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Venkateswaran P. Krishnan

Publications and source records attributed to Venkateswaran P. Krishnan.

30 records · Page 2Linked to original sources
arXiv2020

Microlocal inversion of a 3-dimensional restricted transverse ray transform of symmetric $m$-tensor fields

We study the problem of inverting a restricted transverse ray transform to recover a symmetric $m$-tensor field in $\mathbb{R}^3$ using microlocal analysis techniques. More precisely, we prove that a symmetric $m$-tensor field can be recovered up to a known singular term and a smoothing term if its transverse ray transform is known along all lines intersecting a fixed smooth curve satisfying the Kirillov-Tuy condition.

math.AP↗
arXiv2019

Momentum Ray Transforms, II: Range Characterization In the Schwartz space

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines of ${\R}^n$ with the weight $t^k$: $ (I^k\!f)(x,ξ)=\int_{-\infty}^\infty t^kłf(x+tξ),ξ^m\r\,dt. $ We give the range characterization for the operator $f\mapsto(I^0\!f,I^1\!f,\dots, I^m\!f)$ on the Schwartz space of rank $m$ smooth fast decaying tensor fields. In dimensions $n\ge3$, the range is characterized by certain differential equations of order $2(m+1)$ which generalize the classical John equations. In the two-dimensional case, the range is characterized by certain integral conditions which generalize the classical Gelfand -- Helgason -- Ludwig conditions.

math.AP↗
arXiv2019

An inverse problem for the relativistic Schrödinger equation with partial boundary data

We study the inverse problem of determining the vector and scalar potentials $\mathcal{A}(t,x)=\left(A_{0},A_{1},\cdots,A_{n}\right)$ and $q(t,x)$, respectively, in the relativistic Schrödinger equation \begin{equation*} \Big{(}\left(\partial_{t}+A_{0}(t,x)\right)^{2}-\sum_{j=1}^{n}\left(\partial_{j}+A_{j}(t,x)\right)^{2}+q(t,x)\Big{)}u(t,x)=0 \end{equation*} in the region $Q=(0,T)\timesΩ$, where $Ω$ is a $C^{2}$ bounded domain in $\mathbb{R}^{n}$ for $n\geq 3$ and $T>\mbox{diam}(Ω)$ from partial data on the boundary $\partial Q$. We prove the unique determination of these potentials modulo a natural gauge invariance for the vector field term.

math.AP↗
arXiv2018

On solenoidal-injective and injective ray transforms of tensor fields on surfaces

We first give a constructive answer to the attenuated tensor tomography problem on simple surfaces. We then use this result to propose two approaches to produce vector-valued integral transforms which are fully injective over tensor fields. The first approach is by construction of appropriate weights which vary along the geodesic flow, generalizing the moment transforms. The second one is by changing the pairing with the tensor field to generate a collection of transverse ray transforms.

math.DG↗
arXiv2018

Momentum ray transforms

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines with the weight $t^k$: $ (I^k\!f)(x,ξ)=\int_{-\infty}^\infty t^k\langle f(x+tξ),ξ^m\rangle\,dt. $ In particular, the ray transform $I=I^0$ was studied by several authors since it had many tomographic applications. We present an algorithm for recovering $f$ from the data $(I^0\!f,I^1\!f,\dots, I^m\!f)$. In the cases of $m=1$ and $m=2$, we derive the Reshetnyak formula that expresses $\|f\|_{H^s_t({\mathbb{R}}^n)}$ through some norm of $(I^0\!f,I^1\!f,\dots, I^m\!f)$. The $H^{s}_{t}$-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

