A two-step model for image denoising using a duality strategy and surface fitting

In order to alleviate the staircase effect or the edge blurring in the course of the image denoising, we propose a two-step model based on the duality strategy. In fact, this strategy follows the observation that the dual variable of the restored image can be looked at as the normal vector. So we first obtain the dual variable and then reconstruct the image by fitting the dual variable. Following the augmented Lagrangian strategy, we propose a projection gradient method for solving this two-step model. We also give some convergence analyses of the proposed projection gradient method. Several numerical experiments are tested to compare our proposed model with the ROF model and the LLT model.

[1]  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.

[2]  Tony F. Chan,et al.  Image processing and analysis - variational, PDE, wavelet, and stochastic methods , 2005 .

[3]  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..

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

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

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

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

[8]  Gabriele Steidl,et al.  A Note on the Dual Treatment of Higher-Order Regularization Functionals , 2005, Computing.

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

[10]  Mario Bertero,et al.  Introduction to Inverse Problems in Imaging , 1998 .

[11]  K. Kunisch,et al.  An active set strategy based on the augmented Lagrangian formulation for image restoration , 1999 .

[12]  Curtis R. Vogel,et al.  Iterative Methods for Total Variation Denoising , 1996, SIAM J. Sci. Comput..

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

[14]  D. Dobson,et al.  Convergence of an Iterative Method for Total Variation Denoising , 1997 .

[15]  Wotao Yin,et al.  An Iterative Regularization Method for Total Variation-Based Image Restoration , 2005, Multiscale Model. Simul..

[16]  Pierre Kornprobst,et al.  Mathematical problems in image processing - partial differential equations and the calculus of variations , 2010, Applied mathematical sciences.

[17]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[18]  Kazufumi Ito,et al.  Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces , 2000 .

[19]  Xue-Cheng Tai,et al.  Image Denoising Using TV-Stokes Equation with an Orientation-Matching Minimization , 2009, SSVM.

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

[21]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

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

[23]  Mostafa Kaveh,et al.  Fourth-order partial differential equations for noise removal , 2000, IEEE Trans. Image Process..

[24]  P. G. Ciarlet,et al.  Introduction a l'analyse numerique matricielle et a l'optimisation , 1984 .

[25]  Jian-Feng Cai,et al.  Split Bregman Methods and Frame Based Image Restoration , 2009, Multiscale Model. Simul..

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