Solving variational problems in image processing via projections - A common view on TV denoising and wavelet shrinkage

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Volumn 87, Issue 3, 2007, Pages 247-256

  Lorenz, Dirk A ^a  

^a UNIVERSITY OF BREMEN   (Germany)

Author keywords

Deblurring; Denoising; Texture extraction; Total variation; Wavelet shrinkage

Indexed keywords

EID: 33947409547     PISSN: 00442267     EISSN: 15214001     Source Type: Journal    
DOI: 10.1002/zamm.200610300     Document Type: Article

References (22)

1. K. Bredies and D. A. Lorenz, Iterated hard shrinkage for minimization problems with sparsity constraints, University of Bremen, Preprint Series of the DFG-Schwerpunktprogramm 1114, 2006.
2. K. Bredies, D. A, Lorenz, and P. Maass, Equivalence of a generalized conditional gradient method and the method of surrogate functionals, University of Bremen, Preprint Series of the DFG-Schwerpunktprogramm 1114, 2005.
3. K. Bredies, D.A. Lorenz, and P. Maaß, Mathematical concepts of multiscale smoothing, Appl. Comput. Harmon. Anal. 19(2), 141-161 (2005).
4. A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vis. 20, 89-97 (2004).
5. A. Chambolle, R.A. DeVore, N. Lee, and B. J. Lucier, Nonlinear wavelet image processing: Variational problems, compression and noise removal through wavelet shrinkage, IEEE Trans. Image Process. 7, 319-335 (1998).
6. A. Chambolle and B. J. Lucier, Interpreting translation-invariant wavelet shrinkage as a new image smoothing scale space, IEEE Trans. Image Process. 10, 993-1000 (2001).
7. T. Chan and C. Wong, Convergence of the alternating minimization algorithm for blind deconvolution, Linear Algebr. Appl. 316(1), 259-285 (2000).
8. T. F. Chan and J. Shen. Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods (SIAM, Philadelphia, 2005).
9. A. Cohen, Numerical Analysis of Wavelet Methods (Elsevier Science B.V., Amsterdam, 2003).
10. A. Cohen, W. Dahmen, I. Daubechies, and R.A. DeVore, Harmonic analysis of the space BV. Revista Matematica Iberoamericana 19(1), 235-263 (2003).
11. I. Daubechies, M. Defrise, and C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math. 57(11), 1413-1457 (2004).
12. I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976).
13. D.A. Lorenz, Wavelet Shrinkage in Signal and Image Processing - An Investigation of Relations and Equivalences, PhD thesis, Universität Bremen, February 2005.
14. Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations, American Mathematical Society, Providence, RI, 2001. The fifteenth Dean Jacqueline B. Lewis memorial lectures.
15. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and variational problems, Commun. Pure Appl. Math. 42(5), 577-685 (1989).
16. R. Ramlau and G. Teschke, A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints, Numer. Math. 104(2), 177-203 (2006).
17. L.I. Rudin, S. J. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60, 259-268 (1992).
18. O. Scherzer, Scale-space methods and regularization for denoising and inverse problems, Advances in Imaging and Electron Physics 128, 445-530 (2003).
19. G. Steidl, J. Weickert, T. Brox, P. Mrazek, and M. Welk, On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization and SIDEs, SIAM J. Numer. Anal. 42(2), 42(2), 658-686 (2004).
20. H. Triebel, Theory of Function Spaces II, Monographs in Mathematics (Birkhäuser, Basel, 1992).
21. 2-distance projection onto the canonical simplex: A simple algorithm, ALGO Research Quarterly 4, 53-55 (2001).
22. L. A. Vese, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Opt. 44, 131-161 (2001).