Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 31106
Discontinuous Galerkin Method for Total Variation Minimization on Inpainting Problem

Authors: Xijian Wang

Abstract:

This paper is concerned with the numerical minimization of energy functionals in BV ( ) (the space of bounded variation functions) involving total variation for gray-scale 1-dimensional inpainting problem. Applications are shown by finite element method and discontinuous Galerkin method for total variation minimization. We include the numerical examples which show the different recovery image by these two methods.

Keywords: Finite Element Method, inpainting, discontinuous Galerkin method, total variation minimization

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1332162

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1043

References:


[1] M. Fornasier, Theoretical Foundations and Numerical Methods for Sparse Recovery, De Gruyter, 2010.
[2] L. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992) 259-268.
[3] A. Chambolle, P.-L. Lions, Image recovery via total variation minimization and related problems, Numer Math. 76 (1997) 167-188.
[4] L. Vese, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optim. 44 (2001) 131-161.
[5] G. Aubert, P. Kornprobst, Mathematial Problems in Image Processing. Partial Differential Equations and the Calculus of Variation, Springer, 2006.
[6] T. Chan, J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, 2005.
[7] H. W. Engl, M. Hanke, A. Neubauer, Regularization of inverse problems, Kluwer Academic Publishers, Dordrecht, The Netherlands 76 (1996).
[8] A. Chambolle, J. Darbon, On total variation minimization and surface evolution using parametric maximum flows, International Journal of Computer Vision 84 (2009) no. 3, 288-307.
[9] T. Goldstein, S. Osher, The split bregman method for L 1 regularized problems, SIAM Journal on Imaging Sciences 2 (2009) no. 2, 323-343.
[10] S. Osher, M. Burger, D. Goldfarb, J. Xu, W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul. 4 (2005) no. 2, 460-489.
[11] P. Weiss, L. Blanc-F'eraud, G. Aubert, Efficient schemes for total variation minimization under constraints in image processing, SIAM J. Sci. Comput. 31 (2009) no. 3, 2047-2080.
[12] M. Fornasier, C. Sch┬¿onlieb, Subspace correction methods for total variation and Ôäô1-minimization, SIAM J. Numer. Anal. 47 (2009) no.5, 3397-3428.
[13] B. Cockburn, G. E. Karniadakis, C.-W. Shu, Discontinuous Galerkin Methods: Theory, Compuration and Applications, Springer, 2000.
[14] B. Rivi`ere, Discontinuous Galerkin Methods For Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM, 2008.
[15] M. Fornasier, R. March, Restoration of color images by vector valued BV functions and variational calculus, SIAM J. Appl. Math. 68 (2007) 437-460.