Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T08:38:45.192Z Has data issue: false hasContentIssue false
  • English
  • Français

A COMBINED FIRST-ORDER AND SECOND-ORDER VARIATION APPROACH FOR MULTIPLICATIVE NOISE REMOVAL

Published online by Cambridge University Press:  22 December 2014

LE JIANG ,
JIN HUANG ,
JUN LIU  and
XIAO-GUANG LV
LE JIANG
Affiliation:
School of Science, Huaihai Institute of Technology, Lianyungang, Jiangsu, 222005, China email [email protected], [email protected] School of Mathematical Sciences/Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, China email [email protected], [email protected]
JIN HUANG
Affiliation:
School of Mathematical Sciences/Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, China email [email protected], [email protected]
JUN LIU
Affiliation:
School of Mathematical Sciences/Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, China email [email protected], [email protected]
XIAO-GUANG LV*
Affiliation:
School of Science, Huaihai Institute of Technology, Lianyungang, Jiangsu, 222005, China email [email protected], [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Denoising of images corrupted by multiplicative noise is an important task in various applications, such as laser imaging, synthetic aperture radar and ultrasound imaging. We propose a combined first-order and second-order variational model for removal of multiplicative noise. Our model substantially reduces the staircase effects while preserving edges in the restored images, since it combines advantages of the first-order and second-order total variation. The issues of existence and uniqueness of a minimizer for this variational model are analysed. Moreover, a gradient descent method is employed to solve the associated Euler–Lagrange equation, and several numerical experiments are given to show the efficiency of our model. In particular, a comparison with an existing model in terms of peak signal-to-noise ratio and structural similarity index is provided.

