arXiv subjects
Publications and source records attributed to Linh V. Nguyen.
In this article, we consider the limited angle problem in computed X-ray tomography. A common practice is to use the filtered back projection with the limited data. However, this practice has been shown to introduce the artifacts along some straight lines. In this work, we characterize the strength of these artifacts.
This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the {\bf time} regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latters are frequency dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well-controlled. This idea is originally due to Nguyen and Vogelius in \cite{NguyenVogelius2} for the cloaking context. In this paper, we present the analysis to the GIBCs of orders 0 and 1. To implement the ideas in \cite{NguyenVogelius2}, we revise and extend the work of Haddar, Joly, and Nguyen in \cite{HJNg1}, where the GIBCs were investigated for a fixed frequency in three dimensions. Even though we heavily follow the strategy in \cite{NguyenVogelius2}, our analysis on the stability contains new ingredients and ideas. First, instead of considering the difference between solutions of the exact model and the approximate model, we consider the difference between their derivatives in time. This simple idea helps us to avoid the machinery used in \cite{NguyenVogelius2} concerning the integrability with respect to frequency in the low frequency regime. Second, in the high frequency regime, the Morawetz multiplier technique used in \cite{NguyenVogelius2} does not fit directly in our setting. Our proof makes use of a result by Hörmander in \cite{Hor}. Another important part of the analysis in this paper is the well-posedness in the time domain for the approximate problems imposed with GIBCs on the boundary of the obstacle, which are non-local in time.
Let $\mR$ be the restriction of the spherical Radon transform to the set of spheres centered on a hypersurface $\mS$. We study the inversion of $\mR$ by a closed-form formula. We approach the problem by studying an oscillatory integral, which depends on the observation surface $\mS$ as a parameter. We then derive various microlocal analytic properties of the associated closed-form inversion formula.
We study the spherical mean transform on $\rN^n$. The transform is characterized by the Euler-Poisson-Darboux equation. By looking at the spherical harmonic expansions, we obtain a system of 1+1-dimension hyperbolic equations, which provide a good machinery to attack problems of spherical mean transform. As showcases, we discuss two known problems. The first one is a local uniqueness problem investigated by M. Agranovsky and P. Kuchment, [{\em Memoirs on Differential Equations and Mathematical Physics}, 52:1--16, 2011]. We present a simple proof which works even under a weaker condition. The second problem is to characterize the kernel of spherical mean transform on annular regions, which was studied by C. Epstein and B. Kleiner [{\em Comm. Pure Appl. Math.}, 46(3):441--451, 1993]. We present a short proof that simultaneously obtains the necessity and sufficiency for the characterization. As a consequence, we derive a reconstruction procedure for the transform with additional interior (or exterior) information. We also discuss how the approach works for the hyperbolic and spherical spaces.
We describe the range of a restricted spherical mean transform, which sends a function supported inside a closed ball in a hyperbolic space to its mean values on the geodesics spheres centered at the boundary of the ball. The description resembles that of the same transform on the Euclidean spaces obtained by Mark Agranovsky, David Finch, and Peter Kuchment [Inverse Problems and Imaging, 3(3):373--382, 2009] and Mark Agranovsky and Linh V. Nguyen [J. Anal. Math., 112:351--367, 2010]. We also derive a similar characterization for the corresponding transform on the two dimensional spherical space.
We consider the problem of thermoacoustic tomography (TAT), in which one needs to reconstruct the initial value of a solution of the wave equation from its value on an observation surface. We show that if some geometric rays for the equation do not intersect the observation surface, the reconstruction in TAT is not Hölder stable.
The transform under study is defined by integration of functions over spheres centered on a sphere. Such transform is of interest due to its applications in analysis and (thermoacoustic) tomography. The range of this transform has been described recently. Besides natural smoothness and support conditions, two type of conditions were involved: orthogonality condition and, in even dimension, moment condition. However, the moment condition was later derived from the other ones, which implies that orthogonality condition completely characterizes the range. We present a direct proof of this fact by proving existence of global extension for solutions of a certain boundary value problem for Darboux equation, associated with the spherical means. This extendibility phenomenon seems to be of independent interest.
We present a family of closed form inversion formulas in thermoacoustic tomography in the case of a constant sound speed. The formulas are presented in both time-domain and frequency-domain versions. As special cases, they imply most of the previously known filtered backprojection type formulas.