arXiv subjects
Publications and source records attributed to Alexandru Tamasan.
We characterize the range of the attenuated and non-attenuated $X$-ray transform of compactly supported vector fields in the plane. The characterization is in terms of a Hilbert transform associated with the $A$-analytic functions à la Bukhgeim. As an application we determine necessary and sufficient conditions for the attenuated Doppler and $X$-ray data to be mistaken for each other.
We prove uniqueness for minimizers of the weighted least gradient problem \[\inf \left\lbrace \int_Ω a|Du|: \ \ u\in BV(Ω), \ \ u|_{\partial Ω}=f \right\rbrace.\] The weight function $a$ is assumed to be continuous and it is allowed to vanish in certain subsets of $Ω$. Existence is assumed a priori. Our approach is motivated by the hybrid inverse problem of imaging electric conductivity from interior knowledge (obtainable by MRI) of the magnitude of one current density vector field.
We provide an example of an $L^1$ function on the circle, which cannot be the trace of a function of bounded variation of least gradient in the disk. This shows that in theorems on existence and uniqueness of solutions to the least gradient problem, proven by Sternberg, Williams and Ziemer and published in 1992-1993, the hypothesis that the boundary data is continuous cannot be removed.
We present new necessary and sufficient conditions for a function on $\partialΩ\times S^1$ to be in the range of the attenuated Radon transform of a sufficiently smooth function support in the convex set $\barΩ\subset\mathbb{R}^2$. The approach is based on an explicit Hilbert transform associated with traces of the boundary of A-analytic functions in the sense of Bukhgeim.
For Ωa C^{2}-smooth domain, and a positive bounded continuous map a \in C(Ω), we prove existence of a minimizer of the functional u \mapsto $\int_Ω a|Du| over the space BV(Ω) of functions of bounded variation with fixed trace f \in L^{1}(\partialΩ). The method of proof is constructive.
We consider the problem of recovering an isotropic conductivity outside some perfectly conducting or insulating inclusions from the interior measurement of the magnitude of one current density field $|J|$. We prove that the conductivity outside the inclusions, and the shape and position of the perfectly conducting and insulating inclusions are uniquely determined (except in an exceptional case) by the magnitude of the current generated by imposing a given boundary voltage. We have found an extension of the notion of admissibility to the case of possible presence of perfectly conducting and insulating inclusions. This also makes it possible to extend the results on uniqueness of the minimizers of the least gradient problem $F(u)=\int_Ωa|\nabla u|$ with $u|_{\partial Ω}=f$ to cases where $u$ has flat regions (is constant on open sets).
For anisotropic attenuating media, the albedo operator determines the scattering and the attenuation coefficients up to a gauge transformation. We show that such a determination is stable.
We characterize the non-uniqueness in the inverse problem for the stationary transport model, in which the absorption "a" and the scattering coefficient "k" are to be recovered from the albedo operator. We show that "gauge equivalent" pairs (a,k) yield the same albedo operator, and that we can recover uniquely the class of gauge equivalent pairs. We apply this result to show unique determination of the media when the absorption "a" depends on the line of travel through each point while scattering "k" obeys a symmetry property. Previously known results concerned directional independent absorption.
In this paper we reconstruct convection terms from boundary measurements.We reduce the Beals and Coifman inverse scattering scattering formalism from a first order system to a formalism for the $\bar{\partial}$ equation.
We study the inverse conductivity problem of how to reconstruct an isotropic electrical conductivity distribution $γ$ in an object from static electrical measurements on the boundary of the object. We give an exact reconstruction algorithm for the conductivity $γ\in C^{1+ε}(\ol \Om)$ in the plane domain $Ω$ from the associated Dirichlet to Neumann map on $\partial \Om.$ Hence we improve earlier reconstruction results. The method used relies on a well-known reduction to a first order system, for which the $\ol\partial$-method of inverse scattering theory can be applied.