## A total variation model based on the strictly convex modification for image denoising.(English)Zbl 1474.94032

Summary: We propose a strictly convex functional in which the regular term consists of the total variation term and an adaptive logarithm based convex modification term. We prove the existence and uniqueness of the minimizer for the proposed variational problem. The existence, uniqueness, and long-time behavior of the solution of the associated evolution system is also established. Finally, we present experimental results to illustrate the effectiveness of the model in noise reduction, and a comparison is made in relation to the more classical methods of the traditional total variation (TV), the Perona-Malik (PM), and the more recent D-$$\alpha$$-PM method. Additional distinction from the other methods is that the parameters, for manual manipulation, in the proposed algorithm are reduced to basically only one.

### MSC:

 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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### References:

 [1] Chang, T.; Kuo, C. C. J., Texture analysis and classification with tree-structured wavelet transform, IEEE Transactions on Image Processing, 2, 4, 429-441, (1993) [2] Scholkmann, F.; Revol, V.; Kaufmann, R.; Baronowski, H.; Kottler, C., A new method for fusion, denoising and enhancement of x-ray images retrieved from Talbot-Lau grating interferometry, Physics in Medicine and Biology, 59, 6, 1425, (2014) [3] Ma, J.; Plonka, G., Combined curvelet shrinkage and nonlinear anisotropic diffusion, IEEE Transactions on Image Processing, 16, 9, 2198-2206, (2007) [4] Candes, E.; Donoho, D. L., Curvelets: a surprisingly effective nonadaptive representation for objects with edges, (2000) [5] Candès, E. J.; Donoho, D. L., New tight frames of curvelets and optimal representations of objects with piecewise $$C^2$$ singularities, Communications on Pure and Applied Mathematics, 57, 2, 219-266, (2004) · Zbl 1038.94502 [6] Candès, E.; Demanet, L.; Donoho, D.; Ying, L., Fast discrete curvelet transforms, Multiscale Modeling & Simulation, 5, 3, 861-899, (2006) · Zbl 1122.65134 [7] Perona, P.; Malik, J., Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 7, 629-639, (1990) [8] Rudin, L. I.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60, 1–4, 259-268, (1992) · Zbl 0780.49028 [9] Chan, T. F.; Esedo\=glu, S., Aspects of total variation regularized $$L^1$$ function approximation, SIAM Journal on Applied Mathematics, 65, 5, 1817-1837, (2005) · Zbl 1096.94004 [10] Weickert, J.; Ter Haar Romeny, B. M.; Viergever, M. A., Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Transactions on Image Processing, 7, 3, 398-410, (1998) [11] Guo, Z.; Sun, J.; Zhang, D.; Wu, B., Adaptive Perona-Malik model based on the variable exponent for image denoising, IEEE Transactions on Image Processing, 21, 3, 958-967, (2012) · Zbl 1372.94102 [12] Catté, F.; Lions, P.-L.; Morel, J.-M.; Coll, T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM Journal on Numerical Analysis, 29, 1, 182-193, (1992) · Zbl 0746.65091 [13] Chen, Y.; Rao, M., Minimization problems and associated flows related to weighted $$p$$ energy and total variation, SIAM Journal on Mathematical Analysis, 34, 5, 1084-1104, (2003) · Zbl 1038.49007 [14] Maiseli, B. J.; Liu, Q.; Elisha, O. A.; Gao, H., Adaptive Charbonnier superresolution method with robust edge preservation capabilities, Journal of Electronic Imaging, 22, 4, (2013) [15] Shah, J., Common framework for curve evolution, segmentation and anisotropic diffusion, Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR ’96) [16] Barash, D.; Comaniciu, D., A common framework for nonlinear diffusion, adaptive smoothing, bilateral filtering and mean shift, Image and Vision Computing, 22, 1, 73-81, (2004) [17] Koenderink, J. J., The structure of images, Biological Cybernetics, 50, 5, 363-370, (1984) · Zbl 0537.92011 [18] Witkin, A. P., Scale-space