Multiplicative Noise Removal Based on the Smooth Diffusion Equation

  • Xiujie Shan
  • Jiebao Sun
  • Zhichang GuoEmail author


The multiplicative noise removal problem is of momentous significance in various image processing applications. In this paper, a nonlinear diffusion equation with smooth solution is proposed to remove multiplicative Gamma noise. The diffusion coefficient takes full advantage of two features of multiplicative noise image, namely, gradient information and gray level information, which makes the model has the ability to remove high level noise effectively and protect the edges. The existence of the solution has been analyzed by Schauder’s fixed-point theorem. Some other theoretical properties such as the maximum principle are also presented in the paper. In the numerical aspect, the explicit finite difference method, fast explicit diffusion method, additive operator splitting method and Krylov subspace spectral method are employed to implement the proposed model. Experimental results show that the fast explicit diffusion method achieves a better trade-off between computational time and denoising performance, and the Krylov subspace spectral method gets better restored results in the visual aspect. In addition, the capability of the proposed model for denoising is illustrated by comparison with other denoising models.


Multiplicative noise removal Diffusion equation Smooth solution Fast algorithm 



The authors would like to thank the reviewers for the valuable suggestions to the paper and professor Patrick Guidotti from the University of California, Irvine, for sharing his code. This work is partially supported by the Natural Science Foundation of Heilongjiang Province of China (Grant Nos. LC2018001, A2016003), the National Science Foundation of China (Grant Nos. U1637208, 51476047), the Natural Science Foundation (Grant No. 11871133), the Fundamental Research Funds for the Central Universities and Program for Innovation Research of Science in Harbin Institute of Technology (PIRS OF HIT 201609), and the Fundamental Research Funds for the Central Universities and Program for Innovation Research of Science in Harbin Institute of Technology (PIRS OF HIT 201601).


