Anisotropic Diffusion Partial Differential Equations for Multichannel Image Regularization: Framework and Applications

Advances in Imaging and Electron Physics

Volumn 145, Issue SUPPL., 2007, Pages 149-209

  Tschumperlé, David ^a   Deriche, Rachid ^b  

^a Centre National de la Recherche Scientifique   (France)

^b UNIVERSITÉ PARIS EST   (France)

Author keywords

[No Author keywords available]

Indexed keywords

ANISOTROPY; IMAGE PROCESSING; ITERATIVE METHODS; NONLINEAR SYSTEMS; OPTIMIZATION; SPURIOUS SIGNAL NOISE;

GENERIC DIFFUSION; GEOMETRIC INTERPRETATION; MULTICHANNEL IMAGE REGULARIZATION; NONLINEAR FILTERS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 33847647237     PISSN: 10765670     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1076-5670(06)45004-7     Document Type: Review

References (100)

1. Alvarez L., Guichard F., Lions P.L., and Morel J.M. Axioms and fundamental equations of image processing. Arch. Ration. Mech. Anal. 123 3 (1993) 199-257
2. Alvarez L., Lions P.L., and Morel J.M. Image selective smoothing and edge detection by nonlinear diffusion (II). SIAM J. Numer. Anal. 29 (1992) 845-866
3. Alvarez L., and Mazorra L. Signal and image restoration using shock filters and anisotropic diffusion. SIAM J. Numer. Anal. 31 2 (1994) 590-605
4. Aubert G., and Kornprobst P. Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences vol. 147 (2002), Springer-Verlag
5. Ashikhmin M. Synthesizing natural textures. ACM Symposium on Interactive 3D Graphics (2001), Research Triangle Park, NC 217-226
6. Barash, D. (2000). Bilateral filtering and anisotropic diffusion: Towards a unified viewpoint. Technical Report, HP Laboratories Israel
7. Becker J., Preusser T., and Rumpf M. PDE methods in flow simulation post processing. Comput. Vis. Sci. 3 3 (2000) 159-167
8. Bertalmio, M., Cheng, L.T., Osher, S., Sapiro, G. (2000a). Variational problems and partial differential equations on implicit surfaces: The framework and examples in image processing and pattern formation. UCLA Research Report
9. Bertalmio M., Sapiro G., Caselles V., and Ballester C. Image inpainting. Proceedings of the SIGGRAPH (2000), ACM Press, Addison-Wesley-Longman 417-424
10. Bertalmio M., Cheng L.T., Osher S., and Sapiro G. Variational problems and partial differential equations on implicit surfaces. Comput. Vis. Sci. 174 2 (2001) 759-780
11. Bertalmio M., Vese L., Sapiro G., and Osher S. Simultaneous structure and texture image inpainting. IEEE Trans. Image Process. 12 8 (2003) 882-889
12. Black M.J., Sapiro G., Marimont D.H., and Heeger D. Robust anisotropic diffusion. IEEE Trans. Image Process. 7 3 (1998) 421-432
13. Blanc-Feraud, L., Charbonnier, P., Aubert, G., Barlaud, M. (1995). Nonlinear image processing: Modeling and fast algorithm for regularization with edge detection. In: Proceedings of the International Conference on Image Processing (ICIP), Washington, DC, 1995, pp. 474-477
14. Blomgren, P. (1998). Total Variation Methods for Restoration of Vector Valued Images. PhD dissertation, Department of Mathematics, University of California, Los Angeles
15. 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
16. Buerkle, D., Preusser, T., Rumpf, M. (2001). Transport and diffusion in time-dependent flow visualization. In: Proceedings IEEE Visualization
17. Cabral B., and Leedom L.C. Imaging vector fields using line integral convolution SIGGRAPH'93. Computer Graphics 27 (1993) 263-272
18. Carmona R., and Zhong S. Adaptive smoothing respecting feature directions. IEEE Trans. Image Process. 7 3 (1998) 353-358
19. Chambolle A., and Lions P.L. Image recovery via total variation minimization and related problems. Numer. Math. 76 2 (1997) 167-188
20. Chan T., Kang S.H., and Shen J. Euler's elastica and curvature based inpainting. SIAM J. Appl. Math (2002)
21. Chan T.F., Kang S.H., and Shen J. Total variation denoising and enhancement color images based on the CB and HSV color models. J. Vis. Comm. Image Represent. 12 4 (2000)
