Locally analytic schemes: A link between diffusion filtering and wavelet shrinkage

Applied and Computational Harmonic Analysis

Volumn 24, Issue 2, 2008, Pages 195-224

  Welk, Martin ^a   Steidl, Gabriele ^b   Weickert, Joachim ^a  

^a SAARLAND UNIVERSITY   (Germany)

^b UNIVERSITY OF MANNHEIM   (Germany)

Author keywords

Diffusion filtering; Dynamical systems; Finite difference methods; Image processing; Rotation invariance; Stability; Wavelet shrinkage

Indexed keywords

ANISOTROPY; DIFFUSION; DYNAMICAL SYSTEMS; FINITE DIFFERENCE METHOD; IMAGE ENHANCEMENT; SYSTEM STABILITY; WAVELET ANALYSIS;

NONLINEAR DIFFUSION FILTERING; ROTATION INVARIANCES; WAVELET SHRINKAGE;

NONLINEAR FILTERING;

EID: 39149088668     PISSN: 10635203     EISSN: 1096603X     Source Type: Journal    
DOI: 10.1016/j.acha.2007.05.004     Document Type: Article

References (64)

1. Alvarez L., Guichard F., Lions P.-L., and Morel J.-M. Axioms and fundamental equations in image processing. Arch. Ration. Mech. Anal. 123 (1993) 199-257
2. Andreu F., Caselles V., Diaz J.I., and Mazón J.M. Qualitative properties of the total variation flow. J. Funct. Anal. 188 2 (2002) 516-547
3. Andreu-Vaillo F., Caselles V., and Mazon J.M. Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Progress in Math. vol. 223 (2004), Birkhäuser, Basel
4. Bao Y., and Krim H. Towards bridging scale-space and multiscale frame analyses. In: Petrosian A.A., and Meyer F.G. (Eds). Wavelets in Signal and Image Analysis. Computational Imaging and Vision vol. 19 (2001), Kluwer, Dordrecht chapter 6
5. N. J. Differential Equations 184 2 (2002) 475-525
6. Blomgren P., and Chan T.F. Color TV: Total variation methods for restoration of vector valued images. IEEE Trans. Image Process. 7 3 (1998) 304-309
7. Bredies K., Lorenz D.A., Maaß P., and Teschke G. A partial differential equation for continuous non-linear shrinkage filtering and its application for analyzing MMG data. In: Truchetet F. (Ed). Wavelet Applications in Industrial Processing. Proceedings of SPIE vol. 5266 (2004), SPIE Press, Bellingham 84-93
8. Breuß M., Brox T., Bürgel A., Sonar T., and Weickert J. Numerical aspects of TV flow. Numer. Algorithms 41 1 (2006) 79-101
9. Candès E., Demanet L., Donoho D., and Ying L. Fast discrete curvelet transform. Multiscale Model. Simul. 5 3 (2006) 861-899
10. Candès E.J., and Donoho D.L. Curvelets: A surprisingly effective nonadaptive representation of objects with edges. In: Cohen A., Rabut C., and Schumaker L.L. (Eds). Curve and Surface Fitting. Saint-Malo, 2000 (2000), Vanderbilt Univ. Press
11. Candès E.J., and Guo F. New multiscale transforms, minimum total variation synthesis: Applications to edge-preserving image reconstruction. Signal Process. 82 11 (2002) 1519-1543
12. Catté F., Lions P.-L., Morel J.-M., and Coll T. Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 32 (1992) 1895-1909
13. Chambolle A., DeVore R.A., Lee N., and Lucier B.L. Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7 3 (1998) 319-335
14. Chambolle A., and Lucier B.L. Interpreting translationally-invariant wavelet shrinkage as a new image smoothing scale space. IEEE Trans. Image Process. 10 7 (2001) 993-1000
15. T.F. Chan, H.M. Zhou, Total variation improved wavelet thresholding in image compression, in: Proc. Seventh International Conference on Image Processing, vol. II, Vancouver, Canada, Sept. 2000, pp. 391-394
16. Cohen A., Dahmen W., Daubechies I., and DeVore R. Harmonic analysis in the space BV. Rev. Mat. Iberoamericana 19 (2003) 235-262
17. 2). Amer. J. Math. 121 (1999) 587-628
18. Coifman R.R., and Donoho D. Translation invariant denoising. In: Antoine A., and Oppenheim G. (Eds). Wavelets in Statistics (1995), Springer, New York 125-150
