Parametric Level Set Methods for Inverse Problems

In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, re-initialization and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the path for easier use of Newton and quasi-Newton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which used in the proposed manner provides flexibility in presenting a larger class of shapes with fewer terms. Also they provide a "narrow-banding" advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, X-ray computed tomography and diffuse optical tomography

Automated classification of single airborne particles from two-dimensional angle-resolved optical scattering (TAOS) patterns by non-linear filtering

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Localization of small obstacles from back-scattered data at limited incident angles with full-waveform inversion

International audienceWe investigate numerically the inverse problem of locating small circular obstacles in a homogeneous medium from multi-frequency back-scattered data limited to four angles of incidence. The main novelty of our paper is working with the position of the obstacles as parameter space in the frame work of full-waveform inversion (FWI) procedure. The computational cost of FWI is lowered by using a method based on single-layer potential. Reconstruction results are shown up to twenty-four obstacles, from initial guesses allowed to be far from the target. In experiments with six obstacles, we supplement the reconstruction with an analysis of the performance of the nonlinear conjugate gradient and quasi-Newton methods, in used with various line search algorithms