22. Chan T.F., and Shen J. Non-texture inpainting by curvature-driven diffusions (CDD). J. Vis. Comm. Image Represent. 12 4 (2001) 436-449
23. Chan T., and Shen J. Variational restoration of non-flat image features: Models and algorithms. SIAM J. Appl. Math. 61 4 (2000) 1338-1361
24. Chan, T., Shen, J. (2000b). Mathematical models for local deterministic inpaintings. Technical Report 00-11. Department of Mathematics, UCLA, Los Angeles
25. Charbonnier, P., Aubert, G., Blanc-Féraud, M., Barlaud, M. (1994). Two deterministic half-quadratic regularization algorithms for computed imaging. In: Proceedings of the International Conference on Image Processing (ICIP), vol. II, pp. 168-172
26. Charbonnier P., Blanc-Féraud L., Aubert G., and Barlaud M. Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Process. 6 2 (1997) 298-311
27. Chefd'hotel C., Tschumperlé D., Deriche R., and Faugeras O. Regularizing flows for constrained matrix-valued images. J. Math. Imaging Vision 20 2 (2004) 147-162
28. Chu, M.T. (1990a). A list of matrix flows with applications. Technical Report. Department of Mathematics, North Carolina State University
29. Chu, M.T. (1990b). Matrix differential equations: A continuous realization process for linear algebra problems. Technical Report. Department of Mathematics, North Carolina State University
30. Coulon O., Alexander D.C., and Arridge S.R. A regularization scheme for diffusion tensor magnetic resonance images. 17th International Conference on Information Processing in Medical Imaging. Lecture Notes in Computer Science vol. 2082 (2001), Springer 92-105
31. Criminisi A., Perez P., and Toyama K. Object removal by exemplar-based inpainting. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) vol. 2 (2003) 721-728
32. Deriche R., and Faugeras O. Les EDP en traitement des images et vision par ordinateur. Traitement Signal 13 6 (1997)
33. Diewald U., Preusser T., and Rumpf M. Anisotropic diffusion in vector field visualization on Euclidian domains and surfaces. IEEE Trans. Vis. Comput. Graphics 6 2 (2000) 139-149
34. Di Zenzo S. A note on the gradient of a multi-image. Comput. Vision, Graphics Image Process. 33 (1986) 116-125
35. Gilboa G., Sochen N., and Zeevi Y. Forward-and-backward diffusion processes for adaptative image enhancement and denoising. IEEE Trans. Image Process (2002)
36. Hadamard J. Lectures on the Cauchy Problem in Linear Partial Differential Equations (1923), Yale University Press, New Haven
37. Jia J., and Tang C.K. Image repairing: Robust image synthesis by adaptive ND tensor voting. IEEE Conference on Computer Vision and Pattern Recognition (CVPR) vol. 1 (2003) 643-650
38. Kichenassamy S. The Perona-Malik paradox. SIAM J. Appl. Math. 57 5 (1997) 1328-1342
39. Kimmel R., Malladi R., and Sochen N. Image processing via the Beltrami operator. Proceedings of the 3rd Asian Conference on Computer Vision vol. 1 (1998), Hong Kong 574-581
40. Kimmel R., Malladi R., and Sochen N. Images as embedded maps and minimal surfaces: Movies, color, texture, and volumetric medical images. Int. J. Comput. Vision 39 2 (2000) 111-129
41. Kimmel R., and Sochen N. Geometric-variational approach for color image enhancement and segmentation. Scale-Space Theories in Computer Vision, Scale-Space'99. Lecture Notes in Computer Science vol. 1682 (1999), Springer 295-305
42. Kimmel R., and Sochen N. Orientation diffusion or how to comb a porcupine. J. Vis. Commun. Image Represent. 13 (2002) 238-248
43. Koenderink J.J. The structure of images. Biol. Cybernet. 50 (1984) 363-370
44. Kornprobst, P. (1998). Contributions à la Restauration d'Images et à l'Analyse de Séquences: Approches Variationnelles et Solutions de Viscosité. PhD thesis, Université de Nice-Sophia Antipolis
45. Kornprobst P., Deriche R., and Aubert G. Image restoration via PDEs. First Annual Symposium on Enabling Technologies for Law Enforcement and Security-SPIE Conference 2942 (1996), Boston, Massachusetts
46. Kornprobst, P., Deriche, R., Aubert, G. (1997). Nonlinear operators in image restoration. In: Proceedings of the IEEE International Conference on Computer Vision and Pattern Recognition. Puerto Rico, pp. 325-331
