Santosa, Fadil A level-set approach for inverse problems involving obstacles. (English) Zbl 0870.49016 ESAIM, Control Optim. Calc. Var. 1, 17-33 (1996). Summary: An approach for solving inverse problems involving obstacles is proposed. The approach uses a level-set method which has been shown to be effective in treating problems of moving boundaries, particularly those that involve topological changes in the geometry.We develop two computational methods based on this idea. One method results in a nonlinear time-dependent partial differential equation for the level-set function whose evolution minimizes the residual in the data fit. The second method is an optimization that generates a sequence of level-set functions that reduces the residual. The methods are illustrated in two applications : a deconvolution problem and a diffraction screen reconstruction problem. Cited in 3 ReviewsCited in 94 Documents MSC: 49L20 Dynamic programming in optimal control and differential games Keywords:inverse problems; level-set method; Hamilton-Jacobi equations; surface evolution; optimization; deconvolution; diffraction Software:Matlab × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] MATLAB: High-performance numeric computation and visualization software - Refrence guide, MathWorks, Natick, MA, 1992. [2] V. Casselles, F. Catté, T. Coll, and F. Dibos: A geometric model for active contours in image processing, Numerische Mathematik, 66, 1993, 1-31. Zbl0804.68159 MR1240700 · Zbl 0804.68159 · doi:10.1007/BF01385685 [3] D. Colton and R. Kress: Inverse acoustic and electromagnetic scattering theory, Springer-Verlag, Berlin, 1992. Zbl0760.35053 MR1183732 · Zbl 0760.35053 [4] J. Dennis and R. Schnabel: Numerical methods for unconstrained optimization and nonlinear equations, Prentice-Hall, Englewood Cliffs, 1983. Zbl0579.65058 MR702023 · Zbl 0579.65058 [5] A. Friedman: Detection of mines by electric measurements, SIAM J. Appl Math., 47, 1987, 201-212. Zbl0636.35084 MR873244 · Zbl 0636.35084 · doi:10.1137/0147012 [6] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi: Gradient flows and geometric active contour models, Proc. ICCV, Cambridge, 1995. [7] R. LeVeque and Z. Li: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM ,L Num, Analysis, 31, 1994, 1019-1044. Zbl0811.65083 MR1286215 · Zbl 0811.65083 · doi:10.1137/0731054 [8] R. Magnanini and G. Papi: An inverse problem for the helmholtz equation, Inverse Problems, 1, 1985, 357-370. Zbl0608.35076 MR824135 · Zbl 0608.35076 · doi:10.1088/0266-5611/1/4/007 [9] R. Malladi, J. Sethian, and B. Vemuri: Shape modeling with front propagation: a level set approach, IEEE Trans. Pattern Anal. Machine Intell., 17, 1995, 158-175. [10] S. Osher and J. Sethian: Fronts propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 56, 1988, 12-49. Zbl0659.65132 MR965860 · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2 [11] M. Sondhi: Reconstruction of objects from their sound-diffraction patterns, J. Acoust. Soc. Am., 46, 1969, 1158-1164. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.