arXiv subjects

Rohit Kumar Mishra

Publications and source records attributed to Rohit Kumar Mishra.

At least 19 recordsLinked to original sources

arXiv2026

Gaussian beam Radon transform for tensor fields in $\mathbb{R}^2$

In this article, we introduce and study a set of generalized Gaussian beam Radon transforms (GbRt) acting on tensor fields in $\mathbb{R}^2$. The operators considered include longitudinal, transverse, mixed GbRts, along with their integral moments. These operators extend the corresponding notions of the classical generalized Radon transforms for tensor fields. We establish reconstruction results for vector and symmetric 2-tensor fields using appropriate combinations of the defined transforms. This work extends a recent study on the recovery of scalar functions from their GbRt to the recovery of vector and tensor fields from analogously defined generalized GbRts.

math.CA↗

arXiv2026

An Inverse problem for a fourth order nonlinear Schr\"odinger equation (NLS)

We study an inverse problem for the time-dependent nonlinear fourth-order Schr\"odinger equation on both compact Euclidean domains and compact Riemannian manifolds, (say) $M$. This model arises in nonlinear fiber optics and the theory of optical solitons in gyrotropic media. Our main objective is the identification of unknown coefficients from the associated source-to-solution map, which assigns to each source term $f$, supported in $(0, T)\times \Gamma$, the corresponding solution $u$ restricted to the same set, where $\Gamma \subset M$ is a neighborhood of $\partial M$. We prove that the zeroth-order term, the second-order coefficient, and the nonlinear coefficient are uniquely determined by this map. Moreover, the recovery of the symmetric second-order tensor reduces to the inversion of a divergent beam transform.

math.AP↗

arXiv2026

Radon Transform over Tensor Fields: Injectivity, Range, and Unique Continuation Principle

A central objective in inverse problems arising in integral geometry is to understand the kernel characterization, inversion formulas, stability estimates, range characterization, and unique continuation properties of integral transforms. In this paper, we study all these aspects for Radon transforms acting on symmetric $m$-tensor fields in $\mathbb{R}^n$. Our results show that these transforms admit a coherent analytic structure, extending several key features of the classical Radon transform and tensor ray transforms to a broader geometric setting.

math.AP↗

arXiv2026

A generalized Helmholtz-type decomposition of symmetric tensor fields and applications to ray transforms

We study a solenoidal-potential type decomposition of a symmetric $m$-tensor field in $\Rb^2$, and its implications to injectivity questions for the momentum and elastic ray transforms. For symmetric tensor fields, a general decomposition with a restriction on the dimension and order of the decomposition was proved in [16]. We extend the result to dimension $2$ under a mean-zero assumption. We use the decomposition in $2$ dimensions to prove the injectivity of the momentum and elastic ray transforms. We also prove a connection between the two integral transforms for $2$-tensors. Later, we use our decomposition to prove the injectivity of integral transforms, including longitudinal and elastic ray transforms, without any mean-zero assumption on tensor fields. Next, we explore connections between different integral transforms and use these to relate their corresponding properties.

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arXiv2025

Reconstruction of scalar functions and vector fields from weighted V-line transforms with swinging branches

Weighted V-line transforms map a symmetric tensor field of order $m\ge0$ to a linear combination of certain integrals of those fields along two rays emanating from the same vertex. A significant focus of current research in integral geometry centers on the inversion of V-line transforms in formally determined setups. Of particular interest are the restrictions of these operators in which the vertices of integration trajectories can be anywhere inside the support of the field, while the directions of the pair of rays, often called branches of the V-line, are determined by the vertex location. Such transforms have been thoroughly investigated when the branch directions are either constant or radial. In addition to that, in most of the prior research on this subject, it was assumed that the weights of integration along each branch are the same. In this paper we analyze the transforms defined on scalar functions and vector fields, satisfying a much weaker assumption on the branch directions. The weights restriction is also lifted in all but one setup. Consequently, we extend multiple previously known results on the kernel description, injectivity, and inversion of the transforms with simplifying assumptions and prove pertinent statements for more general setups not studied before.

