Shape identification via metrics constructed from the oriented distance function

• Autorzy: Delfour M. C. , Zolesio J.-P.
• Języki publikacji: EN

Abstrakty

EN

This paper studies the generic identification problem: to find the best non-parametrized object [Omega] which minimizes some weighted sum of distances to I a priori given objects [Omega]_i for metric distances constructed from the W^1,p-norm on the oriented (resp. signed) distance function which occurs in many different fields of applications. It discusses existence of solution to the generic identification problem and investigates the Eulerian shape semiderivatives with special consideration to the non-differentiable terms occurring in their expressions. A simple example for the new cracked sets recently introduced in Delfour and Zolesio (2004b) is also presented. It can be viewed as an approximation of a cracked set by sets whose boundary is made up of pieces of lines or Bezier curves that are not necessarily connected.

Słowa kluczowe

EN

shape identification; sensitivity analysis; variational problems; set-valued and variational analysis; image processing; image enhancing; metric; distance; oriented distance function; signed distance function; Sobolev domains; velocity method; cracked sets

PL

identyfikacja kształtu; analiza wrażliwości; zagadnienie wariacyjne; przetwarzanie obrazów; metryka; dziedzina Soboleva; metoda prędkości; odległość

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2005
• Tom: Vol. 34, no 1
• Strony: 137--164
• Opis fizyczny: Bibliogr. 25 poz., rys.

Twórcy

autor

Delfour M. C.

• Centre de recherches mathematiques et Departement de mathematiques et de statistique, Universite de Montreal C. P. 6128, succ. Centre-ville, Montreal (Qc), Canada H3C 3J7

autor

Zolesio J.-P.

• CNRS and INRIA INRIA, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France

Bibliografia

• Adalsteinsson, D. and Sethian, J.A. (1999) The fast construction of extension velocities in level set methods. J. Comp. Physics 148, 2–22.
• Aubin, J.-P. (1999) Mutational and Morphological Analysis, Tools for Shape Evolution and Morphogenesis. Birkhauser, Boston, Basel, Berlin.
• Caselles, V., Kimmel, R. and Sapiro, G. (1997) Geodesic active contours. Int. J. of Computer Vision 22 (1), 61–79.
• Delfour, M.C. (2000) Tangential differential calculus and functional analysis on a C1,1 submanifold. In: R. Gulliver, W. Littman and R. Triggiani, eds., Differential-geometric methods in the control of partial differential equations, Contemp. Math. 268, AMS Publications, 83–115.
• Delfour, M.C., Doyon, N. and Zolesio, J.-P. (2005a) Uniform cusp property, boundary integral, and compactness for shape optimization. In: J. Cagnol and J.-P. Zolesio, eds., System Modeling and Optimization, Kluwer, 25-40.
• Delfour, M.C., Doyon, N. and J.-P. Zolesio (2005b) Extension of the uniform cusp property in shape optimization. In: O. Emanuvilov, G. Leugering, R. Triggiani, and B. Zhang, eds., Control of Partial Differential Equations, Lectures Notes in Pure and Applied Mathematics 242, Chapman & Hall/CRC, Boca Raton, 71-86.
• Delfour, M.C., Doyon, N. and Zolesio, J.-P. (2005c) The uniform fat segment and cusp properties in shape optimization. In: J. Cagnol and J.-P. Zolesio, eds., Control and Boundary Analysis, Lecture Notes in Pure and Applied Maths., 240, Chapman & Hall/CRC, Boca Raton, 85-96.
• Delfour, M.C. and Zolesio, J.-P. (1994) Shape analysis via oriented distance functions. J. Funct. Anal. 123 (1), 129–201.
• Delfour, M.C. and Zolesio, J.-P. (1998) Shape analysis via distance functions: local theory. In: Boundaries, interfaces, and transitions, CRM Proc. Lecture Notes, 13, Amer. Math. Soc., Providence, RI, 91–123.
• Delfour, M.C. and Zolesio, J.-P. (2001) Shapes and Geometries: Analysis, Differential Calculus and Optimization. SIAM series on Advances in Design and Control, SIAM, Philadelphia.
• Delfour, M.C. and Zolesio, J.-P. (2004a) Oriented distance function and its evolution equation for initial sets with thin boundary. SIAM J. Control and Optim. 42 (6), 2286–2304.
• Delfour, M.C. and Zolesio, J.-P. (2004b) The new family of cracked sets and the image segmentation problem revisited. Communications in Information and Systems 4 (1), 29-52.
• Gilbarg, D. and Trudinger, N.S. (1977) Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, New York.
• Gomes, J. and Faugeras, O. (2000) Reconciling distance functions and level sets. J. Visual Com. and Image Representation 11, 209–223.
• Hoffmann, C.M., Preparata, F.P., Kanellakis, P.C., Micali, S., eds. (1992)Advances in Computing Research: Issues in Robotics and Nonlinear Geometry. JAI Press.
• Hoffmann, C.M. (1990) Algebraic and numerical techniques for offsets and blends. In: W. Dahmen et al., eds., Computation of Curves and Surfaces Kluwer, 499–528.
• Hoffmann, C.M. (1990) How to construct the skeleton of CSG objects. In: A. Bowyer and J. Davenport, eds., Computer-aided surface geometry and design, (Bath, 1990), Inst. Math. Appl. Conf. Ser., New Ser. 48, Oxford Univ. Press, New York, 1994, 421–437.
• Ishii, H. and Souganidis, P. (1995) Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor. Tohoku Math. J. 47 (2), 227–250.
• Malladi, R., Sethian, J.A. and Vemuri, B.C. (1995) Shape Modeling with Front Propagation: A Level Set Approach. IEEE Trans. on Pattern Analysis and Machine Intelligence 17 (2), 158–175.
• Matheron, G. (1988) Examples of Topological Properties of Skeletons. In: J. Serra, ed., Image Analysis and Mathematical Morphology, Academic Press, London, 217–238.
• Osher, S. and Sethian, J.A. (1988) Front propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulation. J. Computational Physics 79, 12–49.
• Serra, J. (1984) Image Analysis and Mathematical Morphology. English version revised by Noel Cressie. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London.
• Serra, J. (1998) Hausdorff distances and interpolations. In: Mathematical morphology and its applications to image and signal processing, Comput. Imaging Vision, 12, Kluwer Acad. Publ., Dordrecht, 107–114.
• Stifter, S. (1992) The Roider method: a method for static and dynamic collision detection. In: C. Hoffmann et al. eds., Advances in Computing Research: Issues in Robotics and Nonlinear Geometry, JAI Press.
• Zolesio, J.-P. (1984) Les derivees par rapport aux noeuds des triangulations en identification de domaines, (French) [Derivatives with respect to triangularization nodes and their uses in domain identification]. Ann. Sci. Math. Quebec 8 (1), 97–120.