Oriented distance function and its evolution equation for initial sets with thin boundary

SIAM Journal on Control and Optimization

Volumn 42, Issue 6, 2004, Pages 2286-2304

  Delfour, Michel C ^a   Zolésio, Jean Paul ^b  

^a UNIVERSITÉ DE MONTRÉAL   (Canada)

^b INRIA   (France)

Author keywords

Design; Extension velocity; Image; Level set method; Narrow band method; Nonlinear evolution equation; Oriented distance function; Shape; Signed distance function; Smooth sets; Submanifold; Vision

Indexed keywords

COMPUTER VISION; DIFFERENTIAL EQUATIONS; FUNCTIONS; IMAGE ANALYSIS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; OPTIMIZATION; VELOCITY MEASUREMENT;

EXTENSION VELOCITY; LEVEL SET METHODS; NARROW-BAND METHODS; NONLINEAR EVOLUTION EQUATION; ORIENTED DISTANCE FUNCTIONS; SIGNED DISTANCE FUNCTIONS; SMOOTH SETS;

NONLINEAR EQUATIONS;

EID: 10244276990     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012902411945     Document Type: Article

References (42)

1. D. ADALSTEINSSON AND J. A. SETHIAN, The fast construction of extension velocities in level set methods, J. Comput. Phys., 148 (1999), pp. 2-22.
2. L. AMBROSIO AND C. MANTEGAZZA, Curvature and distance function from a manifold, J. Geom. Anal., 8 (1998), pp. 723-748.
3. D. H. ARMITAGE AND Ü. KURAN, The convexity of a domain and the superharmonicity of the signed distance function, Proc. Amer. Math. Soc., 93 (1985), pp. 598-600.
4. J.-P. AUBIN, Mutational equations in metric spaces, Set-Valued Anal., 1 (1993), pp. 3-46.
5. J.-P. AUBIN, Mutational and Morphological Analysis, Tools for Shape Evolution and Morphogenesis, Birk̈user, Boston, Basel, Berlin, 1999.
6. D. BUCUR AND J.-P. ZOLÉSIO, Free boundary problems and density perimeter, J. Differential Equations, 126 (1996), pp. 224-243.
7. V. I. BURENKOV, Regularized distance, Trudy Moskov. Inst. Radiotehn. Élektron, i Avtomat. Vyp. 67 Mat. (1973), pp. 113-117, 152 (in Russian).
8. 1,1 submanifold, in Differential Geometric Methods in the Control of Partial Differential Equations, Contemp. Math. 268, R. Gulliver, W. Littman, and R. Triggiani, eds., AMS, Providence, RI, 2000, pp. 83-115.
9. M. C. DELFOUR AND J.-P. ZOLÉSIO, Shape analysis via oriented distance functions, J. Funct. Anal., 123 (1994), pp. 129-201.
10. M. C. DELFOUR AND J.-P. ZOLÉSIO, Shape analysis via distance functions: Local theory, in Boundaries, Interfaces, and Transitions, CRM Proc. Lecture Notes 13, M. C. Delfour, ed., AMS, Providence, RI, 1998, pp. 91-123.
11. M. C. DELFOUR AND J.-P. ZOLÉSIO, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, Adv. Des. Control 4, SIAM, Philadelphia, 2001.
12. M. C. DELFOUR AND J.-P. ZOLÉSIO, Shape Identification via Metrics Constructed from the Oriented Distance Function, CRM Report, July 2002, first revision May 9, 2003, Université de Montréal.
13. M. C. DELFOUR AND J.-P. ZOLÉSIO, The New Family of Cracked Sets and the Image Segmentation Problem Revisited, CRM Report, May 2003, Université de Montréal; Communications in Information and Systems, to appear.
14. C. DELLACHERIE, Ensembles analytiques, capacités, mesures de Hausdorff, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
15. F. R. DESAINT AND J.-P. ZOLÉSIO, Shape boundary derivative of tangential boundary value problems, Z. Angew. Math. Mech., 76, suppl. 2 (1996), pp. 85-88.
16. F. R. DESAINT AND J.-P. ZOLÉSIO, Manifold derivative in the Laplace-Beltrami equation, J. Funct. Anal., 151 (1997), pp. 234-269.