19. Coifman R.R., and Sowa A. Combining the calculus of variations and wavelets for image enhancement. Appl. Comput. Harmon. Anal. 9 1 (2000) 1-18
20. Coifman R.R., and Sowa A. New methods of controlled total variation reduction for digital functions. SIAM J. Numer. Anal. 39 2 (2001) 480-498
21. Dibos F., and Koepfler G. Global total variation minimization. SIAM J. Numer. Anal. 37 2 (2000) 646-664
22. Do M.N., and Vetterli V. Contourlets. In: Stöckler J., and Welland G. (Eds). Beyond Wavelets (2003), Academic Press, New York 1-27
23. Donoho D.L. De-noising by soft thresholding. IEEE Trans. Inform. Theory 41 (1995) 613-627
24. Donoho D.L. Orthonormal ridgelets and linear singularities. SIAM J. Math. Anal. 31 5 (2000) 1062-1099
25. Donoho D.L., and Johnstone I.M. Ideal spatial adaptation by wavelet shrinkage. Biometrica 81 3 (1994) 425-455
26. DuFort E.C., and Frankel S.P. Stability conditions in the numerical treatment of parabolic differential equations. Math. Tables Other Aids Comput. 7 (1953) 135-152
27. Durand S., and Froment J. Reconstruction of wavelet coefficients using total-variation minimization. SIAM J. Sci. Comput. 24 5 (2003) 1754-1767
28. S. Durand, M. Nikolova, Restoration of wavelet coefficients by minimizing a specially designed objective function, in: O. Faugeras, N. Paragios (Eds.), Proc. Second IEEE Workshop on Geometric and Level Set Methods in Computer Vision, Nice, France, INRIA, 2003
29. Feng X., and Prohl A. Analysis of total variation flow and its finite element approximations. ESAIM: Math. Models Methods Appl. Sci. 37 3 (2002) 533-556
30. Filippov A.F. Differential Equations with Discontinuous Righthand Sides (1988), Kluwer, Dordrecht
31. W. Förstner, E. Gülch, A fast operator for detection and precise location of distinct points, corners and centres of circular features, in: Proc. ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data, Interlaken, Switzerland, June 1987, pp. 281-305
32. Gerig G., Kübler O., Kikinis R., and Jolesz F.A. Nonlinear anisotropic filtering of MRI data. IEEE Trans. Med. Imag. 11 (1992) 221-232
33. Jawerth B., Lin P., and Sinzinger E. Lattice Boltzmann models for anisotropic diffusion of images. J. Math. Imag. Vision 11 (1999) 231-237
34. Keeling S.L., and Stollberger R. Nonlinear anisotropic diffusion filters for wide range edge sharpening. Inverse Problems 18 (2002) 175-190
35. Kingsbury N.G. Complex wavelets for shift invariant analysis and filtering of signals. J. Appl. Comput. Harmon. Anal. 10 3 (2001) 234-253
36. Labate D., Lim W., Kutyniok G., and Weiss G. Sparse multidimensional representation using shearlets. In: Papadakis M., Laine A.F., and Unser M.A. (Eds). Wavelets XI. Proceedings of SPIE vol. 5914 (2005), SPIE Press, Bellingham 254-262
37. Lu T., Neittaanmäki P., and Tai X.-C. A parallel splitting up method and its application to Navier-Stokes equations. Appl. Math. Lett. 4 2 (1991) 25-29
38. Malgouyres F. Combining total variation and wavelet packet approaches for image deblurring. Proc. First IEEE Workshop on Variational and Level Set Methods in Computer Vision. Vancouver, Canada, July 2001 (2001), IEEE Computer Society Press 57-64
39. Malgouyres F. Mathematical analysis of a model which combines total variation and wavelet for image restoration. Inverse Problems 2 1 (2002) 1-10
40. Meyer Y. Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. Univ. Lecture Ser. vol. 22 (2001), AMS, Providence
41. Mrázek P., and Weickert J. Rotationally invariant wavelet shrinkage. In: Michaelis B., and Krell G. (Eds). Pattern Recognition. Lecture Notes in Comput. Sci. vol. 2781 (2003), Berlin, Springer 156-163
42. Mrázek P., Weickert J., and Steidl G. Diffusion-inspired shrinkage functions and stability results for wavelet denoising. Int. J. Comput. Vision 64 2/3 (2005) 171-186
