Romina Gaburro

Enhanced angular resolution from multiply scattered waves

Multiply scattered waves are often neglected in imaging methods, largely because of the inability of standard algorithms to deal with the associated non-linear models. This paper shows that by incorporating a known environment into the background model, we not only retain the benefits of imaging techniques based on linear models, but also obtain different views of the target scatterer. The net result is an enhanced angular resolution of the target to be imaged. We carry out our analysis in the context of high-frequency radar imaging, in which a steerable beam from a moving platform is used to produce an image of a region on the earths surface (the target scatterers being buildings, etc). We consider the case where the target we want to image is situated in the vicinity of an a priori known reflecting wall. This is one of the simplest possible multipathing environments for the scatterer, and in the case when the illuminating beam is narrow enough to isolate different scattering paths, we will show that the imaging process achieves enhanced angular resolution. Although we carry out our analysis here in the context of radar, our technique is general enough that it can be adapted to many imaging modalities, such as acoustics, ultrasound, elasticity, etc. The extension of the method to other more complicated environments is also possible.

Determining Conductivity with Special Anisotropy by Boundary Measurements

We prove results of uniqueness and stability at the boundary for the inverse problem of electrical impedance tomography in the presence of possibly anisotropic conductivities. We assume that the unknown conductivity has the form A=A(x,a(x)), where a(x) is an unknown scalar function and A(x,t) is a given matrix-valued function. We also deduce results of uniqueness in the interior among conductivities A obtained by piecewise analytic perturbations of the scalar term a.

The Local Calderòn Problem and the Determination at the Boundary of the Conductivity

We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body Ω ⊂ ℝ n when the so-called Dirichlet-to-Neumann map is locally given on a non empty portion Γ of the boundary ∂Ω. We extend results of uniqueness and stability at the boundary, obtained by the same authors in SIAM J. Math. Anal. 33:153–171, where the Dirichlet-to-Neumann map was given on all of ∂Ω instead. We also obtain a pointwise stability result at the boundary among the class of conductivities which are continuous at some point y ∈ Γ. Our arguments also apply when the local Neumann-to-Dirichlet map is available.

Recovering Riemannian metrics in monotone families from boundary data

We discuss the inverse problem of determining the anisotropic conductivity of a body described by a compact, orientable, Riemannian manifold M with boundary bdy M, when measurements of electric voltages and currents are taken on all of bdy M. Specifically we consider a one parameter family of conductivity tensors, extending results obtained in [AG] where the simpler Euclidean case is considered. Our problem is equivalent to the geometric one of determining a Riemannian metric in monotone one parameter family of metrics from its Dirichlet to Neumann map on bdy M.

Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities

We consider the stability issue of the inverse conductivity problem for a conformal class of anisotropic conductivities in terms of the local Dirichlet–Neumann map. We extend here the stability result obtained by Alessandrini and Vessella (Alessandrini G and Vessella S 2005 Lipschitz stability for the inverse conductivity problem Adv. Appl. Math. 35 207–241), where the authors considered the piecewise constant isotropic case.

Imaging from multiply scattered waves

We consider the problem of imaging in a region where ultrasonic waves are multiply scattered. A transducer emits ultrasonic pulses in tissue where they scatter from a heterogeneity (e.g. a tumor) in the region of interest (ROI). The reflected signals are recorded and used to produce an image of tissue. Many of the conventional imaging methods assume the wave has scattered just once (Born-approximation) from the heterogeneity before returning to the sensor to be recorded. In reality, waves can scatter several times before returning to the detector. The purpose of this paper is to show how this restriction (the Born approximation or weak, single-scattering approximation) can be partially removed by incorporating a-priori known environmental scatterers, such as a cavity wall or bones into the background velocity model in the context of acoustic medical imaging. We also show how the partial removal of the Born approximation assumption leads to an enhanced angular resolution of heterogeneities that are present. We will illustrate our method using a locally planar scatterer, which is one of the simplest possible environments for the scatterer.

Uniqueness for the electrostatic inverse boundary value problem with piecewise constant anisotropic conductivities

We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body

Microlocal analysis of synthetic aperture radar imaging in the presence of a vertical wall

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MICROLOCAL ANALYSIS OF SAR IMAGING OF A DYNAMIC REFLECTIVITY FUNCTION

when the so-called Neumann-to-Dirichlet map is locally given on a non empty curved portion

Lipschitz stability for the electrostatic inverse boundary value problem with piecewise linear conductivities

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