17. D. E. EDMUNDS AND R. M. EDMUNDS, Distance functions and Orlicz-Sobolev spaces, Canad. J. Math., 38 (1986), pp. 1181-1198.
18. H. FEDERER, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), pp. 418-491.
19. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, New York, 1977.
20. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, New York, 1983.
21. J. GOMES AND O. FAUGERAS, Reconciling distance functions and level sets, J. Visual Com. and Image Representation, 11 (2000), pp. 209-223.
22. F. HAUSDORFF, Grundzüge der Mengenlehre, Leipzig, 1914; Set Theory, translation from the German 3rd edition (1937) into English by J. R. Aumann et al., Chelsea, New York, 1957.
23. C. M. HOFFMANN, F. P. PREPARATA, P. C. KANELLAKIS, AND S. MICALI, EDS., Advances in Computing Research: Issues in Robotics and Nonlinear Geometry, JAI Press, Greenwich, CT, 1992.
24. C. M. HOFFMANN, Algebraic and numerical techniques for offsets and blends, in Computation of Curves and Surfaces, W. Dahmen et al., eds., Kluwer, Dordrecht, 1990, pp. 499-528.
25. C. M. HOFFMANN, How to construct the skeleton of CSG objects, in Computer-Aided Surface Geometry and Design (Bath, 1990), Inst. Math. Appl. Conf. Ser. New Ser. 48, A. Bowyer, ed., Oxford University Press, New York, 1994, pp. 421-437.
26. R. B. HOLMES, Smoothness of certain metric projections on Hilbert space, Trans. Amer. Math. Soc., 184 (1973), pp. 87-100.
27. H. ISHII AND P. SOUGANIDIS, Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor, Tohoku Math. J. (2), 47 (1995), pp. 227-250.
28. k hypersurfaces, J. Differential Equations, 40 (1981), pp. 116-120.
29. G. M. LIEBERMAN, Regularized distance and its applications, Pacific J. Math., 117 (1985), pp. 329-352.
30. R. MALLADI, J. A. SETHIAN, AND B. C. VEMURI, Shape modeling with front propagation: A level set approach, IEEE Trans. on Pattern Analysis and Machine Intelligence, 17 (1995), pp. 158-175.
31. G. MATHERON, Examples of topological properties of skeletons, in Image Analysis and Mathematical Morphology, J. Serra, ed., Academic Press, London, 1988, pp. 217-238.
32. S. OSHER AND J. A. SETHIAN, Front propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulation, J. Comput. Phys., 79 (1988), pp. 12-49.
33. M. J. PARKER, The fundamental function of the distance, Bull. London Math. Soc., 19 (1987), pp. 337-342.
34. M. J. PARKER, Convex sets and subharmonicity of the distance function, Proc. Amer. Math. Soc., 103 (1988), pp. 503-506.
35. J.-B. POLY AND G. RABY, Fonction distance et singularités, Bull. Sci. Math. (2), 108 (1984), pp. 187-195.
36. D. POMPÉJU, Ann. de Toulouse (2), 7 (1905).
37. J. SERRA, Image Analysis and Mathematical Morphology, English version revised by N. Cressie, Academic Press, London, 1984.
38. J. SERRA, Hausdorff distances and interpolations, in Mathematical Morphology and Its Applications to Image and Signal Processing (Amsterdam, 1998), Comput. Imaging Vision 12, H. J. A. M. Heijmans and J. B. T. M. Roerdink, eds., Kluwer, Dordrecht, 1998, pp. 107-114.
39. J. SERRIN, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London, Ser. A, 264 (1969), pp. 413-496.
40. S. STIFTER, The Roider method: A method for static and dynamic collision detection, in Advances in Computing Research: Issues in Robotics and Nonlinear Geometry, C. Hoffmann et al., eds., JAI Press, Greenwich, CT, 1992.
41. L. YOUNES, Computable elastic distances between shapes, SIAM J. Appl. Math., 58 (1998), pp. 565-586.
42. L. YOUNES, Optimal matching between shapes via elastic deformations, Image and Vision Computing, 17 (1999), pp. 381-389.