47. Koschan, A. (1995). A comparative study on color edge detection. In: Proceedings of the 2nd Asian Conference on Computer Vision, ACCV'95, pp. 574-578
48. Krissian K. Multiscale analysis: Application to medical imaging and 3D vessel detection. PhD Thesis (2000), INRIA, Sophia Antipolis
49. Lindeberg T. Scale-Space Theory in Computer Vision (1994), Kluwer Academic Publishers
50. Masnou S., and Morel J.-M. Level lines based disocclusion. IEEE Int. Conf. Image Process. 3 (1998) 259-263
51. Meyer Y. Oscillatory Patterns in Image Processing and Nonlinear Evolution Equations. University Lecture Series vol. 22 (2001), American Mathematical Society, Providence, RI
52. Nielsen M., Florack L., and Deriche R. Regularization, scale-space and edge detection filters. J. Math. Imaging Vision 7 4 (1997) 291-308
53. Nikolova, M., (2001). Image restoration by minimizing objective functions with nonsmooth data-fidelity terms. In: IEEE Workshop on Variational and Level Set Methods. Vancouver, Canada, pp. 11-19
54. Nikolova, M., Ng, M. (2001). Fast image reconstruction algorithms combining half-quadratic regularization and preconditioning. In: Proceedings of the International Conference on Image Processing. IEEE Signal Processing Society
55. Papadopoulo, T., Lourakis, M.I.A. (2000). Estimating the Jacobian of the Singular Value Decomposition: Theory and Applications. Research Report 3961. INRIA, Sophia Antipolis
56. Pardo, A., Sapiro, G. (2000). Vector probability diffusion. In: Proceedings of the International Conference on Image Processing, IEEE Signal Processing Society
57. Perona P. Orientation diffusions. IEEE Trans. Image Process. 7 3 (1998) 457-467
58. Perona P., and Malik J. Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Machine Intell. 12 7 (1990) 629-639
59. Poynton, C.A. (1995). Poynton's colour FAQ, www.inforamp.net/poynton [Web page]
60. Preusser, T., Rumpf, M. (1999). Anisotropic nonlinear diffusion in flow visualization. In: IEEE Visualization Conference
61. Press W.H., Flannery B.P., Teukolsky S.A., and Vetterling W.T. Runge-Kutta method. Numerical Recipes in FORTRAN: The Art of Scientific Computing (1992), Cambridge University Press 704-716
62. ter Haar Romeny B.M. Geometry-driven diffusion in computer vision. Computational Imaging and Vision (1994), Kluwer Academic Publishers
63. Rudin L., Osher S., and Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D 60 (1992) 259-268
64. Sapiro, G. (1996). Vector-valued active contours. In: Proceedings of the International Conference on Computer Vision and Pattern Recognition. San Francisco, pp. 680-685
65. Sapiro G. Color snakes. Comput. Vision Image Understanding 68 2 (1997)
66. Sapiro G. Geometric Partial Differential Equations and Image Analysis (2001), Cambridge University Press
67. Sapiro G., and Ringach D.L. Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans. Image Process. 5 11 (1996) 1582-1585
68. Sochen N., Kimmel R., and Bruckstein A.M. Diffusions and confusions in signal and image processing. J. Math. Imaging Vision 14 3 (2001) 195-209
69. Sochen, N. (2001). On affine invariance in the Beltrami framework for vision. In: IEEE Workshop on Variational and Level Set Methods. Vancouver, Canada, pp. 51-56
70. Sochen N., Kimmel R., and Malladi R. A geometrical framework for low level vision. IEEE Trans. Image Process. [Special Issue on PDE based Image Processing] 7 3 (1998) 310-318
71. Stalling, D., Hege, H.C. (1995). Fast and resolution independent line integral convolution. In: ACM SIGGRAPH, 22nd Annual Conference on Computer Graphics and Interactive Technique, pp. 249-256
72. Tang, B., Sapiro, G., Caselles, V. (1998). Direction diffusion. In: International Conference on Computer Vision
73. Tang B., Sapiro G., and Caselles V. Diffusion of general data on non-flat manifolds via harmonic maps theory: The direction diffusion case. Int. J. Comput. Vision 36 2 (2000) 149-161
