splitting methods. J. Math. Imaging Vis. 36(2), 168–184 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Teuber, T., Lang, A.: A new similarity measure for nonlocal filtering in the presence of multiplicative noise. Comput. Stat. Data Anal. 56(12), 3821–3842 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Tian, D., Du, Y., Chen, D.: An adaptive fractional-order variation method for multiplicative noise removal. J. Inf. Sci. Eng. 32(3), 747–762 (2016)

    MathSciNet  Google Scholar 

  • Ulaby, F., Dobson, M.C., Álvarez-Pérez, J.L.: Handbook of Radar Scattering Statistics for Terrain. Artech House, Norwood (2019)

    Google Scholar 

  • Ullah, A., Chen, W., Khan, M.A.: A new variational approach for restoring images with multiplicative noise. Comput. Math. Appl. 71(10), 2034–2050 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Ullah, A., Chen, W., Khan, M.A., Sun, H.: A new variational approach for multiplicative noise and blur removal. PloS One 12(1), e0161787 (2017)

    Article  Google Scholar 

  • Wang, P., Zhang, H., Patel, V.M.: SAR image despeckling using a convolutional neural network. IEEE Signal Process. Lett. 24(12), 1763–1767 (2017)

    Article  Google Scholar 

  • Wang, G., Pan, Z., Zhang, Z.: Deep CNN Denoiser prior for multiplicative noise removal. Multimed. Tools Appl. 78(20), 29007–29019 (2019)

    Article  Google Scholar 

  • Xiao, L., Huang, L.-L., Wei, Z.-H.: A Weberized total variation regularization-based image multiplicative noise removal algorithm. EURASIP J. Adv. Signal Process. 2010, 1–15 (2010)

    Google Scholar 

  • Xie, H., Pierce, L.E., Ulaby, F.T.: Statistical properties of logarithmically transformed speckle. IEEE Trans. Geosci. Remote Sens. 40(3), 721–727 (2002)

    Article  Google Scholar 

  • Yun, S., Woo, H.: A new multiplicative denoising variational model based on mth root transformation. IEEE Trans. Image Process. 21(5), 2523–2533 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, X.-L., Wang, F., Ng, M.K.: A new convex optimization model for multiplicative noise and blur removal. SIAM J. Imaging Sci. 7(1), 456–475 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, C.-P., Feng, X.-C., Jia, X.-X., He, R.-Q., Xu, C.: Root-transformation based multiplicative denoising model and its statistical analysis. Neurocomputing 275, 2666–2680 (2018)

    Article  Google Scholar 

  • Zhang, K., Zuo, W., Chen, Y., Meng, D., Zhang, L.: Beyond a gaussian denoiser: residual learning of deep CNN for image denoising. IEEE Trans. Image Process. 26(7), 3142–3155 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, China

    Xiangchu Feng & Xiaolong Zhu

Authors
  1. Xiangchu Feng
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  2. Xiaolong Zhu
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Corresponding author

Correspondence to Xiangchu Feng .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Liverpool, Liverpool, UK

    Ke Chen

  2. Dept Appl Math & Theoretical Phys, University of Cambridge, Cambridge, UK

    Carola-Bibiane Schönlieb

  3. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Xue-Cheng Tai

  4. Department of Applied Mathematics & Stat, Johns Hopkins University, Baltimore, MD, USA

    Laurent Younces

Section Editor information

  1. No affiliation provided

    Ke Chen

  2. No affiliation provided

    Laurent Younes

  3. No affiliation provided

    Xue-Cheng Tai

  4. No affiliation provided

    Carola-Bibiane Schönlieb

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Feng, X., Zhu, X. (2021). Models for Multiplicative Noise Removal. In: Chen, K., Schönlieb, CB., Tai, XC., Younces, L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham. https://doi.org/10.1007/978-3-030-03009-4_60-1

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  • DOI: https://doi.org/10.1007/978-3-030-03009-4_60-1

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