A direct variational approach to a problem arising in image reconstruction

Interfaces and Free Boundaries

Volumn 5, Issue 1, 2003, Pages 63-81

  Ambrosio, Luigi ^a   Masnou, Simon ^b  

^a SCUOLA NORMALE SUPERIORE   (Italy)

^b UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

Author keywords

BV; Image processing; Image reconstruction; Mean curvature; Relaxation; Varifolds

Indexed keywords

EID: 84892197297     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/72     Document Type: Article

References (45)

1. ALLARD, W. K. On the first variation of a varifold. Ann. of Math. (2) 95 (1972), 417-491. MR 53 #1379 MR 46 #6136 Zbl 0252.49028
2. ALVAREZ, L., GOUSSEAU, Y., & MOREL, J.-M. The size of objects in natural and artificial images. Adv. Imaging Electr. Phys. 111 (1999), 167-242.
3. AMBROSIO, L., CASELLES, V., MASNOU, S., & MOREL, J.-M. Connected components of sets of finite perimeter and applications to image processing. J. Europ. Math. Soc. 3 (2001), 39-92. MR 2002g:49072 Zbl 0981.49024
4. AMBROSIO, L., FUSCO, N., & PALLARA, D. Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). MR 2003a:49002 Zbl 0957.49001
5. ASHIKHMIN, M. Synthesizing natural textures. Proc. ACM Symp. Interactive 3D Graphics (Research Triangle Park, 2001), 217-226.
6. BALLESTER, C., BERTALMIO, M., CASELLES, V., SAPIRO, G., & VERDERA, J. Filling-in by joint interpolation of vector fields and gray levels. IEEE Trans. Image Process. 10 (2001), 1200-1211. MR 2003a:94007
7. BELLETTINI, G., DAL MASO, G., & PAOLINI, M. Semicontinuity and relaxation properties of a curvature depending functional in 2D. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), 247-297. MR 94g:49101 Zbl 0797.49013
8. BRAKKE, K. The Motion of a Surface by its Mean Curvature. Princeton University Press (1978). MR 82c:49035
9. CASELLES, V., COLL, T., & MOREL, J.-M. Geometry and color in natural images. J. Math. Imaging Vision 2 (2002), 89-105. MR 1 892 677 Zbl 0994.68167
10. CHAN, T. & SHEN, J. Mathematical models for local deterministic inpaintings. SIAM J. Appl. Math. 62 (2001), 1019-1043. Zbl 01730720
11. COHEN, A., DAHMEN, W., DAUBECHIES, I., & DEVORE, R. Harmonic analysis of the space BV. Rev. Math. Iberoamericana, submitted (2002).
12. DAL MASO, G. An Introduction to -Convergence. Progr. Nonlinear Differential Equations Appl. 8, Birkhäuser (1993). MR 94a:49001 Zbl 0816.49001
13. DUGGAN, J. P. W2,p regularity for varifolds with mean curvature. Comm. Partial Differential Equations 11 (1986), 903-926. MR 87m:49086 Zbl 0634.35022
14. EFROS, A. & FREEMAN, W. Image quilting for texture synthesis and transfer. Proc. of SIGGRAPH'01 (Los Angeles, 2001), 341-346.
15. EFROS, A. & LEUNG, T. Texture synthesis by non-parametric sampling. Internat. Conf. on Comput. Vision, Vol. 2, 1999, 1033-1038.
16. EULER, L. Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Lausanne (1744). Zbl 0788.01072
17. EVANS, L. C. & SPRUCK, J. Motion of level sets by mean curvature II. Trans. Amer. Math. Soc. 330 (1991), 635-681. MR 92f:58050 Zbl 0829.53040
18. FANTONI, C. & GERBINO, W. Probing the amodal completion of asymmetric occluded angles: a test of interpolation models. J. Vision 1 (2001), 461a; available at http://journalofvision.org/1/3/461.
19. FEDERER, H. Geometric Measure Theory. Springer (1969). MR 41 #1976 Zbl 0874.49001
20. GIUSTI, E. Minimal Surfaces and Functions of Bounded Variation. Birkhäuser (1994). MR 87a:58041 Zbl 0545.49018
21. HORN, B. K. P. The curve of least energy. ACM Trans. Math. Software 9 (1983), 441-460. MR 87c:65009 Zbl 0533.41007
