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Rohit Kumar Mishra

Publications and source records attributed to Rohit Kumar Mishra.

28 records · Page 2Linked to original sources
arXiv2020

Generalized V-line transforms in 2D vector tomography

We study the inverse problem of recovering a vector field in $\mathbb{R}^2$ from a set of new generalized $V$-line transforms in three different ways. First, we introduce the longitudinal and transverse $V$-line transforms for vector fields in $\mathbb{R}^2$. We then give an explicit characterization of their respective kernels and show that they are complements of each other. We prove invertibility of each transform modulo their kernels and combine them to reconstruct explicitly the full vector field. In the second method, we combine the longitudinal and transverse V-line transforms with their corresponding first moment transforms and recover the full vector field from either pair. We show that the available data in each of these setups can be used to derive the signed V-line transform of both scalar component of the vector field, and use the known inversion of the latter. The final major result of this paper is the derivation of an exact closed form formula for reconstruction of the full vector field in $\mathbb{R}^2$ from its star transform with weights. We solve this problem by relating the star transform of the vector field to the ordinary Radon transform of the scalar components of the field.

math.CA↗
arXiv2020

Sparse reconstruction of log-conductivity in current density impedance tomography

A new non-linear optimization approach is proposed for the sparse reconstruction of log-conductivities in current density impedance imaging. This framework comprises of minimizing an objective functional involving a least squares fit of the interior electric field data corresponding to two boundary voltage measurements, where the conductivity and the electric potential are related through an elliptic PDE arising in electrical impedance tomography. Further, the objective functional consists of a $L^1$ regularization term that promotes sparsity patterns in the conductivity and a Perona-Malik anisotropic diffusion term that enhances the edges to facilitate high contrast and resolution. This framework is motivated by a similar recent approach to solve an inverse problem in acousto-electric tomography. Several numerical experiments and comparison with an existing method demonstrate the effectiveness of the proposed method for superior image reconstructions of a wide-variety of log-conductivity patterns.

math.OC↗
arXiv2020

A sparsity-based nonlinear reconstruction method for two-photon photoacoustic tomography

We present a new nonlinear optimization approach for the sparse reconstruction of single-photon absorption and two-photon absorption coefficients in photoacoustic tomography (PAT). This framework comprises of minimizing an objective functional involving a least squares fit of the interior pressure field data corresponding to two boundary source functions, where the absorption coefficients and the photon density are related through a semi-linear elliptic partial differential equation (PDE) arising in PAT. Further, the objective functional consists of an $L^1$ regularization term that promotes sparsity patterns in absorption coefficients. The motivation for this framework primarily comes from some recent works related to solving inverse problems in acousto-electric tomography and current density impedance tomography. We provide a new proof of existence and uniqueness of a solution to the semi-linear PDE. Further, a proximal method, involving a Picard solver for the semi-linear PDE and its adjoint, is used to solve the optimization problem. Several numerical experiments are presented to demonstrate the effectiveness of the proposed framework.

math.OC↗
arXiv2020

Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature

For a one-parameter family of simple metrics of constant curvature ($4κ$ for $κ\in (-1,1)$) on the unit disk $M$, we first make explicit the Pestov-Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also translates into moment conditions {\it à la} Helgason-Ludwig or Gel'fand-Graev. We then derive an explicit Singular Value Decomposition for the geodesic X-ray transform. Computations dictate a specific choice of weighted $L^2-L^2$ setting which is equivalent to the $L^2(M, dVol_κ)\to L^2(\partial_+ SM, dΣ^2)$ one for any $κ\in (-1,1)$.

math.AP↗
arXiv2020

Injectivity and range description of first $(k+1)$ integral moment transforms over $m$-tensor fields in $\mathbb{R}^n$

In this work, we prove a new decomposition result for rank $m$ symmetric tensor fields which generalizes the well known solenoidal and potential decomposition of tensor fields. This decomposition is then used to describe the kernel and to prove an injectivity result for first $(k+1)$ integral moment transforms of symmetric $m$-tensor fields in $\mathbb{R}^n$. Additionally, we also present a range characterization for first $(k+1)$ integral moment transforms in terms of the John's equation.

math.AP↗
arXiv2019

Full reconstruction of a vector field from restricted Doppler and first integral moment transforms in $\mathbb{R}^n$

We show that a vector field in $\mathbb{R}^n$ can be reconstructed uniquely from the knowledge of restricted Doppler and first integral moment transforms. The line complex we consider consists of all lines passing through a fixed curve $γ\subset \mathbb{R}^n$. The question of reconstruction of a symmetric $m$-tensor field from the knowledge of the first $m+1$ integral moments was posed by Sharafutdinov in his book (see pp. 78), "Integral geometry of tensor fields," Inverse and Ill-posed problems series, De Grutyer. In this work, we provide an answer to Sharafutdinov's question for the case of vector fields from restricted data comprising of the first $2$ integral moment transforms.

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

Support Theorems and an Injectivity Result for Integral Moments of a Symmetric m-Tensor Field

In this work, we show an injectivity result and support theorems for integral moments of a m-tensor field on a simple, real analytic, Riemannian manifold. Integral moments of m-tensor field were first introduced by Sharafutdinov. At first we generalize a Helgason type support theorem proven by Krishnan and Stefanov in "A support theorem for the geodesic ray transform of symmetric tensor fields", Inverse Problems and Imaging, 3(3):453-464,2009. We use this extended result along with the first m+1-integral moments of a m-tensor field to prove the aforementioned results.

math.DG↗