Keywords

68U10multiplicative noise removaldenoisingtotal variationEuler–Lagrange equationstructural similarity index
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 2 , October 2014 , pp. 116 - 137
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Andrew, H. C. and Hunt, B. R., Digital image restoration (Prentice Hall, Englewood Cliffs, NJ, 1977).Google Scholar
Aubert, G. and Aujol, J. F., “A variational approach to remove multiplicative noise”, SIAM J. Appl. Math. 68 (2008) 925–946; doi:10.1137/060671814.CrossRefGoogle Scholar
Banham, M. R. and Kataggelos, A. K., “Digital image restoration”, IEEE Signal Processing Mag. 14(2) (1997) 24–41; doi:10.1109/79.581363.Google Scholar
Bertero, M. and Boccacci, P., Introduction to inverse problems in imaging (Institute of Physics Publishing, London, 1998); doi:10.1887/0750304359.CrossRefGoogle Scholar
Bioucas-Dias, J. M. and Figueiredo, M. A. T., “Multiplicative noise removal using variable splitting and constrained optimization”, IEEE Trans. Image Process. 19(7) (2010) 1720–1730 ;doi:10.1109/TIP.2010.2045029.Google Scholar
Buades, A., Coll, B. and Morel, J. M., “A review of image denoising algorithms, with a new one”, SIAM Multiscale Model. Simul. 4 (2005) 490–530; doi:10.1137/040616024.CrossRefGoogle Scholar
Chambolle, A., “An algorithm for total variation minimization and applications”, J. Math. Imaging Vision 20 (2004) 89–97; doi:10.1023/B:JMIV.0000011325.36760.1e.Google Scholar
Chan, T. F., Marquina, A. and Mulet, P., “High-order total variation-based image restoration”, SIAM J. Sci. Comput. 22 (2000) 503–516; doi:10.1137/S1064827598344169.Google Scholar
Chan, T. F. and Shen, J. H., Image processing and analysis: variational, PDE, wavelet, and stochastic methods (SIAM, Philadelphia, 2005); doi:10.1137/1.9780898717877.Google Scholar
Chen, D. Q. and Cheng, L. Z., “Spatially adapted total variation model to remove multiplicative noise”, IEEE Trans. Image Process 21 (2007) 1650–1662; doi:10.1109/TIP.2011.2172801.Google Scholar
Chen, H. Z., Song, J. P. and Tai, X. C., “A dual algorithm for minimization of the LLT model”, Adv. Comput. Math. 31 (2009) 115–130; doi:10.1007/s10444-008-9097-0.Google Scholar
Crimmins, T. R., “Geometric filter for reducing speckle”, Opt. Eng. 25 (1986) 651–654 ;doi:10.1117/12.949543.Google Scholar
Feinberg, E. A., Kasyanov, P. O. and Zadoianchuk, N. V., ‘Fatou’s lemma for weakly converging probabilities’, Preprint, 2012, arXiv:1206.4073.Google Scholar
Fox, C., An introduction to the calculus of variations (Dover Publications, New York, 1987).Google Scholar
Goldstein, T. and Osher, S., “The split Bregman method for L1-regularized problems”, SIAM J. Imaging Sci. 2 (2009) 323–343; doi:10.1137/080725891.Google Scholar
Goodman, J., Speckle phenomena in optics: theory and applications (Roberts & Company Publications, Greenwood Village, CO, 2007).Google Scholar
Green, P. J., “Reversible jump MCMC computation and Bayesian model, determination”, Biometrika 82 (1995) 711–732.Google Scholar
Hansen, P. C., Nagy, J. G. and O’Leary, D.  P., Deblurring images: matrices, spectra, and filtering (SIAM, Philadelphia, 2006); doi:10.1137/1.9780898718874.Google Scholar
Herman, G. T., Fundamentals of computerized tomography: image reconstruction from projections, 2nd edn (Springer, New York, 2009); doi:10.1007/978-1-84628-723-7.Google Scholar
Huang, Y. M., Ng, M. K. and Wen, Y. W., “A new total variation method for multiplicative noise removal”, SIAM J. Imag. Sci. 2 (2009) 20–40; doi:10.1137/080712593.Google Scholar
Kornprobst, P., Deriche, R. and Aubert, G., “Image sequence analysis via partial differential equations”, J. Math. Imaging Vision 11 (1999) 5–26; doi:10.1007/s11263-005-4882-4.Google Scholar
Krissian, K., Westin, C. F., Kikinis, R. and Vosburgh, K. G., “Oriented speckle reducing anisotropic diffusion”, IEEE Trans. Image Process. 16 (2007) 1412–1424; doi:10.1109/TIP.2007.891803.Google Scholar
Laobeul, N. D., ‘Regularization methods for ill-posed Poisson imaging problems’, Ph.D. Thesis, University of Montana, 2008.Google Scholar
Lee, P. M., Bayesian statistics: an introduction (Oxford University Press, New York, 1989).Google Scholar
Li, F., Ng, M. and Shen, C. M., “Multiplicative noise removal with spatial-varying regularization parameters”, SIAM J. Imaging Sci. 3 (2010) 1–20; doi:10.1137/090748421.Google Scholar
Li, F., Shen, C. M., Fan, J. S. and Shen, C. L., “Image restoration combining a total variational filter and a fourth-order filter”, J. Vis. Commun. Image Representation 18 (2007) 322–330 ;doi:10.1016/j.jvcir.2007.04.005.CrossRefGoogle Scholar
Lysaker, M., Lundervold, A. and Tai, X. C., “Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time”, IEEE Trans. Image Process. 12 (2003) 1579–1590; doi:10.1109/TIP.2003.819229.Google Scholar
Lysaker, M. and Tai, X.  C., “Iterative image restoration combining total varition minimization and a second-order functional”, Int. J. Comput. Vision 66 (2006) 5–18; doi:10.1007/s11263-005-3219-7.Google Scholar
Marquina, A. and Osher, S., “Explicit algorithms for a new time dependent model based on level set motion for nonliear deblurring and noise removal”, SIAM J Sci. Comput. 22(2) (2000) 387–405 ;doi:10.1137/S1064827599351751.Google Scholar
Nagy, J. G., Palmer, K. and Perrone, L., “Iterative methods for image deblurring: a Matlab object-oriented approach”, Numer. Algorithms 36 (2004) 73–93 ;doi:10.1023/B:NUMA.0000027762.08431.64.Google Scholar
Naylor, A. W. and Sell, G. R., Linear operator theory in engineering and science (Springer, Berlin, 1982); doi:10.1007/978-1-4612-5773-8.Google Scholar
Oliver, C. and Quegan, S., Understanding synthetic aperture radar images (SciTech Publishing, Raleigh, NC, 2004).Google Scholar
Papafitsoros, K. and Schönlieb, C. B., “A combined first and second order variational approach for image reconstruction”, J. Math. Imaging Vision 48 (2014) 308–338 ;doi:10.1007/s10851-013-0445-4.Google Scholar
Riley, K.  F., Hobson, M.  P. and Bence, S.  J., Mathematical methods for physics and engineering (Cambridge University Press, Cambridge, 2006); doi:10.1017/CBO9780511810763.Google Scholar
Rudin, L., Lions, P.  L. and Osher, S., “Multiplicative denoising and deblurring: theory and algorithms”, in: Geometric level sets in imaging, vision and graphics (eds Osher, S. and Paragios, N.), (Springer, New York, 2003) 103–119; doi:10.1007/0-387-21810-6_6.Google Scholar
Rudin, L., Osher, S. and Fatemi, E., “Nonlinear total variation based noise removal algorithms”, Physica D 60 (1992) 259–268; doi:10.1016/0167-2789(92)90242-F.Google Scholar
Shi, J. and Osher, S., “A nonlinear inverse scale space method for a convex multiplicative noise model”, SIAM J. Imag. Sci. 1 (2008) 294–321; doi:10.1137/070689954.Google Scholar
Steidl, G., “A note on the dual treatment of higher order regularization functionals”, Computing 76 (2005) 135–148; doi:10.1007/s00607-005-0129-z.Google Scholar
Steidl, G. and Teuber, T., “Removing multiplicative noise by Douglas-Rachford splitting methods”, J. Math. Imaging Vision 36 (2010) 168–184; doi:10.1007/s10851-009-0179-5.CrossRefGoogle Scholar
Wagner, R. F., Smith, S. W. and Sandrik, J. M., “Statistics of speckle in ultrasound B-scans”, IEEE Trans. Sonics Ultrason 30 (1983) 156–163; doi:10.1109/T-SU.1983.31404.Google Scholar
Wang, Z., Bovik, A.  C., Sheikh, H.  R. and Simoncelli, E.  P., “Image quality assessment: from error visibility to structural similarity”, IEEE Trans. Image Process. 13(4) (2004) 600–612; doi:10.1109/TIP.2003.819861.Google Scholar
Wu, Y. and Maitre, H., “Smoothing speckled synthetic aperture radar images by using maximum homogeneous region filters”, Opt. Eng. 31 (1992) 1785–1792; doi:10.1117/12.59897.Google Scholar
Yu, Y. and Acton, S. T., “Speckle reducing anisotropic diffusion”, IEEE Trans. Image Process. 11 (2002) 1260–1270; doi:10.1109/TIP.2002.804276.Google Scholar