Riccardo Aramini

The linear sampling method without sampling

We present a new implementation of the linear sampling method in which the set of discretized far-field equations for all sampling points is replaced by a single-functional equation formulated in a Hilbert space defined as a direct sum of L2 spaces. The squared norm of the regularized solution of such an equation is used as an indicator function and is analytically determined together with its Fourier transform. This provides some theoretical hints about the spatial resolution achievable by the method.

The linear sampling method and energy conservation

In this paper we explain the linear sampling method and its performances under various scattering conditions by means of an analysis of the far-field equation based on the principle of energy conservation. Specifically, we consider the conservation of energy along the flow strips of the Poynting vector associated with the scattered field whose far-field pattern is one of the two terms in the far-field equation. The behavior of these flow lines is numerically investigated and theoretically described. Appropriate assumptions on the flow lines, based on the numerical results, allow characterizing a set of approximate solutions of the far-field equation which can be used to visualize the boundary of the scatterer in the framework of the linear sampling method. In particular, under the same assumptions, we can show that Tikhonov regularized solutions belong to this set of approximate solutions for appropriate choices of the regularization parameter.

A fully no-sampling formulation of the linear sampling method for three-dimensional inverse electromagnetic scattering problems

We describe a very fast and automatic formulation of the linear sampling method for three-dimensional electromagnetic inverse scattering problems. This formulation is an extension of a no-sampling implementation recently proposed for two-dimensional configurations. In this 3D framework, regularization occurs independently not only of the sampling point but even of the polarization of the fundamental solution used as a known term. Furthermore, a very effective automatic procedure for the selection of the optimal surface describing the scatterer is introduced.

Bayesian multi-dipole modelling of a single topography in MEG by adaptive sequential Monte Carlo samplers

In this paper, we develop a novel Bayesian approach to the problem of estimating neural currents in the brain from a fixed distribution of magnetic field (called topography), measured by magnetoencephalography. Differently from recent studies that describe inversion techniques, such as spatio-temporal regularization/filtering, in which neural dynamics always plays a role, we face here a purely static inverse problem. Neural currents are modelled as an unknown number of current dipoles, whose state space is described in terms of a variable-dimension model. Within the resulting Bayesian framework, we set up a sequential Monte Carlo sampler to explore the posterior distribution. An adaptation technique is employed in order to effectively balance the computational cost and the quality of the sample approximation. Then, both the number and the parameters of the unknown current dipoles are simultaneously estimated. The performance of the method is assessed by means of synthetic data, generated by source configurations containing up to four dipoles. Eventually, we describe the results obtained by analysing data from a real experiment, involving somatosensory evoked fields, and compare them to those provided by three other methods.

A Visualization Method for Breast Cancer Detection Using Microwaves

This paper proposes a qualitative approach to the inverse scattering problem of microwave tomography for breast cancer detection. In a two-dimensional framework, the tumor inside the breast is regarded as an unknown scatterer placed inside an inhomogeneous and lossy background formed by skin and fat. We first present in detail the mathematical formulation of the method, which is based on the reciprocity gap functional: in particular, the physical and geometrical properties of the healthy breast are coded into the Greens function of the corresponding scattering equation, while any other object outside the array of receiving antennas can be neglected. Then, we propose a “no-sampling” implementation of the method, which allows a very fast visualization of the breast slices (the computational time is around 1s). Finally, we test the resulting algorithm against synthetic but realistic and noisy scattering data, by considering different plausible clinical situations.

Hybrid approach to the inverse scattering problem by using ant colony optimization and no-sampling linear sampling

A hybrid approach to microwave imaging has been presented by combining the computational effectiveness of the nLSM with the global optimization capabilities of ACO. The reported numerical results validate the ability of the method to satisfactorily solve diagnostic problems by exploiting strengths of the two ones.

The linear sampling method in a lossy background: an energy perspective

In this article we propose a physical approach to the linear sampling method (LSM) for a possibly inhomogeneous and lossy background, whereby the modified far-field equation at the basis of the method can be regarded as a constraint on the power fluxes carried by the Poynting vector associated with the scattered field. Under appropriate assumptions on the flow lines of this Poynting vector, the general theorem inspiring the LSM (and concerning the existence of approximate solutions to the modified far-field equation) can be reformulated in a different way, which is more appropriate to explain the performance of the method.

A new formulation of the linear sampling method: spatial resolution and post-processing

A new formulation of the linear sampling method is described, which requires the regularized solution of a single functional equation set in a direct sum of L2 spaces. This new approach presents the following notable advantages: it is computationally more effective than the traditional implementation, since time consuming samplings of the Tikhonov minimum problem and of the generalized discrepancy equation are avoided; it allows a quantitative estimate of the spatial resolution achievable by the method; it facilitates a post-processing procedure for the optimal selection of the scatterer profile by means of edge detection techniques. The formulation is described in a two-dimensional framework and in the case of obstacle scattering, although generalizations to three dimensions and penetrable inhomogeneities are straightforward.

The Role of Point Sources and Their Power Fluxes in the Linear Sampling Method

In this paper we investigate the linear sampling method by focusing on energy conservation inside a lossless background, in the case of a three-dimensional, impenetrable, and acoustic scattering set-up. We analyze, from both a numerical and a theoretical viewpoint, how average power fluxes are carried throughout the host medium by the flow tubes of radiating fields. As a result, the far-field equation, which is the core of the linear sampling method, can be regarded as a physical constraint linking the power flux of the scattered wave with that of the field radiated by a point source. Then, we show that this constraint, together with appropriate assumptions on the flow tubes of the scattered field, gives rise to a physical framework whereby some theoretical flaws of the linear sampling method can be overcome.

On the use of the Reciprocity Gap Functional in inverse scattering with near-field data: An application to mammography

Microwave tomography is a non-invasive approach to the early diagnosis of breast cancer. However the problem of visualizing tumors from diffracted microwaves is a difficult nonlinear ill-posed inverse scattering problem. We propose a qualitative approach to the solution of such a problem, whereby the shape and location of cancerous tissues can be detected by means of a combination of the Reciprocity Gap Functional method and the Linear Sampling method. We validate this approach to synthetic near-fields produced by a finite element method for boundary integral equations, where the breast is mimicked by the axial view of two nested cylinders, the external one representing the skin and the internal one representing the fat tissue.