Augmented Lagrangian Method for Generalized TV-Stokes Model

In this paper, we propose a general form of TV-Stokes models and provide an efficient and fast numerical algorithm based on the augmented Lagrangian method. The proposed model and numerical algorithm can be used for a number of applications such as image inpainting, image decomposition, surface reconstruction from sparse gradient, direction denoising, and image denoising. Comparing with properties of different norms in regularity term and fidelity term, various results are investigated in applications. We numerically show that the proposed model recovers jump discontinuities of a data and discontinuities of the data gradient while reducing stair-case effect.

[1]  Rama Chellappa,et al.  Enforcing integrability by error correction using l1-minimization , 2009, CVPR.

[2]  Arvid Lundervold,et al.  Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time , 2003, IEEE Trans. Image Process..

[3]  Jean-François Aujol,et al.  Some First-Order Algorithms for Total Variation Based Image Restoration , 2009, Journal of Mathematical Imaging and Vision.

[4]  A. Chambolle.  Practical, Unified, Motion and Missing Data Treatment in Degraded Video , 2004, Journal of Mathematical Imaging and Vision.

[5]  Yves Meyer,et al.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures , 2001 .

[6]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[7]  Mila Nikolova,et al.  Minimizers of Cost-Functions Involving Nonsmooth Data-Fidelity Terms. Application to the Processing of Outliers , 2002, SIAM J. Numer. Anal..

[8]  Rachid Deriche,et al.  Regularizing Flows for Constrained Matrix-Valued Images , 2004 .

[9]  Stanley Osher,et al.  Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing , 2003, J. Sci. Comput..

[10]  Michael K. Ng,et al.  Solving Constrained Total-variation Image Restoration and Reconstruction Problems via Alternating Direction Methods , 2010, SIAM J. Sci. Comput..

[11]  Ross T. Whitaker,et al.  Geometric surface processing via normal maps , 2003, TOGS.

[12]  Stanley Osher,et al.  A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration , 2010, J. Sci. Comput..

[13]  Xue-Cheng Tai,et al.  Augmented Lagrangian Method, Dual Methods, and Split Bregman Iteration for ROF, Vectorial TV, and High Order Models , 2010, SIAM J. Imaging Sci..

[14]  Simon Setzer,et al.  Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage , 2009, SSVM.

[15]  Talal Rahman,et al.  A Modified TV-Stokes Model for Image Processing , 2011, SIAM J. Sci. Comput..

[16]  Michael S. Brown,et al.  ShapePalettes: interactive normal transfer via sketching , 2007, SIGGRAPH 2007.

[17]  Michael Hintermüller,et al.  An Infeasible Primal-Dual Algorithm for Total Bounded Variation-Based Inf-Convolution-Type Image Restoration , 2006, SIAM J. Sci. Comput..

[18]  M. Nikolova.  An Algorithm for Total Variation Minimization and Applications , 2004 .

[19]  Yiqiu Dong,et al.  An Efficient Primal-Dual Method for L1TV Image Restoration , 2009, SIAM J. Imaging Sci..

[20]  Guillermo Sapiro,et al.  Filling-in by joint interpolation of vector fields and gray levels , 2001, IEEE Trans. Image Process..

[21]  Gabriele Steidl,et al.  Removing Multiplicative Noise by Douglas-Rachford Splitting Methods , 2010, Journal of Mathematical Imaging and Vision.

[22]  Wotao Yin,et al.  Bregman Iterative Algorithms for (cid:2) 1 -Minimization with Applications to Compressed Sensing ∗ , 2008 .

[23]  Tom Goldstein,et al.  The Split Bregman Method for L1-Regularized Problems , 2009, SIAM J. Imaging Sci..

[24]  Guillermo Sapiro,et al.  Navier-stokes, fluid dynamics, and image and video inpainting , 2001, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001.

[25]  Knut-Andreas Lie,et al.  Image Processing Based on Partial Differential Equations , 2007 .

[26]  Junfeng Yang,et al.  An Efficient TVL1 Algorithm for Deblurring Multichannel Images Corrupted by Impulsive Noise , 2009, SIAM J. Sci. Comput..

[27]  Max A. Viergever,et al.  Efficient and reliable schemes for nonlinear diffusion filtering , 1998, IEEE Trans. Image Process..

[28]  Tai-Pang Wu,et al.  Surface-from-Gradients with Incomplete Data for Single View Modeling , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[29]  Knut-Andreas Lie,et al.  Scale Space and Variational Methods in Computer Vision, Second International Conference, SSVM 2009, Voss, Norway, June 1-5, 2009. Proceedings , 2009, SSVM.

[30]  Tony F. Chan,et al.  Aspects of Total Variation Regularized L[sup 1] Function Approximation , 2005, SIAM J. Appl. Math..

[31]  Xue-Cheng Tai,et al.  Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model , 2009, SSVM.

[32]  Stanley Osher,et al.  Image Decomposition and Restoration Using Total Variation Minimization and the H1 , 2003, Multiscale Model. Simul..

[33]  Jean-Michel Morel,et al.  Level lines based disocclusion , 1998, Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269).

[34]  Zhen Liu,et al.  An Improved LOT Model for Image Restoration , 2009, Journal of Mathematical Imaging and Vision.

[35]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[36]  Talal Rahman,et al.  A Dual Formulation of the TV-Stokes Algorithm for Image Denoising , 2009, SSVM.

[37]  T. Chan,et al.  Fast dual minimization of the vectorial total variation norm and applications to color image processing , 2008 .

[38]  Rama Chellappa,et al.  A Method for Enforcing Integrability in Shape from Shading Algorithms , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[39]  Xuecheng Tai,et al.  AUGMENTED LAGRANGIAN METHOD FOR TOTAL VARIATION RESTORATION WITH NON-QUADRATIC FIDELITY , 2011 .

[40]  Thomas B. Moeslund,et al.  Long-Term Occupancy Analysis Using Graph-Based Optimisation in Thermal Imagery , 2013, 2013 IEEE Conference on Computer Vision and Pattern Recognition.

[41]  Gene H. Golub,et al.  A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration , 1999, SIAM J. Sci. Comput..

[42]  Xue-Cheng Tai,et al.  Noise removal using smoothed normals and surface fitting , 2004, IEEE Transactions on Image Processing.

[43]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[44]  Talal Rahman,et al.  A TV-Stokes Denoising Algorithm , 2007, SSVM.

[45]  Tony F. Chan,et al.  Euler's Elastica and Curvature-Based Inpainting , 2003, SIAM J. Appl. Math..

[46]  Stefano Alliney,et al.  Digital filters as absolute norm regularizers , 1992, IEEE Trans. Signal Process..

[47]  Alfred M. Bruckstein,et al.  Orientation-Matching Minimization for Image Denoising and Inpainting , 2011, International Journal of Computer Vision.

[48]  Junfeng Yang,et al.  A New Alternating Minimization Algorithm for Total Variation Image Reconstruction , 2008, SIAM J. Imaging Sci..

[49]  T. Chan,et al.  Edge-preserving and scale-dependent properties of total variation regularization , 2003 .

[50]  Stéphanie Jehan-Besson,et al.  An Augmented Lagrangian Method for TVg+L1-norm Minimization , 2010, Journal of Mathematical Imaging and Vision.

[51]  Thomas S. Huang,et al.  Image processing , 1971 .