An efficient algorithm for adaptive total variation based image decomposition and restoration

Liu, X. ; Huang, L.

International Journal of Applied Mathematics and Computer Science, Tome 24 (2014) no. 2, pp. 405-415.

Résumé

With the aim to better preserve sharp edges and important structure features in the recovered image, this article researches an improved adaptive total variation regularization and H^-1 norm fidelity based strategy for image decomposition and restoration. Computationally, for minimizing the proposed energy functional, we investigate an efficient numerical algorithm—the split Bregman method, and briefly prove its convergence. In addition, comparisons are also made with the classical OSV (Osher–Sole–Vese) model (Osher et al., 2003) and the TV-Gabor model (Aujol et al., 2006), in terms of the edge-preserving capability and the recovered results. Numerical experiments markedly demonstrate that our novel scheme yields significantly better outcomes in image decomposition and denoising than the existing models.

Détail

Keywords: image decomposition, image restoration, adaptive total variation, H-1 norm, split Bregman method
Mots-clés : dekompozycja obrazu, odtworzenie obrazu, wariacja zupełna adaptacyjna, metoda Bregmana

Bibliographie

[1] Aujol, J.-F., Aubert, G., Blanc-Féraud, L. and Chambolle, A. (2005). Image decomposition into a bounded variation component and an oscillating component, Journal of Mathematical Imaging and Vision 22(1): 71–88.

[2] Aujol, J.-F., Gilboa, G., Chan, T. and Osher, S. (2006). Structure-texture image decomposition—modeling, algorithms, and parameter selection, International Journal of Computer Vision 67(1): 111–136.

[3] Aujol, J.-F. and Gilboa, G. (2006). Constrained and SNR-based solutions for TV-Hilbert space image denoising, Journal of Mathematical Imaging and Vision 26(1–2): 217–237.

[4] Barcelos, C.A.Z. and Chen Y. (2000). Heat flows and related minimization problem in image restoration, Computers Mathematics with Applications 39(5–6): 81–97.

[5] Cai, J.F., Osher, S. and Shen, Z. (2009). Split Bregman methods and frame based image restoration, CAM Report 09-28, UCLA, Los Angeles, CA.

[6] Chambolle, A. (2004). An algorithm for total variation minimization and application, Journal of Mathematical Imaging and Vision 20(1–2): 89–97.

[7] Chan, T.F., Golub, G.H. and Mulet, P. (1999). A nonlinear primal-dual method for total variation-based image restoration, SIAM Journal on Scientific Computing 20(6): 1964–1977.

[8] Chan, T.F., Esedoglu, S. and Park, F.E. (2007). Image decomposition combining staircase reducing and texture extraction, Journal of Visual Communication and Image Representation 18(6): 464–486.

[9] Chen, Y. and Wunderli, T. (2002). Adaptive total variation for image restoration in BV space, Journal of Mathematical Analysis and Applications 272(1): 117–137.

[10] Daubechies, I. and Teschke, G. (2005). Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring and denoising, Applied and Computational Harmonic Analysis 19(1): 1–16.

[11] Goldstein, T. and Osher, S. (2009). The split Bregman algorithm for L1 regularized problems, SIAM Journal on Imaging Sciences 2(2): 323–343.

[12] Hajiaboli, M.R. (2010). A self-governing fourth-order nonlinear diffusion filter for image noise removal, IPSJ Transactions on Computer Vision and Applications 2: 94–103.

[13] Jia, R.-Q., Zhao, H. and Zhao, W. (2009). Convergence analysis of the Bregman method for the variational model of image denoising, Applied and Computational Harmonic Analysis 27(3): 367–379.

[14] Liu, X. and Huang, L. (2010). Split Bregman iteration algorithm for total bounded variation regularization based image deblurring, Journal of Mathematical Analysis and Applications 372(2): 486–495.

[15] Liu, X., Huang, L. and Guo, Z. (2011). Adaptive fourth-order partial differential equation filter for image denoising, Applied Mathematics Letters 24(8): 1282–1288.

[16] Liu, X. and Huang, L. (2012). Total bounded variation based Poissonian images recovery by split Bregman iteration, Mathematical Methods in the Applied Sciences 35(5): 520–529.

[17] Liu, X. and Huang, L. (2013). Poissonian image reconstruction using alternating direction algorithm, Journal of Electronic Imaging 22(3): 033007.

[18] Liu, X. and Huang, L. (2014). A new nonlocal total variation regularization algorithm for image denoising, Mathematics and Computers in Simulation 97: 224–233.

[19] Meyer, Y. (2001). Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, Vol. 22, AMS, Providence, RI.

[20] Ng, M.K., Yuan, X.M. and Zhang, W.X. (2013). A coupled variational image decomposition and restoration model for blurred cartoon-plus-texture images with missing pixels, IEEE Transactions on Image Processing 22(6): 2233–2246.

[21] Osher, S., Solé, A. and Vese, L. (2003). Image decomposition and restoration using total variation minimization and H−1 norm, Multiscale Modeling Simulation 1(3): 349–370.

[22] Prasath, V.B.S. (2011). A well-posed multiscale regularization scheme for digital image denoising, International Journal of Applied Mathematics and Computer Science 21(4): 769–777, DOI: 10.2478/v10006-011-0061-7.

[23] Rudin, L., Osher, S. and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms, Physica D 60(1–4): 259–268.

[24] Setzer, S., Steidl, G. and Teuber T. (2010). Deblurring Poissonian images by split Bregman techniques, Journal of Visual Communication and Image Representation 21(3): 193–199.

[25] Strong, D.M. and Chan, T.F. (1996). Spatially and scale adaptive total variation based regularization and anisotropic diffusion in image processing, CAMReport 96-46, UCLA, Los Angeles, CA.

[26] Szlam, A., Guo, Z. and Osher S. (2010). A split Bregman method for non-negative sparsity penalized least squares with applications to hyperspectral demixing, CAM Report 10-06, UCLA, Los Angeles, CA.

[27] Vese, L. and Osher, S. (2003). Modeling textures with total variation minimization and oscillating patterns in image processing, Journal of Scientific Computing 19(1–3): 553–572.

[28] Wang, Y., Yang, J., Yin, W. and Zhang, Y. (2007). A new alternating minimization algorithm for total variation image reconstruction, CAAM Technical Report, TR07–10, Rice University, Houston, TX.

[29] Zhang, X., Burger, M., Bresson, X. and Osher, S. (2009). Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, CAM Report 09-03, UCLA, Los Angeles, CA.