filtering [19] Weickert, J., Anisotropic Diffusion in Image Processing, 1, (1998), Stuttgart, Germany: Teubner, Stuttgart, Germany · Zbl 0886.68131 [20] Chen, Y.; Wunderli, T., Adaptive total variation for image restoration in BV space, Journal of Mathematical Analysis and Applications, 272, 1, 117-137, (2002) · Zbl 1020.68104 [21] Chan, T. F.; Shen, J., Mathematical models for local nontexture inpaintings, SIAM Journal on Applied Mathematics, 62, 3, 1019-1043, (2001) · Zbl 1050.68157 [22] Vogel, C. R., Total variation regularization for Ill-posed problems, (1993), Department of Mathematical Sciences, Montana State University [23] Vese, L., Problemes variationnels et EDP pour lA analyse dA images et lA evolution de courbes [Ph.D. thesis], (1996), Universite de Nice Sophia-Antipolis [24] Chen, Y.; Levine, S.; Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM Journal on Applied Mathematics, 66, 4, 1383-1406, (2006) · Zbl 1102.49010 [25] Chan, T.; Marquina, A.; Mulet, P., High-order total variation-based image restoration, SIAM Journal on Scientific Computing, 22, 2, 503-516, (2000) · Zbl 0968.68175 [26] Andreu-Vaillo, F.; Caselles, V.; Mazón, J. M., Parabolic QuasiLinear Equations Minimizing Linear Growth Functionals, 223, (2004), Springer · Zbl 1053.35002 [27] Acar, R.; Vogel, C. R., Analysis of bounded variation penalty methods for ill-posed problems, Inverse Problems, 10, 6, 1217-1229, (1994) · Zbl 0809.35151 [28] Strong, D. M.; Chan, T. F., Spatially and scale adaptive total variation based regularization and anisotropic diffusion in image processing, Diusion in Image Processing, (1996), UCLA Math Department CAM Report, Cite-seer [29] Chambolle, A.; Lions, P.-L., Image recovery via total variation minimization and related problems, Numerische Mathematik, 76, 2, 167-188, (1997) · Zbl 0874.68299 [30] You, Y.-L.; Xu, W.; Tannenbaum, A.; Kaveh, M., Behavioral analysis of anisotropic diffusion in image processing, IEEE Transactions on Image Processing, 5, 11, 1539-1553, (1996) [31] Vese, L., A study in the BV space of a denoising-deblurring variational problem, Applied Mathematics and Optimization, 44, 2, 131-161, (2001) · Zbl 1003.35009 [32] Marquina, A.; Osher, S., Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM Journal on Scientific Computing, 22, 2, 387-405, (2000) · Zbl 0969.65081 [33] Zhou, X., An evolution problem for plastic antiplanar shear, Applied Mathematics and Optimization, 25, 3, 263-285, (1992) · Zbl 0758.73014 [34] Aubert, G.; Kornprobst, P., Mathematical Problems in Image Processing, 147, (2006), New York, NY, USA: Springer, New York, NY, USA · Zbl 1019.94002 [35] Doob, J. L., Measure Theory, 143, (1994), New York, NY, USA: Springer, New York, NY, USA [36] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems, 254, (2000), Oxford, UK: Clarendon Press, Oxford, UK · Zbl 0957.49001 [37] Niculescu, C. P.; Persson, L.-E., Convex Functions and Their Applications: : A Contemporary Approach, 23, (2006), Springer [38] Renardy, M.; Rogers, R. C., An Introduction to Partial Differential Equations, 13, (2004), Springer · Zbl 1072.35001 [39] Brézis, H., Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, 50, (1973), North-Holland · Zbl 0252.47055 [40] Durand, S.; Fadili, J.; Nikolova, M., Multiplicative noise removal using $$L^1$$ fidelity on frame coefficients, Journal of Mathematical Imaging and Vision, 36, 3, 201-226, (2010) [41] Wang, Z.; Bovik, A. C.; Sheikh, H. R.; Simoncelli, E. P., Image quality assessment: from error visibility to structural similarity, IEEE Transactions on Image Processing, 13, 4, 600-612, (2004) [42] Ogada, E. A.; Guo, Z.; Wu, B., An alternative variational framework for image denoising, Abstract and Applied Analysis, 2014, (2014) · Zbl 1474.94024 [43] Yu, Y.; Acton, S. T., Speckle reducing anisotropic diffusion, IEEE Transactions on Image Processing, 11, 11, 1260-1270, (2002)
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