The density function reconstruction of surface sources from a single Cauchy measurement (original) (raw)

The inverse problem of reconstructing a source term from boundary measurements, for the wave equation, is revisited. We propose a novel approach to recover the unknown source through measuring the wave fields after injecting small particles, enjoying a high contrast, into the medium. For this purpose, we first derive the asymptotic expansion of the wave field, based on the time-domain Lippmann-Schwinger equation. The dominant term in the asymptotic expansion is expressed as an infinite series in terms of the eigenvalues {λn}n∈N of the Newtonian operator (for the pure Laplacian). Such expansions are useful under a certain scale between the size of the particles and their contrast. Second, we observe that the relevant eigenvalues appearing in the expansion have non-zero averaged eigenfunctions. We prove that the family {sin( c1 √ λn t), cos( c1 √ λn t)}, for those relevant eigenvalues, with c1 as the contrast of the small particle, defines a Riesz basis (contrary to the family corresp...

We discuss recent results from [10] on sparse recovery for inverse potential problem with source term in divergence form. The notion of sparsity which is set forth is measure- theoretic, namely pure 1-unrectifiability of the support. The theory applies when a superset of the support is known to be slender, meaning it has measure zero and all connected components of its complement has infinite measure in ℝ3. We also discuss open issues in the non-slender case.

Estimation of the continuous current-source density in bulk tissue from a finite set of electrode measurements is a daunting task. Here we present a methodology which allows such a reconstruction by generalizing the one-dimensional inverse CSD method. The idea is to assume a particular plausible form of CSD within a class described by a number of parameters which can be estimated from available data, for example a set of cubic splines in 3D spanned on a fixed grid of the same size as the set of measurements. To avoid specificity of particular choice of reconstruction grid we add random jitter to the points positions and show that it leads to a correct reconstruction. We propose different ways of improving the quality of reconstruction which take into account the sources located outside the recording region through appropriate boundary treatment. The efficiency of the traditional CSD and variants of inverse CSD methods is compared using several fidelity measures on different test data to investigate when one of the methods is superior to the others. The methods are illustrated with reconstructions of CSD from potentials evoked by stimulation of a bunch of whiskers recorded in a slab of the rat forebrain on a grid of 4×5×7 positions.

Sensing electrodes arranged in or around a display can provide input function for interactive displays. Commercially this is interesting because the sensing electrodes and electronics can be made in the same manufacturing process as that of the display itself thus reducing cost. In engineering terms the electrodes measure capacitance changes resulting from the presence and movement of objects such as hands and fingers in front of the display. At the quasi static frequencies used (100kHz) the human body is conductive and the hands or fingers provide a screen between the capacitive electrodes. There is no need to touch the actual display and the overall system constitutes a touchless gesture input system. Determining the shape of the hand or fingers is a boundary condition reconstruction problem of finding the boundary of an earthed conductive object D from electrostatic measurements. This is the ill-posed problem of recovering the zero-surface of a solution to Laplace’s equation from...

A new method is developed in this work to solve the inverse electromagnetic scattering problem in inhomogeneous media using near-field measurements. The modeling is based on the formulation as contrast source integral equations of the full three-dimensional time-harmonic Maxwell-model. This inverse problem is ill-posed and nonlinear. The known idea of using equivalent sources splits inverse scattering into two subproblems: the inverse source problem, which is linear and ill-posed, and the inverse medium problem, which is more stable but nonlinear. We introduce the concept of generalized induced source to recast the system of intertwined vector equations, describing the electromagnetic inverse source problem, into decoupled scalar scattering problems. We utilize the method of the approximate inverse to recover the induced source for each experiment. We consider in three-dimensional setting the spherical scattering operator introduced by Abbdullah and Louis [Abd98] for 2-D acoustic wa...