A variational algorithm for the detection of line segments

In this paper we propose an algorithm for the detection of edges in images that is based on topological asymptotic analysis. Motivated from the Mumford--Shah functional, we consider a variational functional that penalizes oscillations outside some approximate edge set, which we represent as the union of a finite number of thin strips, the width of which is an order of magnitude smaller than their length. In order to find a near optimal placement of these strips, we compute an asymptotic expansion of the functional with respect to the strip size. This expansion is then employed for defining a (topological) gradient descent like minimization method. As opposed to a recently proposed method by some of the authors, which uses coverings with balls, the usage of strips includes some directional information into the method, which can be used for obtaining finer edges and can also result in a reduction of computation times

Polarization tensors of planar domains as functions of the admittivity contrast

(Electric) polarization tensors describe part of the leading order term of asymptotic voltage perturbations caused by low volume fraction inhomogeneities of the electrical properties of a medium. They depend on the geometry of the support of the inhomogeneities and on their admittivity contrast. Corresponding asymptotic formulas are of particular interest in the design of reconstruction algorithms for determining the locations and the material properties of inhomogeneities inside a body from measurements of current flows and associated voltage potentials on the body's surface. In this work we consider the two-dimensional case only and provide an analytic representation of the polarization tensor in terms of spectral properties of the double layer integral operator associated with the support of simply connected conductivity inhomogeneities. Furthermore, we establish that an (infinitesimal) simply connected inhomogeneity has the shape of an ellipse, if and only if the polarization tensor is a rational function of the admittivity contrast with at most two poles whose residues satisfy a certain algebraic constraint. We also use the analytic representation to provide a proof of the so-called Hashin-Shtrikman bounds for polarization tensors; a similar approach has been taken previously by Golden and Papanicolaou and Kohn and Milton in the context of anisotropic composite materials

Maximizing the electromagnetic chirality of thin dielectric tubes

Any time-harmonic electromagnetic wave can be uniquely decomposed into a left and a right circularly polarized component. The concept of electromagnetic chirality (em-chirality) describes differences in the interaction of these two components with a scattering object or medium. Such differences can be quantified by means of em-chirality measures. These measures attain their minimal value zero for em-achiral objects or media that interact essentially in the same way with left and right circularly polarized waves. Scattering objects or media with positive em-chirality measure interact qualitatively different with left and right circularly polarized waves, and maximally em-chiral scattering objects or media would not interact with fields of either positive or negative helicity at all. This paper examines a shape optimization problem, where the goal is to determine thin tubular structures consisting of dielectric isotropic materials that exhibit large measures of em-chirality at a given frequency. We develop a gradient based optimization scheme that uses an asymptotic representation formula for scattered waves due to thin tubular scattering objects. Numerical examples suggest that thin helical structures are at least locally optimal among this class of scattering objects

Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities

The main result of this paper establishes optimal pointwise bounds for the polarization tensors that appear in representation formulae for the voltage perturbations caused by low volume fraction inhomogeneities. Furthermore it is demonstrated how these pointwise bounds may be used to derive a particularly simple version of the representation formulae in the case of thin inhomogeneities. Copyright 2006 - IOS Press and the authors. All rights reserved

Asymptotic Expansions for Higher Order Elliptic Equations with an Application to Quantitative Photoacoustic Tomography

In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of piecewise smooth functions. This algorithm can be used for edge detection in imaging, topological optimization, and for inverse problems, such as Quantitative Photoacoustic Tomography, for which we demonstrate the effectiveness of our asymptotic expansion method numerically