math.CA↗

arXiv2025

The head-wave transform

We introduce and study a new integral ray transform called the head-wave transform. The head-wave transform integrates a function along a piecewise linear (in general geodesic) path consisting of three parts. The geometry of such paths corresponds to ray paths of head-waves propagating in a medium with sharp changes in sound speed. The middle part of the ray paths corresponds to gliding along the so-called gliding surface. As our main results, we prove inversion formulas and null space characterizations under multiple different sets of assumptions on the geometry of the gliding surface and the integrand function.

math.CA↗

arXiv2025

Inversion of generalized Radon transform over symmetric $m$-tensor fields in $\mathbb{R}^n$

In this work, we study a set of generalized Radon transforms over symmetric $m$-tensor fields in $\mathbb{R}^n$. The longitudinal/transversal Radon transform and corresponding weighted integral transforms for symmetric $m$-tensor field are introduced. We give the kernel descriptions for the longitudinal and transversal Radon transform. Further, we also prove that a symmetric $m$-tensor field can be recovered uniquely from certain combinations of these integral transforms of the unknown tensor field. This generalizes a recent study done for the recovery of vector fields from its weighted Radon transform data to recovery of a symmetric $m$-tensor field from analogously defined weighted Radon transforms.

math.AP↗

arXiv2024

Tensor tomography using V-line transforms with vertices restricted to a circle

In this article, we study the problem of recovering symmetric $m$-tensor fields (including vector fields) supported in a unit disk $\mathbb{D}$ from a set of generalized V-line transforms, namely longitudinal, transverse, and mixed V-line transforms, and their integral moments. We work in a circular geometric setup, where the V-lines have vertices on a circle, and the axis of symmetry is orthogonal to the circle. We present two approaches to recover a symmetric $m$-tensor field from the combination of longitudinal, transverse, and mixed V-line transforms. With the help of these inversion results, we are able to give an explicit kernel description for these transforms. We also derive inversion algorithms to reconstruct a symmetric $m$-tensor field from its first $(m+1)$ moment longitudinal/transverse V-line transforms.

math.NA↗

arXiv2024

V-line tensor tomography: numerical results

This article presents the numerical verification and validation of several inversion algorithms for V-line transforms (VLTs) acting on symmetric 2-tensor fields in the plane. The analysis of these transforms and the theoretical foundation of their inversion methods were studied in a recent work [G. Ambartsoumian, R. K. Mishra, and I. Zamindar, Inverse Problems, 40 (2024), 035003]. We demonstrate the efficient recovery of an unknown symmetric 2-tensor field from various combinations of the longitudinal, transverse, and mixed VLTs, their corresponding first moments, and the star VLT. The paper examines the performance of the proposed algorithms in different settings and illustrates the results with numerical simulations on smooth and non-smooth phantoms.

math.NA↗

arXiv2024

Inverse problem for a time-dependent Convection-diffusion equation in admissible geometries

We consider a partial data inverse problem for a time-dependent convection-diffusion equation on an admissible manifold. We prove that the time-dependent convection term and time-dependent density can be recovered uniquely modulo a known gauge invariance. There have been several works on inverse problems related to the steady state convection-diffusion operator in Euclidean as well as in Riemannian geometry settings; however, inverse problems related to time-dependent convection-diffusion equation on a manifold are not studied in the prior works, which is the main aim of this paper. In fact, to the best of our knowledge, the problem studied here is the first work related to a partial data inverse problem for recovering both first and zeroth-order time-dependent perturbations of evolution equations in the Riemannian geometry setting.

