arXiv subjects

Gaik Ambartsoumian

Publications and source records attributed to Gaik Ambartsoumian.

At least 19 recordsLinked to original sources

arXiv2026

A simple range characterization for spherical mean transform in odd dimensions and its applications

This article provides a novel and simple range description for the spherical mean transform of functions supported in the unit ball of an odd dimensional Euclidean space. The new description comprises a set of symmetry relations between the values of certain differential operators acting on the coefficients of the spherical harmonics expansion of the function in the range of the transform. As a central part of the proof of our main result, we derive a remarkable cross product identity for the spherical Bessel functions of the first and second kind, which may be of independent interest in the theory of special functions. Finally, as one application of the range characterization, we construct an explicit counterexample proving that unique continuation type results cannot hold for the spherical mean transform in odd dimensional spaces.

math.CA↗

arXiv2026

Microlocal analysis of Radon transforms over quadric surfaces

We study the microlocal properties of generalized Radon transforms over a family of quadric hypersurfaces whose centers lie on an orientable hypersurface $S$. The quadric surfaces we consider are level sets of the quadratic form associated to a symmetric, invertible matrix $A$, with real entries. We study the singularities of the right and left projections of the canonical relation associated with these operators and show that they are determined by the signature of the matrix $A$ and the hypersurface $S$. If the matrix is positive/negative definite (i.e., the surface of integration is an ellipsoid) and $S$ is strictly convex, we prove that the singularities are folds. If the matrix is indefinite (i.e., the surface of integration is a hyperboloid-type quadric) and $S$ is either strictly convex or a cylinder, then cusp, fold, or blowdown singularities are present. We also study the case when the surface of integration is a paraboloid and show that the Bolker condition is satisfied.

math.CA↗

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

Symmetric star transforms and the algebraic geometry of their dual differential operators

The star transform is a generalized Radon transform mapping a function on $\mathbb{R}^n$ to the function whose value at a point is the integral along a union of rays emanating from the point in a fixed set of directions, called branch vectors. We show that the injectivity and inversion properties of the star transform are connected to its dual differential operator, an object introduced in this paper. We prove that if the set of branch vectors forms a symmetric shape with respect to the action of a finite rotation group $G$, then the symbol of its dual differential operator belongs to the ring of $G$-invariant polynomials. Furthermore, we show that star transforms with degenerate symmetry correspond to linear subspaces contained in the zero set of certain elementary symmetric polynomials, and we investigate the associated real algebraic Fano varieties. In particular, non-invertible star transforms in dimension 2 correspond to certain real lines on the Cayley nodal cubic surface.

math.AG↗

arXiv2025

A simple range characterization for spherical mean transform in even dimensions

The paper presents a new and simple range characterization for the spherical mean transform of functions supported in the unit ball in even dimensions. It complements the previous work of the same authors, where they solved an analogous problem in odd dimensions. The range description in even dimensions consists of symmetry relations, using a special kind of elliptic integrals involving the coefficients of the spherical harmonics expansion of the function in the range of the transform. The article also introduces a pair of original identities involving normalized Bessel functions of the first and the second kind. The first result is an integral cross-product identity for Bessel functions of integer order, complementing a similar relation for Bessel functions of half-integer order obtained in the aforementioned work of the same authors. The second result is a new Nicholson-type identity. Both of these relations can be considered as important standalone results in the theory of special functions. Finally, as part of the proof of one of the theorems, the authors derive an interesting equality involving elliptic integrals, which may be of independent interest.

math.CA↗

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

On the null space of the backprojection operator and Rubin's conjecture for the spherical mean transform

The spherical mean transform associates to a function $f$ its integral averages over all spheres. We consider the spherical mean transform for functions supported in the unit ball $\mathbb{B}$ in $\mathbb{R}^n$ for odd $n$, with the centers of integration spheres restricted to the unit sphere $\mathbb{S}^{n-1}$. In this setup, Rubin employed properties of Erdélyi-Kober fractional integrals and analytic continuation to re-derive the explicit inversion formulas proved earlier by Finch, Patch, and Rakesh using wave equation techniques. As part of his work, Rubin stated a conjecture relating spherical mean transform, its associated backprojection operator and the Riesz potential. Furthermore, he pointed to the necessity of a detailed analysis of injectivity of the backprojection operator as a crucial step toward the resolution of his conjecture. This article addresses both questions posed by Rubin by providing a characterization of the null space of the backprojection operator, and disproving the conjecture through the construction of an explicit counterexample. Crucial to the proofs is the range characterization for the spherical mean transform in odd dimensions derived recently by the authors.

