The Topological Asymptotic for the Helmholtz Equation

Samet, Bessem; Amstutz, Samuel; Masmoudi, Mohamed

doi:10.1137/s0363012902406801

Cited by 94 publications

(79 citation statements)

References 13 publications

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“…This has been tested in a wide range of physical settings, including acoustics, electromagnetism, elastodynamics, electrical impedance tomography, fluorescence optical tomography, and photothermal imaging [1,5,7,8,10,15,20].…”

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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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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“…The classification model that we propose is based on the topological gradient method [27]. In fact, to assign each pixel of the original image to one of the classes C i , 1 ≤ i ≤ n, it suffices to suppose first that all pixels are assigned to the same class, and then to find subsets of pixels that should be reassigned to the other classes.…”

Section: Topological Gradient For the Image Classificationmentioning

confidence: 99%

“…In order to apply the topological asymptotic theory, we have to verify the following hypothesis [6,27]. …”

Section: Image Restoration and Smoothing Using The Topological Gradiementioning

confidence: 99%

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A topological asymptotic analysis for the regularized grey-level image classification problem

2007

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Abstract. The aim of this article is to propose a new method for the grey-level image classification problem. We first present the classical variational approach without and with a regularization term in order to smooth the contours of the classified image. Then we present the general topological asymptotic analysis, and we finally introduce its application to the grey-level image classification problem.

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“…Our crack detection technique [3] is based on the topological gradient approach [17,20,19,25,27,29,30]. Similar ideas have been applied by the authors to image restoration and classification [6,21,5].…”

Section: Introductionmentioning

confidence: 99%

A one-shot inpainting algorithm based on the topological asymptotic analysis

Auroux¹,

Masmoudi²

2006

Mat. apl. comput.

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Abstract. The aim of this article is to propose a new method for the inpainting problem.Inpainting is the problem of filling-in holes in images. We consider in this article the crack localization problem, which can be solved using the Dirichlet to Neumann approach and the topological gradient. In a similar way, we can define a Dirichlet and a Neumann inpainting problem. We then define a cost function measuring the discrepancy between the two corresponding solutions. The minimization is done using the topological asymptotic analysis, and is performed in only one iteration. The optimal solution provides the best localization of the missing edges, and it is then easy to inpaint the holes.Mathematical subject classification: 68U10, 94A08, 34E05.

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The Topological Asymptotic for the Helmholtz Equation

Cited by 94 publications

References 13 publications

Domain and Parameter Reconstruction in Photothermal Imaging

Domain and Parameter Reconstruction in Photothermal Imaging

A topological asymptotic analysis for the regularized grey-level image classification problem

A one-shot inpainting algorithm based on the topological asymptotic analysis

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