  1. 1.
    Abd-Elmoniem, K., Youssef, A.B., Kadah, Y.: Real-time speckle reduction and coherence enhancement in ultrasound imaging via nonlinear anisotropic diffusion. IEEE Trans. Biomed. Eng. 49(9), 997–1014 (2002)Google Scholar
  2. 2.
    Achim, A., Tsakalides, P., Bezerianos, A.: SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling. IEEE Trans. Geosci. Remote Sens. 41(8), 1773–1784 (2003)Google Scholar
  3. 3.
    Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65, pp. 44–173. Academic Press, Boston (1975)Google Scholar
  4. 4.
    Argenti, F., Lapini, A., Bianchi, T., Alparone, L.: A tutorial on speckle reduction in synthetic aperture radar images. IEEE Geosci. Remote Sens. Mag. 1(3), 6–35 (2013)Google Scholar
  5. 5.
    Aubert, G., Aujol, J.F.: A variational approach to removing multiplicative noise. SIAM J. Appl. Math. 68(4), 925–946 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Balocco, S., Gatta, C., Pujol, O., Mauri, J., Radeva, P.: SRBF: speckle reducing bilateral filtering. Ultrasound Med. Biol. 36(8), 1353–1363 (2010)Google Scholar
  7. 7.
    Bhuiyan, M.I.H., Ahmad, M.O., Swamy, M.N.S.: Spatially adaptive wavelet-based method using the Cauchy prior for denoising the SAR images. IEEE Trans. Circuits Syst. Video Technol. 17(4), 500–507 (2007)Google Scholar
  8. 8.
    Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Catté, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chan, T., Marquina, A., Mulet, P.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)Google Scholar
  11. 11.
    Chan, T.F., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chen, Y., Huang, T.Z., Deng, L.J., Zhao, X.L., Wang, M.: Group sparsity based regularization model for remote sensing image stripe noise removal. Neurocomputing 267, 95–106 (2017)Google Scholar
  13. 13.
    Deledalle, C.A.: Image denoising beyond additive Gaussian noise-Patch-based estimators and their application to SAR imagery. Ph.D. thesis, Télécom ParisTech (2011)Google Scholar
  14. 14.
    Dey, N., BlancFeraud, L., Zimmer, C., Roux, P., Kam, Z., OlivoMarin, J., Zerubia, J.: Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution. Microsc. Res. Tech. 69(4), 260–266 (2006)Google Scholar
  15. 15.
    Dobson, D.C., Santosa, F.: Recovery of blocky images from noisy and blurred data. SIAM J. Appl. Math. 56(4), 1181–1198 (1996)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Dong, G., Guo, Z., Wu, B.: A convex adaptive total variation model based on the gray level indicator for multiplicative noise removal. Abstr. Appl. Anal. 2013, 912373 (2013). MathSciNetGoogle Scholar
  17. 17.
    Dong, Y., Zeng, T.: A convex variational model for restoring blurred images with multiplicative noise. SIAM J. Imaging Sci. 6(3), 1598–1625 (2013)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Durand, S., Fadili, J., Nikolova, M.: Multiplicative noise removal using \(L_{1}\) fidelity on frame coefficients. J. Math. Imaging Vis. 36(3), 201–226 (2010)Google Scholar
  19. 19.
    Evans, L.C.: Weak convergence methods for nonlinear partial differential equations, pp. 4–14. Loyola University of Chicago, Chicago (1988)Google Scholar
  20. 20.
    Evans, L.C.: Partial differential equations. Humphreys, J.E., Saltman, D.J., Sattinger, D., Shaneson, J.L. (eds.) Graduate Studies in Mathematics, vol. 19, 2nd edn., pp. 301–305. American Mathematical Society, Providence (2010)Google Scholar
  21. 21.
    Fabbrini, L., Greco, M., Messina, M., Pinelli, G.: Improved edge enhancing diffusion filter for speckle-corrupted images. IEEE Geosci. Remote Sens. Lett. 11(1), 99–103 (2014)Google Scholar
  22. 22.
    Fan, J., Wu, Y., Wang, F., Zhang, Q., Liao, G., Li, M.: SAR image registration using phase congruency and nonlinear diffusion-based SIFT. IEEE Geosci. Remote Sens. Lett. 12(3), 562–566 (2015)Google Scholar
  23. 23.
    Frost, V.S., Stiles, J.A., Shanmugan, K.S., Holtzman, J.C.: A model for radar images and its application to adaptive digital filtering of multiplicative noise. IEEE Trans. Pattern Anal. Mach. Intell. 4(2), 157–166 (1982)Google Scholar
  24. 24.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Gilboa, G., Sochen, N., Zeevi, Y.Y.: Image enhancement and denoising by complex diffusion processes. IEEE Trans. Pattern Anal. Mach. Intell. 26(8), 1020–1036 (2004)Google Scholar
  26. 26.
    Guidotti, P., Kim, Y., Lambers, J.: Image restoration with a new class of forward-backward-forward diffusion equations of Perona–Malik type with applications to satellite image enhancement. SIAM J. Imaging Sci. 6(3), 1416–1444 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Krissian, K., Westin, C.F., Kikinis, R., Vosburgh, K.G.: Oriented speckle reducing anisotropic diffusion. IEEE Trans. Image Process. 16(5), 1412–1424 (2007)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kuan, D.T., Sawchuk, A.A., Strand, T.C., Chavel, P.: Adaptive noise smoothing filter for images with signal-dependent noise. IEEE Trans. Pattern Anal. Mach. Intell. PAMI–7(2), 165–177 (1985)Google Scholar
  29. 29.
    Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasi-linear equations of parabolic type. In: Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence, RI (1968)Google Scholar
  30. 30.
    Lee, J.S.: Digital image enhancement and noise filtering by use of local statistics. IEEE Trans. Pattern Anal. Mach. Intell. 2(2), 165–168 (1980)Google Scholar
  31. 31.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations, pp. 100–108. Springer, Berlin (1971)Google Scholar
  32. 32.
    Oliver, C., Quegan, S.: Understanding Synthetic Aperture Radar Images. SciTech Publishing, Raleigh (2004)Google Scholar
  33. 33.
    Palchak, E.M., Cibotarica, A., Lambers, J.V.: Solution of time-dependent PDE through rapid estimation of block Gaussian quadrature nodes. Linear Algebra Appl. 468, 233–259 (2015)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)Google Scholar
  35. 35.
    Rudin, L.I., 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
  36. 36.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D: Nonlinear Phenom. 60(1–4), 259–268 (1992)MathSciNetzbMATHGoogle Scholar
  37. 37.
    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)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Simon, J.: Compact sets in the space \(L_{p}(O, T; B)\). Ann. Mat. Pura Appl. 146(1), 65–96 (1986)Google Scholar
  39. 39.
    Steidl, G., Teuber, T.: Removing multiplicative noise by Douglas–Rachford splitting methods. J. Math. Imaging Vis. 36(2), 168–184 (2010)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Teuber, T., Steidl, G., Chan, R.H.: Minimization and parameter estimation for seminorm regularization models with \(I\)-divergence constraints. Inv. Prob. 29(3), 035007 (2013)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Tychonoff, A.: Ein fixpunktsatz. Math. Ann. 111, 767–776 (1935)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Weickert, J.: Anisotropic Diffusion in Image Processing, pp. 14–27. B. G. Teubner Stuttgart, Leipzig (1998)zbMATHGoogle Scholar
  43. 43.
    Weickert, J., Grewenig, S., Schroers, C., Bruhn, A.: Cyclic schemes for PDE-based image analysis. Int. J. Comput. Vis. 118(3), 275–299 (2016)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Weickert, J., Romeny, B.M.T.H., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7(3), 398–410 (1998)Google Scholar
  45. 45.
    Yu, Y., Acton, S.T.: Speckle reducing anisotropic diffusion. IEEE Trans. Image Process. 11(11), 1260–1270 (2002)MathSciNetGoogle Scholar
  46. 46.
    Yun, S., Woo, H.: A new multiplicative denoising variational model based on \(m\)th root transformation. IEEE Trans. Image Process. 21(5), 2523–2533 (2012)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Zhang, J., Wei, Z., Xiao, L.: Adaptive fractional-order multi-scale method for image denoising. J. Math. Imaging Vis. 43(1), 39–49 (2012)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Zhang, Q., Wu, Y., Wang, F., Fan, J., Zhang, L., Jiao, L.: Anisotropic-scale-space-based salient-region detection for SAR images. IEEE Geosci. Remote Sens. Lett. 13(3), 457–461 (2016)Google Scholar
  49. 49.
    Zhou, Z., Guo, Z., Dong, G., Sun, J., Zhang, D., Wu, B.: A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal. IEEE Trans. Image Process. 24(1), 249–260 (2015)MathSciNetGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinChina

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