Integrodifferential equations for continuous multiscale wavelet shrinkage

Inverse Problems and Imaging

Volumn 1, Issue 1, 2007, Pages 47-62

  Didas, Stephan ^a   Weickert, Joachim ^a  

^a SAARLAND UNIVERSITY   (Germany)

Author keywords

Image processing; Integrodifferential equations; Nonlinear diffusion filtering; Wavelet shrinkage

Indexed keywords

EID: 47749103562     PISSN: 19308337     EISSN: 19308345     Source Type: Journal    
DOI: 10.3934/ipi.2007.1.47     Document Type: Article

References (49)

1. F. Andreu, C. Ballester, V. Caselles, and J. M. Mazón. Minimizing total variation flow, Differential and Integral Equations, 14(2001) 321–360.
2. Y. Bao and H. Krim. Towards bridging scale-space and multiscale frame analyses, In A. A. Petrosian and F. G. Meyer, editors, Wavelets in Signal and Image Analysis, volume 19 of Computational Imaging and Vision, chapter 6. Kluwer, Dordrecht, 2001.
3. K. Bredies, D. A. Lorenz, and P. Maass. Mathematical concepts of multiscale smoothing, Applied and Computational Harmonic Analysis, 19(2005), 141–161.
4. E. J. Candés and F. Guo. New multiscale transforms, minimum total variation synthesis: Applications to edge-preserving image reconstruction, Signal Processing, 82(2002), 1519–1543.
5. F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll. Image selective smoothing and edge detection by nonlinear diffusion, SIAM Journal on Numerical Analysis, 29(1992), 182–193.
6. A. Chambolle, R. DeVore, N.-Y. Lee, and B. J. Lucier. Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Transactions on Image Processing, 7(1998), 319–335.
7. A. Chambolle and B. L. Lucier. Interpreting translation-invariant wavelet shrinkage as a new image smoothing scale space, IEEE Transactions on Image Processing, 10(2001), 993–1000.
8. T. F. Chan and H. M. Zhou. Total variation improved wavelet thresholding in image compression, In “Proc. Seventh International Conference on Image Processing,” volume II, pages 391–394, Vancouver, Canada 2000.
9. P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud. Two deterministic half-quadratic regularization algorithms for computed imaging, Proc. IEEE International Conference on Image Processing (ICIP-94, Austin, Nov. 13-16, 1994), 2(1994), 168–172.
10. A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore. Harmonic analysis in the space BV, Revista Matematica Iberoamericana, 19(2003), 235–262.
11. 2), American Journal of Mathematics, 121(1999), 587–628.
12. R. R. Coifman and A. Sowa. Combining the calculus of variations and wavelets for image enhancement, Applied and Computational Harmonic Analysis, 9(2000), 1–18.
13. R. R. Coifman and A. Sowa. New methods of controlled total variation reduction for digital functions, SIAM Journal on Numerical Analysis, 39(2001), 480–498.
14. I. Daubechies. “Ten Lectures on Wavelets,” SIAM, Philadelphia, 1992.
15. I. Daubechies and G. Teschke. Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring and denoising, Applied and Computational Harmonic Analysis, 19(2005), 1–16.
16. S. Didas, J. Weickert, and B. Burgeth. Stability and local feature enhancement of higher order nonlinear diffusion filtering, in “Pattern Recognition”(eds. W. Kropatsch, R. Sablatnig, and A. Hanbury), volume 3663 of Lecture Notes in Computer Science, pages 451–458. Springer, 2005.
17. D. L. Donoho. De-noising by soft thresholding, IEEE Transactions on Information Theory, 41(1995), 613–627.
18. D. L. Donoho and I. M. Johnstone. Ideal spatial adaption by wavelet shrinkage, Biometrica, 81(1994), 425–455.
19. R. Duits, L. Florack, J. de Graaf, and B. Ter Haar Romeny. On the axioms of scale space theory, Journal of Mathematical Imaging and Vision, 20(2004), 267–298.
20. S. Durand and J. Froment. Reconstruction of wavelet coefficients using total-variation minimization, SIAM Journal on Scientific Computing, 24(2003), 1754–1767.
21. S. Durand and M. Nikolova. Restoration of wavelet coefficients by minimizing a specially designed objective function, in “Proc. Second IEEE Workshop on Geometric and Level Set Methods in Computer Vision” (eds. O. Faugeras and N. Paragios), Nice, France, INRIA, 2003.
22. H.-Y. Gao. Wavelet shrinkage denoising using the non-negative garrote, Journal of Computational and Graphical Statistics, 7(1998), 469–488.