22. HUTCHINSON, J. E. C1,-multiple function regularity and tangent cone behaviour for varifolds with second fundamental form in Lp. Geometric Measure Theory and the Calculus of Variations (Arcata, CA, 1984), W. K. Allard and F. J. Almgren (eds.), Proc. Sympos. Pure Math. 44, Amer. Math. Soc. (1986), 281-306. MR 87i:49059 Zbl 0635.49020
23. KANIZSA, G. Organisation in Vision. Praeger, New York (1979).
24. LEUNG, T. & MALIK, J. Contour continuity in region based image segmentation. Fifth Euro. Conf. Computer Vision (Freiburg, 1998), 1, 544-559.
25. LIANG, L., LIU, C., XU, Y., GUO, B., & SHUM, H.-Y. Real-Time texture synthesis by patch-based sampling. Microsoft Research Technical Report, MSR-TR-2001-40, March 2001.
26. MASNOU, S. Filtrage et désocclusion d'images par méthodes d'ensembles de niveau. PhD thesis, Université Paris-Dauphine (1998).
27. MASNOU, S. Disocclusion: a variational approach using level lines. IEEE Trans. Image Process. 11 (2002), 68-76. MR 2002m:68111
28. MASNOU, S. & MOREL, J.-M. Level lines based disocclusion. Proc. ICIP'98 IEEE Internat. Conf. on Image Processing (Chicago, 1998), Vol. 3, 259-263.
29. MASNOU, S. & MOREL, J.-M. Construction of solutions to the disocclusion problem. Forthcoming (2002).
30. MUMFORD, D. Elastica and computer vision. Algebraic Geometry and Applications, C. Bajaj (ed.), Springer (1994), 491-506. MR 95a:92026 Zbl 0798.53003
31. NITZBERG, M., MUMFORD, D., & SHIOTA, T. Filtering, Segmentation and Depth. Lecture Notes in Comput. Sci. 662, Springer (1993). MR 94h:92012 Zbl 0801.68171
32. PAGET, R. & LONGSTAFF, I. Texture synthesis via a noncausal nonparametric multiscale Markov random field. IEEE Trans. Image Process. 7 (1998), 925-931.
33. POPAT, A. C. Conjoint probabilistic subband modeling. PhD thesis, Massachussets Institute of Technology (1997).
34. RUDIN, L. & OSHER, S. Total variation based image restoration with free local constraints. Proc. IEEE ICIP'94 (Austin, TX, 1994), Vol. 1, 31-35.
35. SCHÄTZLE, R. Hypersurfaces with mean curvature given by a trace. Habilitationsschrift, Albert-Ludwigs-Universität, Freiburg im Breisgau (1999).
36. SCHÄTZLE, R. Quadratic tilt-excess decay and strong maximum principle for varifolds. Preprint (2000).
37. SHARON, E., BRANDT, A., & BASRI, R. Completion energies and scale. Proc. IEEE Conf. Comput. Vision and Pattern Recognition (San Juan, Puerto Rico, 1997), 884-890.
38. SIMON, L. Lectures on Geometric Measure Theory. Proc. Centre for Math. Anal. 3, Australian Nat. Univ. (1983). MR 87a:49001 Zbl 0546.49019
39. THORNBER, K. K. & WILLIAMS, L. R. Analytic solution of stochastic completion fields. Biological Cybernetics 75 (1996), 141-151. Zbl 0869.92030
40. THORNBER, K. K. & WILLIAMS, L. R. Characterizing the distribution of completion shapes with corners using a mixture of random processes. Pattern Recognition 33 (2000), 543-553.
41. ULLMAN, S. Filling-in the gaps: the shape of subjective contour and a model for their generation. Biological Cybernetics 25 (1976), 1-6.
42. WEI, L.-Y. & LEVOY, M. Fast texture synthesis using tree-structured vector quantization. Proc. SIGGRAPH 2000, 479-488.
43. WILLIAMS, L. R. & JACOBS, D. W. Stochastic completion fields: a neural model of illusory contour shape and salience. Neural Computation 9 (1997), 837-858.
44. ZHU, S., WU, Y., & MUMFORD, D. Filters, random fields and maximum entropy (FRAME)-Towards a unified theory for texture modeling. Internat. J. Comput. Vision 27 (1998), 107-126.
45. ZIEMER, W. P. Weakly Differentiable Functions. Springer (1989). MR 91e:46046 Zbl 0692.46022