74. Teboul S., Blanc-Féraud L., Aubert G., and Barlaud M. Variational approach for edge-preserving regularization using coupled PDEs. IEEE Trans. Image Process. [Special Issue on PDE based Image Processing] 7 3 (1998) 387-397
75. Tikhonov A.N. Regularization of incorrectly posed problems. Sov. Math. Dokl. 4 (1963) 1624-1627
76. Tomasi, C., Manduchi, R. (1998). Bilateral filtering for gray and color images. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 839-846
77. Tschumperlé, D. (2002). PDE's Based Regularization of Multivalued Images and Applications. PhD thesis. Université de Nice Sophia Antipolis
78. Tschumperlé D. Fast anisotropic smoothing of multi-valued images using curvature-preserving PDEs. Int. J. Comput. Vision 68 1 (2006) 65-82
79. Tschumperlé, D., Deriche, R. (2001a). Constrained and unconstrained PDE's for vector image restoration. In: Proceedings of the 10th Scandinavian Conference on Image Analysis, Bergen, Norway, pp. 153-160
80. Tschumperlé, D., Deriche, R. (2001b). Diffusion tensor regularization with constraints preservation. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition. Kauai, Hawaii
81. Tschumperlé, D., Deriche, R. (2001c). Regularization of orthonormal vector sets using coupled PDEs. In: IEEE Workshop on Variational and Level Set Methods. Vancouver, Canada, pp. 3-10
82. Tschumperlé D., and Deriche R. Diffusion PDE's on vector-valued images: Local approach and geometric viewpoint. IEEE Signal Process. Mag. 19 5 (2002) 16-25
83. Tschumperlé, D., Deriche, R. (2002b). Orthonormal vector sets regularization with PDE's and aplications. Int. J. Comput. Vision
84. Tschumperlé D., and Deriche R. Vector-valued image regularization with PDEs: A common framework for different applications. IEEE Conference on Computer Vision and Pattern Recognition vol. 1 (2003) 651-656
85. Tschumperlé D., and Deriche R. Vector-valued image regularization with PDEs: A common framework for different applications. IEEE Trans. Pattern Anal. Machine Intell. 27 4 (2005)
86. Vese L.A., and Osher S. Numerical methods for p-harmonic flows and applications to image processing. CAM Report 01-22. UCLA (2001)
87. Weickert, J. (1994), Anisotropic diffusion filters for image processing based quality control. In: 7th European Conference on Mathematics in Industry, pp. 355-362
88. Weickert J. (1996a). Anisotropic diffusion in image processing. PhD thesis. University of Kaiserslautern, Laboratory of Technomathematics. Germany
89. Weickert J. Theoretical foundations of anisotropic diffusion in image processing. Comput. Suppl. 11 (1996) 221-236
90. Weickert, J. (1997a). Coherence-enhancing diffusion of colour images. In: 7th National Symposium on Pattern Recognition and Image Analysis
91. Weickert J. A Review of Nonlinear Diffusion Filtering. Scale-Space Theory in Computer Vision. Lecture Notes in Computer Science vol. 1252 (1997), Springer, Berlin 3-28
92. Weickert J. Anisotropic Diffusion in Image Processing (1998), Teubner-Verlag, Stuttgart
93. Weickert J. Coherence-enhancing diffusion of colour images. Image Vision Comput. 17 (1999) 199-210
94. Weickert J., and Benhamouda B. A semidiscrete nonlinear scale-space theory and its relation to the Perona-Malik paradox. Adv. Comput. Vision (1997), Springer, Wien 1-10
95. Weickert, J., Benhamouda, B. (1997). Why the Perona-Malik Filter Works. Technical Report 97/22. Department of Computer Science, University of Copenhagen
96. Weickert J., and Brox T. Diffusion and regularization of vector and matrix-valued images. Inverse Problems, Image Analysis, and Medical Imaging, Contemp. Math. 313 (2002) 251-268
97. Wei L.Y., and Levoy M. Fast texture synthesis using tree-structured vector quantization. ACM SIGGRAPH, International Conference on Computer Graphics and Interactive Techniques (2000) 479-488
98. Wesseling P. Principles of Computational Fluid Dynamics (2000), Springer, Berlin
99. Witkin, A.P. (1983). Scale-space filtering. In: International Joint Conference on Artificial Intelligence, pp. 1019-1021
100. Yezzi A. Modified curvature motion for image smoothing and enhancement. IEEE Trans. Image Process. 7 3 (1998) 345-352