Topological Derivatives for Shape Reconstruction

Carpio, Ana; Rapún, María-Luisa

doi:10.1007/978-3-540-78547-7_5

Cited by 45 publications

(120 citation statements)

References 21 publications

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“…[8,11]), obtained when (2.7) is applied to the scattered field data at wavelengths (2π/k) that are smaller than the characteristic size of an obstacle, L o . In this vein, the ensuing analysis assumes a separation of scales in that 2π/k L o , where → 0 is the size of a vanishing perturbation in (2.4) while 2π/k, however small, is fixed in the limiting process.…”

Section: High-frequency Behaviour Of Topological Sensitivitymentioning

confidence: 99%

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Why the high-frequency inverse scattering by topological sensitivity may work

Guzina

Pourahmadian

2015

Proc. R. Soc. A.

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This study deciphers the topological sensitivity (TS) as a tool for the reconstruction and characterization of impenetrable anomalies in the high-frequency regime. It is assumed that the anomaly is simply connected and convex, and that the measurements of the scattered field are of the far-field type. In this setting, the formula for TS-which quantifies the perturbation of a cost functional due to a point-like impenetrable scatterer-is expressed as a pair of nested surface integrals: one taken over the boundary of a hidden obstacle, and the other over the measurement surface. Using multipole expansion, the latter integral is reduced to a set of antilinear forms featuring Green's function and its gradient. The remaining expression is distilled by evaluating the scattered field on the surface of an obstacle via Kirchhoff approximation, and pursuing an asymptotic expansion of the resulting Fourier integral. In this way, the TS is found to survive upon three asymptotic lynchpins, namely (i) the nearboundary approximation for sampling points close to the 'exposed' surface of an obstacle; (ii) uniform expansions synthesizing the diffraction catastrophes for sampling points near caustic surfaces, lines and points; and (iii) stationary phase approximation. Within the framework of catastrophe theory, it is shown that, in the case of the full source aperture, the TS is asymptotically dominated by the (explicit) near-boundary term-which explains the previously reported reconstruction capabilities of this class of indicator functionals. The analysis further shows that, when the (Dirichlet or Neumann) character of an anomaly is unknown beforehand, the latter can be effectively exposed by assuming point-like Dirichlet perturbation and considering the sign of the leading term inside the reconstruction.

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Section: High-frequency Behaviour Of Topological Sensitivitymentioning

confidence: 99%

“…half that of the illuminating wave. In this setting, the key question is that of the conditions under which the most pronounced negative values of TS are localized in a narrow region 'about the boundary' [11] of an obstacle.…”

Section: Imaging Ability Of the Topological Sensitivity Indicator Funmentioning

confidence: 99%

Why the high-frequency inverse scattering by topological sensitivity may work

Guzina

Pourahmadian

2015

Proc. R. Soc. A.

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“…Note also that the number of components does not have to be known from the beginning and can change from one iteration to the next (see figure 3). The method can also deal with components that separate or merge without problems (see [6] for an example in a scattering context).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Even if the admittivity is unknown, we may locate the regions where it undergoes notable changes by computing the topological derivative of the shape functional. We refer the reader to [6] and references therein for an introduction to topological derivative methods in inverse scattering and shape optimization. Once the first guess for £2; = U d -=1 £2;j is found, we may approximate the admittivity y\ optimizing J{Q, \ £2;, y,) with respect to y\ for &i Axed.…”

Section: Reconstruction Of Inclusionsmentioning

confidence: 99%

Hybrid topological derivative and gradient-based methods for electrical impedance tomography

Carpio

Rapún

2012

Inverse Problems

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We present a technique to reconstruct the electromagnetic properties of a medium or a set of objects buried inside it from boundary measurements when applying electric currents through a set of electrodes. The electromagnetic parameters may be recovered by means of a gradient method without a priori information on the background. The shape, location and size of objects, when present, are determined by a topological derivative-based iterative procedure. The combination of both strategies allows improved reconstructions of the objects and their properties, assuming a known background.

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“…Using iterative TD based methods, new objects may be created in the course of the iterations, existing contours may merge and holes inside existing objects may be detected, see [2,3]. Furthermore, even if the number of inclusions is known (assumption that in most practical applications is not realistic), TD-iterative methods are a powerful alternative to these classical methods, providing accurate reconstructions at a low computational cost, as extensively checked by the authors in different contexts (see [2][3][4][5][6] and references therein).…”

Section: Iterative Methods To Reconstruct Inclusions and Parametersmentioning

confidence: 99%

Domain and Parameter Reconstruction in Photothermal Imaging

Carpio

Rapún

2016

Mathematics in Industry

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In this work we address the inverse problem of reconstructing inclusions and their thermal parameters given temperature measurements at the accessible side of a material. We describe an iterative descent method that combines topological derivative computations to reconstruct the geometry of the defects with gradient iterations to approximate the material parameters. A numerical experiment showing the ability of the method to obtain reasonable reconstructions in a few iterations is presented.

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Topological Derivatives for Shape Reconstruction

Cited by 45 publications

References 21 publications

Why the high-frequency inverse scattering by topological sensitivity may work

Why the high-frequency inverse scattering by topological sensitivity may work

Hybrid topological derivative and gradient-based methods for electrical impedance tomography

Domain and Parameter Reconstruction in Photothermal Imaging

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