Variational methods for denoising matrix fields

Mathematics and Visualization

Volumn 0, Issue 9783540883777, 2009, Pages 341-360

  Setzer, S ^a   Steidl, G ^a   Popilka, B ^a   Burgeth, B ^b  

^a UNIVERSITY OF MANNHEIM   (Germany)

^b SAARLAND UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

CALCULATIONS; DIFFERENTIATION (CALCULUS); EQUATIONS OF MOTION; IMAGE DENOISING; NUMERICAL METHODS; STEEPEST DESCENT METHOD;

EULER-LAGRANGE EQUATIONS; HIGHER-ORDER METHODS; MATRIX MULTIPLICATION; NUMERICAL COMPUTATIONS; REGULARIZATION TERMS; SECOND-ORDER CONE PROGRAMMING; VARIATIONAL APPROACHES; WELL-ESTABLISHED TECHNIQUES;

MATRIX ALGEBRA;

EID: 85028683285     PISSN: 16123786     EISSN: 2197666X     Source Type: Book Series    
DOI: 10.1007/978-3-540-88378-4_17     Document Type: Chapter

References (29)

1. A. Barvinok. A Course in Convexity, Graduate Studies in Mathematics. AMS, Providence, RI, 2002.
2. J. M. Borwein and A. S. Lewis. Convex Analysis and Nonlinear Optimization. Springer, New York, 2000.
3. T. Brox, J. Weickert, B. Burgeth, and P. Mrázek. Nonlinear structure tensors. Image and Vision Computing, 24(1):41–55, 2006.
4. B. Burgeth, A. Bruhn, S. Didas, J. Weickert, and M. Welk. Morphology for matrix-data: Ordering versus PDE-based approach. Image and Vision Computing, 25(4):496–511, 2007.
5. B. Burgeth, A. Bruhn, N. Papenberg, M. Welk, and J. Weickert. Mathematical morphology for matrix fields induced by the Loewner ordering in higher dimensions. Signal Processing, 87(2):277–290, 2007.
6. B. Burgeth, S. Didas, L. Florack, and J. Weickert. Singular PDEs for the processing of matrix-valued data. In F. Sgallari, A. Murli, and N. Paragios, editors, Scale Space and Variational Methods in Computer Vision, volume 4485 of Lecture Notes in Computer Science, pages 556–567. Springer, Berlin, 2007.
7. B. Burgeth, S. Didas, and J. Weickert. A generic approach to diffusion filtering of matrix-fields. Computing, 81:179–197, 2007.
8. A. Chambolle. An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20:89–97, 2004.
9. A. Chambolle and P. -L. Lions. Image recovery via total variation minimization and related problems. Numerische Mathematik, 76:167–188, 1997.
10. C. Chefd’Hotel, D. Tschumperlé, R. Deriche, and O. Faugeras. Constrained flows of matrix-valued functions: Application to diffusion tensor regularization. In A. Heyden, G. Sparr, M. Nielsen, and P. Johansen, editors, Computer Vision – ECCV 2002, volume 2350 of Lecture Notes in Computer Science, pages 251–265. Springer, Berlin, 2002.
11. P. G. Ciarlet. Introduction to Numerical Linear Algebra and Optimisation. Cambridge University Press, Cambridge, 1989.
12. 1 regularization. Advances in Computational Mathematics, 30(1), 2008, (available online at www.springerlink.com).
13. D. Goldfarb and W. Yin. Second-order cone programming methods for total variation-based image restoration. SIAM J. Scientific Computing, 2(27):622–645, 2005.
14. Y. Gur and N. Sochen. Fast invariant Riemannian DT-MRI regularization. In IEEE 11th International Conference on Computer Vision, Rio de Janeiro, Brasil, 2007.
15. R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, UK, 1990.
16. M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret. Applications of second-order cone programming. Linear Algebra and its Applications, 284(1):193–228, 1998.
17. J. R. Magnus and H. Neudecker. Matrix Differential Calculus with Applications in Statistics and Econometrics. J. Wiley and Sons, Chichester, 1988.
18. H. Mittelmann. An independent bechmarking of SDP and SOCP solvers. Mathematical Programming Series B, 95(2):407–430, 2003.
19. X. Pennec, P. Fillard, and N. Ayache. A Riemannian framework for tensor computing. International Journal of Computer Vision, 66(1):41–66, 2006.
20. L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268, 1992.
21. O. Scherzer and J. Weickert. Relations between regularization and diffusion filtering. Journal of Mathematical Imaging and Vision, 12(1):43–63, 2000.
22. G. Steidl, S. Setzer, B. Popilka, and B. Burgeth. Restoration of matrix fields by second-order cone programming. Computing, 81:161 – 178, 2007.
23. D. Tschumperlé and R. Deriche. Diffusion tensor regularization with con-traints preservation. In Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition, volume 1, pages 948–953, Kauai, HI, 2001.
24. Z. Wang, B. C. Vemuri, W. Chen, and T. H. Mareci. A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI. IEEE Transactions on Medical Imaging, 23(8):930–939, 2004.
25. J. Weickert. Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.
26. J. Weickert and T. Brox. Diffusion and regularization of vector- and matrix-valued images. In M. Z. Nashed and O. Scherzer, editors, Inverse Problems, Image Analysis, and Medical Imaging, volume 313 of Contemporary Mathematics, pages 251–268. AMS, Providence, 2002.
27. J. Weickert and H. Hagen, editors. Visualization and Processing of Tensor Fields. Springer, Berlin, 2006.
28. Q. Xu, A. W. Anderson, J. C. Gore, and Z. Ding. Diffusion tensor image smoothing using efficient and effective anisotropic filtering. In IEEE 11th International Conference on Computer Vision, Rio de Janeiro, Brasil, 2007.
29. M. Zéraïand M. Moakher. Riemannian curvature-driven flows for tensor-valued data. In F. Sgallari, A. Murli, and N. Paragios, editors, Scale Space and Variational Methods in Computer Vision, volume 4485 of Lecture Notes in Computer Science, pages 592–602, Springer, Berlin, 2007.