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.

math.CA↗

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↗

arXiv2021

Inversion and Symmetries of the Star Transform

The star transform is a generalized Radon transform mapping a function of two variables to its integrals along "star-shaped" trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star transform. We describe the necessary and sufficient conditions for invertibility of the star transform, introduce a new inversion formula and discuss its stability properties. As an unexpected bonus of our approach, we prove a conjecture from algebraic geometry about the zero sets of elementary symmetric polynomials.

math-ph↗

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↗

arXiv2019

The V-line transform with some generalizations and cone differentiation

The paper studies various properties of the V-line transform (VLT) in the plane and conical Radon transform (CRT) in $\mathbb{R}^n$. VLT maps a function to a family of its integrals along trajectories made of two rays emanating from a common point. The CRT considered in this paper maps a function to a set of its integrals over surfaces of polyhedral cones. These types of operators appear in mathematical models of single scattering optical tomography, Compton camera imaging and other applications. We derive new explicit inversion formulae for VLT and CRT, as well as proving some previously known results using more intuitive geometric ideas. Using our inversion formula for VLT, we describe the range of that transformation when applied to a fairly broad class of functions and prove some support theorems. The efficiency of our method is demonstrated on several numerical examples. As an auxiliary result that plays a big role in this article, we derive a generalization of the Fundamental Theorem of Calculus, which we call Cone Differentiation Theorem.

math.CA↗

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

Numerical inversion of a broken ray transform arising in single scattering optical tomography

The article presents an efficient image reconstruction algorithm for single scattering optical tomography (SSOT) in circular geometry of data acquisition. This novel medical imaging modality uses photons of light that scatter once in the body to recover its interior features. The mathematical model of SSOT is based on the broken ray (or V-line Radon) transform (BRT), which puts into correspondence to an image function its integrals along V-shaped piecewise linear trajectories. The process of image reconstruction in SSOT requires inversion of that transform. We implement numerical inversion of a broken ray transform in a disc with partial radial data. Our method is based on a relation between the Fourier coefficients of the image function and those of its BRT recently discovered by Ambartsoumian and Moon. The numerical algorithm requires solution of ill-conditioned matrix problems, which is accomplished using a half-rank truncated singular value decomposition method. Several numerical computations validating the inversion formula are presented, which demonstrate the accuracy, speed and robustness of our method in the case of both noise-free and noisy data.

math.NA↗

arXiv2013

Exact inversion of the conical Radon transform with a fixed opening angle

We study a new class of Radon transforms defined on circular cones called the conical Radon transform. In $\mathbb{R}^3$ it maps a function to its surface integrals over circular cones, and in $\mathbb{R}^2$ it maps a function to its integrals along two rays with a common vertex. Such transforms appear in various mathematical models arising in medical imaging, nuclear industry and homeland security. This paper contains new results about inversion of conical Radon transform with fixed opening angle and vertical central axis in $\mathbb{R}^2$ and $\mathbb{R}^3$. New simple explicit inversion formulae are presented in these cases. Numerical simulations were performed to demonstrate the efficiency of the suggested algorithm in 2D.

math.NA↗

arXiv2013

Exterior/interior problem for the circular means transform with applications to intravascular imaging

Exterior inverse problem for the circular means transform (CMT) arises in the intravascular photoacoustic imaging (IVPA), in the intravascular ultrasound imaging (IVUS), as well as in radar and sonar. The reduction of the IPVA to the CMT is quite straightforward. As shown in the paper, in IVUS the circular means can be recovered from measurements by solving a certain Volterra integral equation. Thus, a tomography reconstruction in both modalities requires solving the exterior problem for the CMT. Numerical solution of this problem usually is not attempted due to the presence of "invisible" wavefronts, which results in severe instability of the reconstruction. The novel inversion algorithm proposed in this paper yields a stable partial reconstruction: it reproduces the "visible" part of the image and blurs the "invisible" part. If the image contains little or no invisible wavefronts (as frequently happens in the IVPA and IVUS) the reconstruction is quantitatively accurate. The presented numerical simulations demonstrate the feasibility of tomography-like reconstruction in these modalities.

math.AP↗

arXiv2011

Microlocal analysis of an ultrasound transform with circular source and receiver trajectories

In this article, we consider a generalized Radon transform that comes up in ultrasound reflection tomography. In our model, the ultrasound emitter and receiver move at a constant distance apart along a circle. We analyze the microlocal properties of the transform $R$ that arises from this model. As a consequence, we show that for distributions with support sufficiently inside the circle, $R^*R$ is an elliptic pseudodifferential operator of order $-1$ and hence all the singularities of such distributions can be recovered.

math.AP↗