math.AP↗
arXiv2018

A fully non-linear optimization approach to acousto-electric tomography

This paper considers the non-linear inverse problem of reconstructing an electric conductivity distribution from the interior power density in a bounded domain. Applications include the novel tomographic method known as acousto-electric tomography, in which the measurement setup in Electrical Impedance Tomography is modulated by ultrasonic waves thus giving rise to a method potentially having both high contrast and high resolution. We formulate the inverse problem as a regularized non-linear optimization problem, show the existence of a minimizer, and derive optimality conditions. We propose a non-linear conjugate gradient scheme for finding a minimizer based on the optimality conditions. All our numerical experiments are done in two-dimensions. The experiments reveal new insight into the non-linear effects in the reconstruction. One of the interesting features we observe is that, depending on the choice of regularization, there is a trade-off between high resolution and high contrast in the reconstructed images. Our proposed non-linear optimization framework can be generalized to other hybrid imaging modalities.

math.OC↗
arXiv2017

Image reconstruction from radially incomplete spherical Radon data

We study inversion of the spherical Radon transform with centers on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of radially incomplete data, we show that the spherical Radon transform can be uniquely inverted recovering the image function in spherical shells. Our result is valid when the support of the image function is inside the data acquisition sphere, outside that sphere, as well as on both sides of the sphere. Furthermore, in addition to the uniqueness result our method of proof provides reconstruction formulas for all those cases. We present a robust computational algorithm based on our inversion formula and demonstrate its accuracy and efficiency on several numerical examples.

math.CA↗
arXiv2017

Singular FIOs in SAR Imaging, II: Transmitter and Receiver at Different Speeds

In this article, we consider two bistatic cases arising in synthetic aperture radar imaging: when the transmitter and receiver are both moving with different speeds along a single line parallel to the ground in the same direction or in the opposite directions. In both cases, we classify the forward operator $\Fc$ as a Fourier integral operator with fold/blowdown singularities. Next we analyze the normal operator $\Fc^*\Fc$ in both cases (where $\Fc^{*}$ is the $L^{2}$-adjoint of $\Fc$). When the transmitter and receiver move in the same direction, we prove that $\Fc^*\Fc$ belongs to a class of operators associated to two cleanly intersecting Lagrangians, $I^{p,l} (Δ, C_1)$. When they move in opposite directions, $\Fc^*\Fc$ is a sum of such operators. In both cases artifacts appear and we show that they are, in general, as strong as the bona-fide part of the image. Moreover, we demonstrate that as soon as the source and receiver start to move in opposite directions, there is an interesting bifurcation in the type of artifact that appears in the image.

math.AP↗
arXiv2016

Inverse Diffusivity Problem via Homogenization Theory

Polarization tensor corresponding to near zero volume inhomogeneities was introduced in the pioneering work by Capdeboscq-Vogelius \cite{CV1,CV2}. A beautiful application of the polarization tensor to an inverse problem involving inhomogeneities was also given by them. In this article, we take an approach toward polarization tensor via homogenized tensor. Accordingly, we introduce polarization tensor corresponding to inhomogeneities with positive volume fraction.A relation between this tensor and the homogenized tensor is found. Next, we proceed to examine the sense in which this tensor is continuous as the volume fraction tends to zero. Our approach has its own advantages, as we will see. In particular, it provides another method to deduce optimal estimates on polarization tensors in any dimension from those on homogenized tensors, along with the information on underlying microstructures.

math.AP↗
arXiv2015

Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials

In this article, stability estimates are given for the determination of the zeroth-order bounded perturbations of the biharmonic operator when the boundary Neumann measurements are made on the whole boundary and on slightly more than half the boundary, respectively. For the case of measurements on the whole boundary, the stability estimates are of ln-type and for the case of measurements on slightly more than half of the boundary, we derive estimates that are of ln ln-type.

math.AP↗
arXiv2007

A generalization of inversion formulas of Pestov and Uhlmann

In this note, we give a generalization of the inversion formulas of Pestov-Uhlmann for the geodesic ray transform of functions and vector fields on simple 2-dimensional manifolds of constant curvature. The inversion formulas given here hold for 2-dimensional simple manifolds whose curvatures close to a constant.

math.DG↗