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arXiv2024

Inversion of generalized V-line transforms of vector fields in $\mathbb{R}^2$

This article studies the inverse problem of recovering a vector field supported in $\mathbb{D}_R$, the disk of radius $R$ centered at the origin, through a set of generalized broken ray/V-line transforms, namely longitudinal and transverse V-line transforms. Geometrically, we work with broken lines that start from the boundary of a disk and break at a fixed angle after traveling a distance along the diameter. We derive two inversion algorithms to recover a vector field in $\mathbb{R}^2$ from the knowledge of its longitudinal and transverse V-line transforms over two different subsets of aforementioned broken lines in $\mathbb{R}^2$.

math.CA↗

arXiv2024

Inversion formula, Unique continuation property, and range characterization of the mixed ray transform in $\mathbb{R}^2$

In this article, we study various aspects of the mixed ray transform of $(k + \ell)$-tensor fields that are symmetric in its first $k$ and last $\ell$ indices. As a first result, we derive an inversion algorithm to recover the solenoidal part of the unknown tensor field using the normal operator of the mixed ray transform. Next, we establish a set of unique continuation results. In addition to these, we discuss the range characterization of the mixed ray transform as the final result.

math.AP↗

arXiv2024

Inversion of a restricted transverse ray transform with sources on a curve

In this paper, a restricted transverse ray transform acting on vector and symmetric $m$-tensor fields is studied. We developed inversion algorithms using restricted transverse ray transform data to recover symmetric $m$-tensor fields in $\mathbb{R}^3$ and vector fields in $\mathbb{R}^n$. We restrict the transverse ray transform to all lines going through a fixed curve $γ$ that satisfies the Kirillov-Tuy condition. We show that the known restricted data can be used to reconstruct a specific weighted Radon transform of the unknown vector/tensor field's components, which we then use to explicitly recover the unknown field.

math.CA↗

arXiv2024

V-line 2-tensor tomography in the plane

In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in $\mathbb{R}^2$. The operators of interest include the longitudinal, transverse, and mixed VLTs, their integral moments, and the star transform. With the exception of the star transform, all these operators are natural generalizations to the broken-ray trajectories of the corresponding well studied concepts defined for straight-line paths of integration. We characterize the kernels of the VLTs and derive exact formulas for reconstruction of tensor fields from various combinations of these transforms. The star transform on tensor fields is an extension of the corresponding concepts that have been previously studied on vector fields and scalar fields (functions). We describe all injective configurations of the star transform on symmetric 2-tensor fields and derive an exact, closed-form inversion formula for that operator.

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arXiv2023

Numerical implementation of generalized V-line transforms on 2D vector fields and their inversions

The paper discusses numerical implementations of various inversion schemes for generalized V-line transforms on vector fields introduced in [6]. It demonstrates the possibility of efficient recovery of an unknown vector field from five different types of data sets, with and without noise. We examine the performance of the proposed algorithms in a variety of setups, and illustrate our results with numerical simulations on different phantoms.

math.NA↗

arXiv2022

The $C^\infty$-isomorphism property for a class of singularly-weighted X-ray transforms

We study a one-parameter family of self-adjoint normal operators for the X-ray transform on the closed Euclidean disk ${\mathbb D}$, obtained by considering specific singularly weighted $L^2$ topologies. We first recover the well-known Singular Value Decompositions in terms of orthogonal disk (or generalized Zernike) polynomials, then prove that each such realization is an isomorphism of $C^\infty({\mathbb D})$. As corollaries: we give some range characterizations; we show how such choices of normal operators can be expressed as functions of two distinguished differential operators. We also show that the isomorphism property also holds on a class of constant-curvature, circularly symmetric simple surfaces. These results allow to design functional contexts where normal operators built out of the X-ray transform are provably invertible, in Fréchet and Hilbert spaces encoding specific boundary behavior.

math.AP↗

arXiv2022

The generalized Saint Venant operator and integral moment transforms

In this article, we work with a generalized Saint Venant operator introduced by Vladimir Sharafutdinov to describe the kernel of the integral moment transforms over symmetric m-tensor fields in n-dimensional Euclidean space. We also provide an equivalence between the injectivity question for the integral moment transforms and the generalized Saint Venant operator over symmetric tensor fields of the Schwartz class.

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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.

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