Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set

We are concerned with the retrieval of the unknown cross section of a homogeneous cylindrical obstacle embedded in a homogeneous medium and illuminated by time-harmonic electromagnetic line sources. The dielectric parameters of the obstacle and embedding materials are known and piecewise constant. That is, the shape (here, the contour) of the obstacle is sufficient for its full characterization. The inverse scattering problem is then to determine the contour from the knowledge of the scattered field measured for several locations of the sources and/or frequencies. An iterative process is implemented: given an initial contour, this contour is progressively evolved such as to minimize the residual in the data fit. This algorithm presents two main important points. The first concerns the choice of the transformation enforced on the contour. We will show that this involves the design of a velocity field whose expression only requires the resolution of an adjoint problem at each step. The second concerns the use of a level-set function in order to represent the obstacle. This level-set function will be of great use to handle in a natural way splitting or merging of obstacles along the iterative process. The evolution of this level-set is controlled by a Hamilton-Jacobi-type equation which will be solved by using an appropriate finite-difference scheme. Numerical results of inversion obtained from both noiseless and noisy synthetic data illustrate the behaviour of the algorithm for a variety of obstacles.

[1]  J. Cea.  Une Methode Numerique pour la Recherche d’un Domaine Optimal , 1976 .

[2]  Edward J. Haug,et al.  Optimization of distributed parameter structures , 1981 .

[3]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[4]  R. Kleinman,et al.  An inverse transmission problem for the Helmholtz equation , 1987 .

[5]  Mohamed Masmoudi,et al.  Outils pour la conception optimale de forme , 1987 .

[6]  J. Sethian.  Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws , 1990 .

[7]  P. M. Berg,et al.  A modified gradient method for two-dimensional problems in tomography , 1992 .

[8]  J. Zolésio,et al.  Introduction to shape optimization : shape sensitivity analysis , 1992 .

[9]  Rainer Kress,et al.  On the numerical solution of the three-dimensional inverse obstacle scattering problem , 1992 .

[10]  J. Sethian,et al.  Crystal growth and dendritic solidification , 1992 .

[11]  Michel C. Delfour,et al.  Structure of shape derivatives for nonsmooth domains , 1992 .

[12]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[13]  Anton G. Tijhuis,et al.  Transient scattering by a lossy dielectric cylinder: marching-on-in-frequency approach , 1993 .

[14]  V. Caselles,et al.  A geometric model for active contours in image processing , 1993 .

[15]  Rainer Kress,et al.  Uniqueness in inverse obstacle scattering (acoustics) , 1993 .

[16]  A. Kirsch.  The domain derivative and two applications in inverse scattering theory , 1993 .

[17]  Peter Monk,et al.  The detection and monitoring of leukemia using electromagnetic waves: mathematical theory , 1994 .

[18]  F. Hettlich.  Frechet derivatives in inverse obstacle scattering , 1995 .

[19]  F. Santosa.  A Level-set Approach Inverse Problems Involving Obstacles , 1995 .

[20]  A. Charalambopoulos.  On the Frechet differentiability of boundary integral operators in the inverse elastic scattering problem , 1995 .

[21]  Baba C. Vemuri,et al.  Shape Modeling with Front Propagation: A Level Set Approach , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[22]  Dominique Lesselier,et al.  Mapping defects in a conductive half-space by simulated annealing with connectivity and size as constraints , 1996 .

[23]  D. Lesselier,et al.  Wavefield inversion of objects in stratified environments. From backpropagation schemes to full solutions. , 1996 .

[24]  K. Belkebir,et al.  Using multiple frequency information in the iterative solution of a two-dimensional nonlinear inverse problem , 1996 .

[25]  W. Rundell,et al.  Iterative methods for the reconstruction of an inverse potential problem , 1996 .

[26]  R. Kleinman,et al.  Modified gradient approach to inverse scattering for binary objects in stratified media , 1996 .

[27]  R. Potthast,et al.  Fréchet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain , 1996 .

[28]  Thomas S. Angell,et al.  A distributed source method for inverse acoustic scattering , 1997 .

[29]  Inverse 2D electromagnetic obstacle scattering for multiple scatterers using an iterative adaptation technique , 1997, IEEE Antennas and Propagation Society International Symposium 1997. Digest.

[30]  Dominique Lesselier,et al.  Shape retrieval of an obstacle immersed in shallow water from single-frequency farfields using a complete family method , 1997 .

[31]  L. Couchman,et al.  Scattering and inverse scattering of sound-hard obstacles via shape deformation , 1997 .

[32]  Deux methodes d'inversion pour la caracterisation electromagnetique ou acoustique d'objets enfouis : transformee de fourier-laplace inverse et deformation d'ensembles de niveaux , 1997 .

[33]  Guy Chavent,et al.  Inverse Problems of Wave Propagation and Diffraction , 1997 .

[34]  F. Santosa,et al.  A level-set approach for eddy current imaging of defects in a conductive half-space , 1997 .