23. T. Iijima. Basic theory on normalization of pattern (in case of typical one-dimensional pat-tern)(in Japanese), Bulletin of the Electrotechnical Laboratory, 26(1962), 368–388.
24. S. L. Keeling and R. Stollberger. Nonlinear anisotropic diffusion filtering for multiscale edge enhancement, Inverse Problems, 18(2002), 175–190.
25. M. Lysaker, A. Lundervold, and X.-C. Tai. Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Transactions on Image Processing, 12(2003), 1579–1590.
26. F. Malgouyres. Mathematical analysis of a model which combines total variation and wavelet for image restoration, Inverse Problems, 2(2002), 1–10.
27. S. Mallat. “A Wavelet Tour of Signal Processing,” Academic Press, San Diego, second edition, 1999.
28. Y. Meyer. “Oscillating Patterns in Image Processing and Nonlinear Evolution Equations,” volume 22 of University Lecture Series, AMS, Providence, 2001.
29. P. Mrázek and J. Weickert. Rotationally invariant wavelet shrinkage. In B. Michaelis and G. Krell, editors, “Pattern Recognition,” volume 2781 of Lecture Notes in Computer Science, pages 156–163. Springer, Berlin, 2003.
30. P. Mrázek, J. Weickert, and G. Steidl. Diffusion-inspired shrinkage functions and stability results for wavelet denoising, International Journal of Computer Vision, 64(2005), 171–186.
31. P. Mrázek, J. Weickert, G. Steidl, and M. Welk. On iterations and scales of nonlinear filters, in “Proc. of the Computer Vision Winter Workshop 2003” (ed. O. Drbohlav), pages 61–66. Czech Pattern Recognition Society, 2003.
32. M. Nielsen, L. Florack, and R. Deriche. Regularization, scale-space and edge detection filters, Journal of Mathematical Imaging and Vision, 7(1997), 291–307.
33. W. J. Niessen, B. M. ter Haar Romeny, L. M. Florack, and M. A. Viergever. A general framework for geometry-driven evolution equations, International Journal of Computer Vision, 21(1997), 187–205.
34. N. Nordström. Biased anisotropic diffusion – a unified regularization and diffusion approach to edge detection, Image and Vision Computing, 8(1990), 318–327.
35. P. Perona and J. Malik. Scale space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(1990), 629–639.
36. L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms, Physica D, 60(1992), 259–268.
37. O. Scherzer. Denoising with higher order derivatives of bounded variation and an application to parameter estimation, Computing, 60(1998), 1–27.
38. O. Scherzer and J. Weickert. Relations between regularization and diffusion filtering, Journal of Mathematical Imaging and Vision, 12(2000), 43–63.
39. C. Schnörr. Unique reconstruction of piecewise smooth images by minimizing strictly convex non-quadratic functionals, Journal of Mathematical Imaging and Vision, 4(1994)189–198.
40. J.-L. Starck, M. Elad, and D. L. Donoho. Image decomposition via the combination of sparse representations and a variational approach, IEEE Transactions on Image Processing, 14(2005):1570–1582.
41. G. Steidl, J. Weickert, T. Brox, P. Mrázek, and M. Welk. On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs, SIAM Journal on Numerical Analysis, 42(2004), 686–713.
42. M. E. Taylor. “Pseudodifferential Operators,” Princeton University Press, Princeton, New Jersey, 1981.
43. M. E. Taylor. “Partial Differential Equations I – Basic Theory,” Springer, New York, 1996.
44. J. Weickert. “Anisotropic Diffusion in Image Processing,” B. G. Teubner, Stuttgart, 1998.
45. J. Weickert, G. Steidl, P. Mrázek, M. Welk, and T. Brox. Diffusion filters and wavelets: What can they learn from each other?, in “Handbook of Mathematical Models in Computer Vision” (eds. N. Paragios, Y. Chen, and O. Faugeras). Springer, New York, 2006.
46. M. Welk, G. Steidl, and J. Weickert. Locally analytic schemes: A link between diffusion filtering and wavelet shrinkage, Technical Report 2100, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN, U.S.A., 2006.
47. D. Werner. “Funktional Analysis,” Springer, Berlin, third edition, 2000.
48. A. P. Witkin. Scale-space filtering, In “Proceedings of the 8th International Joint Conference on Artificial Intelligence” volume 2, pages 945–951, Karlsruhe, Germany, August 1983.
49. Y.-L. You and M. Kaveh. Fourth-order partial differential equations for noise removal, IEEE Transactions on Image Processing, 9(2000), 1723–1730.