On novel developments of controlled evolution of level sets in the field of inverse shape problems

Abstract

[1] Novel developments of the so-called controlled evolution of level sets [Ramananjaona et al., 2001b], which is devoted to shape identification of homogeneous scattering obstacles buried in a known space from time-harmonic wave field data, are considered herein. The emphasis is twofold: regularization of the geometry of the sought shapeenforced via a speed of motion of the level set in ( pseudo)time and space resulting from the minimization of a properly penalized objective functional; and improvement of the convergence of the scheme itself by a choice of a time step adapted to the evolution of the level set and to the sought decrease of the objective functional. Key elements of the theoretical analysis are given, and several numerical examples illustrate pros and cons.INDEX TERMS: 0629 Electromagnetics: Inverse scattering; 0644 Electromagnetics: Numerical methods; 0669 Electromagnetics: Scattering and diffraction; KEYWORDS: wavefield inversion, level-set, speed method, evolution, regularization Citation: Ramananjaona, C., M. Lambert, D. Lesselier, and J.-P. Zolésio, On novel developments of controlled evolution of level sets in the field of inverse shape problems,

Level Set Methods for Structural Inversion and Image Reconstruction

“…It de facto applies as soon as one has well-posedness of the direct and adjoint problems. For more details, refer to [29,30,78,79].…”

“…For both representations (69) and (72), minimizing the cost by a gradient method leads to curvature-driven flow equations, which is V.0/ D Än. This curvature-dependent velocity has been widely used to regularize the computation of motion of fronts via the level set method [48], as well in the field of image processing [70], and has been introduced also recently for regularizing inverse problems; see, e.g., [41,79,82]. Two popular concepts related to the above shape evolution are the MumfordShah and the total variation functionals, which are frequently employed in image segmentation applications.…”

A survey on level set methods for inverse problems and optimal design

“…This gradient evolution can be discretized in time using a forward Euler method to obtain the standard gradient method, which still decreases the objective functional if the time step is sufficiently small. This gradient-type approach was used by several authors to different shape optimization and reconstruction problems [47,48,49,79,112,113,114,144,145]. The approach works well in these cases, but is still limited to problems allowing the above representation formula of the shape sensitivity.…”

“…One of the first application of the level set method was electromagnetic scattering, where several problems have been solved using gradient-type methods by Litman et al [92], and later by Ramananjaona et al [112,113,114]. Dorn et al [47,49,48] applied level set based gradient methods to electromagnetic tomography in different situations.…”

Level Set Techniques For Structural Inversion In Medical Imaging

“…Examples are microwave medical imaging [5,7,11,[38][39][40][41][42][43], electrical impedance tomography [9,12,14,27,[44][45][46][47][48], and electrocardiography [49,50]. Further examples for results discussed in diffuse optical tomography can be found in [35,[51][52][53].…”