43. Mrázek P., Weickert J., Steidl G., and Welk M. On iterations and scales of nonlinear filters. In: Drbohlav O. (Ed). Proc. Eighth Computer Vision Winter Workshop. Valtice, Czech Republic (2003), Czech Pattern Recognition Society 61-66
44. Perona P., and Malik J. Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990) 629-639
45. Petrovic A., Divorra Escoda O., and Vandergheynst P. Multiresolution segmentation of natural images: From linear to non-linear scale-space representations. IEEE Trans. Image Process. 13 8 (2004) 1104-1114
46. Pollak I., Willsky A.S., and Krim H. Image segmentation and edge enhancement with stabilized inverse diffusion equations. IEEE Trans. Image Process. 9 2 (2000) 256-266
47. Preußer T., and Rumpf M. An adaptive finite element method for large scale image processing. J. Vis. Commun. Image Represent. 11 2 (2000) 183-195
48. Richardson J.L., Ferrell R.C., and Long L.N. Unconditionally stable explicit algorithms for nonlinear fluid dynamics problems. J. Comput. Phys. 104 (1993) 69-74
49. Rudin L.I., Osher S., and Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D 60 (1992) 259-268
50. Sapiro G., and Ringach D.L. Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans. Image Process. 5 11 (1996) 1582-1586
51. Shen J. A note on wavelets and diffusion. J. Comput. Anal. Appl. 5 1 (2003) 147-159
52. Shen J., and Strang G. On wavelet fundamental solutions to the heat equation-Heatlets. J. Differential Equations 161 2 (2000) 403-421
53. E. P. Simoncelli, Design of multidimensional derivative filters, in: Proc. 1994 IEEE International Conference on Image Processing, vol. 1, Austin, TX, 1994, pp. 790-793
54. Sochen N., Kimmel R., and Bruckstein F. Diffusions and confusions in signal and image processing. J. Math. Imag. Vision 14 3 (2001) 195-210
55. Steidl G., Weickert J., Brox T., Mrázek P., and Welk M. On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs. SIAM J. Numer. Anal. 42 2 (2004) 686-713
56. Tsurkov V.I. An analytical model of edge protection under noise suppression by anisotropic diffusion. J. Comput. Systems Sci. Int. 39 3 (2000) 437-440
57. Wang W. On the design of optimal derivative filters for coherence-enhancing diffusion filtering. Proc. 2004 International Conference on Computer Graphics, Imaging and Visualization. Penang, Malaysia, July 2004 (2004), IEEE Computer Society Press 35-40
58. J. Weickert, Scale-space properties of nonlinear diffusion filtering with a diffusion tensor, technical report 110, Laboratory of Technomathematics, University of Kaiserslautern, Germany, 1994
59. Weickert J. Anisotropic Diffusion in Image Processing (1998), Teubner, Stuttgart
60. Weickert J., and Brox T. Diffusion and regularization of vector- and matrix-valued images. In: Nashed M.Z., and Scherzer O. (Eds). Inverse Problems, Image Analysis, and Medical Imaging. Contemp. Math. vol. 313 (2002), AMS, Providence 251-268
61. Weickert J., and Scharr H. A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance. J. Vis. Commun. Image Represent. 13 1/2 (2002) 103-118
62. Weickert J., ter Haar Romeny B.M., and Viergever M.A. Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans. Image Process. 7 3 (1998) 398-410
63. Welk M., Weickert J., and Steidl G. A four-pixel scheme for singular differential equations. In: Kimmel R., Sochen N., and Weickert J. (Eds). Scale-Space and PDE Methods in Computer Vision. Lecture Notes in Comput. Sci. vol. 3459 (2005), Berlin, Springer 585-597
64. Welk M., Weickert J., and Steidl G. From tensor-driven diffusion to anisotropic wavelet shrinkage. In: Bischof H., Leonardis A., and Pinz A. (Eds). Computer Vision-ECCV 2006. Lecture Notes in Comput. Sci. vol. 3951 (2006), Berlin, Springer 391-403