Abstract Total variation model is widely used in machine vision due to its strong ability of capturing the details of the images and the videos. Curvelet transform can capture the edges and curved lines of the 2D signals easily. Combining both advantages, a class of joint sparse representation model is proposed, i.e. total variation and curvelet (TVC). This model can represent the characteristics of the 2D signals more effectively. Primal-dual (PD) scheme is used to solve the model, which is called PDTVC algorithm. Experimental results show that PDTVC outperforms the existing algorithms in both subjective visual effect and objective image qualities. PDTVC can be applied to various challenging image processing tasks as well, such as deblurring and super resolution.
[1] Gonzalez R C, Richard E. Digital Image Processing. New York, USA: Prentice-Hall, 2002 [2] Rudin L I, Osher S. Total Variation Based Image Restoration with Free Local Constraints // Proc of the IEEE International Conference on Image Processing. Austin, USA, 1994, I: 31-35 [3] Rudin L I, Osher S, Fatemi E. Nonlinear Total Variation Based Noise Removal Algorithms. Physica D: Nonlinear Phenomena, 1992, 60(1/2/3/4): 259-268 [4] Candes E J, Demanet L, Donoho D L, et al. Fast Discrete Curvelet Transforms. Multiscale Modeling and Simulation, 2006, 5(3): 861-899 [5] Donoho D L. Compressed Sensing. IEEE Trans on Information Theory, 2006, 52(4): 1289-1306 [6] Stephen B, Vandenberghe L. Convex Optimization. Cambridge, UK: Cambridge University Press, 2004 [7] Vinje W E, Gallant J L. Sparse Coding and Decorrelation in Primary Visual Cortex during Natural Vision. Science, 2000, 287(5456): 1273-1276 [8] Olshausen B A, Field D J. Sparse Coding of Sensory Inputs. Cu-rrent Opinion in Neurobiology, 2004, 14(4): 481-487 [9] Aujol J F. Some First-Order Algorithms for Total Variation Based Image Restoration. Journal of Mathematical Imaging and Vision, 2009, 34(3): 307-327 [10] Scherzer O. Handbook of Mathematical Methods in Imaging. New York, USA: Springer-Verlag, 2010 [11] Zhang Xiaoqun, Burger M, Osher S. A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration. Journal of Scientific Computing, 2011, 46(1): 20-46 [12] Setzer S. Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage // Proc of the 2nd International Conference on Scale Space and Variational Methods in Computer Vision. Voss, Norway, 2009: 464-476 [13] Chambolle A, Pock T. A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging. Journal of Mathematical Imaging and Vision, 2011, 40(1): 120-145 [14] Fadili M J, Starck J L, Murtagh F. Inpainting and Zooming Using Sparse Representations. The Computer Journal, 2009, 52(1): 64-79 [15] Starck J L, Murtagh F, Fadili M J. Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity. Cambridge, UK: Cambridge University Press, 2010 [16] Cai Jianfeng, Chan R H, Shen Zuowei. A Framelet-Based Image Inpainting Algorithm. Applied and Computational Harmonic Analysis, 2008, 24(2): 131-149 [17] Beck A, Teboulle M. A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2011, 2(1): 183-202 [18] Bredies K, Lorenz D A. Iterated Hard Shrinkage for Minimization Problems with Sparsity Constraints. SIAM Journal on Scientific Computing, 2008, 30(2): 657-680 [19] Bredies K, Lorenz D A. Linear Convergence of Iterative Soft-Thresholding. Journal of Fourier Analysis and Applications, 2008, 14(5/6): 813-837 [20] Blumensath T, Davies M E. Iterative Hard Thresholding for Compressed Sensing. Applied and Computational Harmonic Analysis, 2009, 27(3): 265-274 [21] Donoho D L, Johnstone J M. Ideal Spatial Adaptation by Wavelet Shrinkage. Biometrika, 1994, 81(3): 425-455
2013, Vol. 26 
