Gaik Ambartsoumian | University of Texas at Arlington (original) (raw)
Papers by Gaik Ambartsoumian
SIAM journal on imaging sciences, Mar 7, 2024
Inverse Problems, Jan 16, 2024
arXiv (Cornell University), May 5, 2020
The star transform is a generalized Radon transform mapping a function of two variables to its in... more 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.
arXiv (Cornell University), Feb 23, 2017
In this article, we consider two bistatic cases arising in synthetic aperture radar imaging: when... more 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 F as a Fourier integral operator with fold/blowdown singularities. Next we analyze the normal operator F * F in both cases (where F * is the L 2-adjoint of F). When the transmitter and receiver move in the same direction, we prove that F * F belongs to a class of operators associated to two cleanly intersecting Lagrangians, I p,l (∆, C 1). When they move in opposite directions, F * F 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.
arXiv (Cornell University), Jun 22, 2023
In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-t... more In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in 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.
Contemporary mathematics, Jul 1, 2013
This article is dedicated to Sigurdur Helgason on the occasion of his eighty fifth birthday. We t... more This article is dedicated to Sigurdur Helgason on the occasion of his eighty fifth birthday. We thank him for creating so much beautiful mathematics and for being a friend and mentor to so many people in the field.
arXiv (Cornell University), May 15, 2023
Inverse Problems, Dec 30, 2019
Proceedings of SPIE, Feb 8, 2007
ABSTRACT
arXiv (Cornell University), Dec 14, 2016
The paper studies various properties of the V-line transform (VLT) in the plane and the conical R... more The paper studies various properties of the V-line transform (VLT) in the plane and the conical Radon transform (CRT) in R n. The 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 the VLT and the CRT, as well as proving some previously known results using more intuitive geometric ideas. Using our inversion formula for the 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 the Cone Differentiation Theorem.
Nucleation and Atmospheric Aerosols, 2020
Proceedings of SPIE, Feb 26, 2009
Reconstruction algorithms for interior and exterior spherical Radon transform-based ultrasound im... more Reconstruction algorithms for interior and exterior spherical Radon transform-based ultrasound imaging. [Proceedings of SPIE 7265, 72651I (2009)]. Ravi Shankar Vaidyanathan, Matthew A. Lewis, Gaik Ambartsoumian, Tuncay Aktosun. Abstract. ...
Computers & mathematics with applications, Aug 1, 2012
Several novel imaging modalities proposed during the last couple of years are based on a mathemat... more Several novel imaging modalities proposed during the last couple of years are based on a mathematical model, which uses the V-line Radon transform (VRT). This transform, sometimes called broken-ray Radon transform, integrates a function along V-shaped piecewise linear trajectories composed of two intervals in the plane with a common endpoint. Image reconstruction problems in these modalities require inversion of the VRT. While there are ample results about inversion of the regular Radon transform integrating along straight lines, very little is known for the case of the V-line Radon transform. In this paper, we derive an exact inversion formula for the VRT of functions supported in a disc of arbitrary radius. The formula uses a two-dimensional restriction of VRT data, namely the incident ray is normal to the boundary of the disc, and the breaking angle is fixed. Our method is based on the classical filtered back-projection inversion formula of the Radon transform, and has similar features in terms of stability, speed, and accuracy.
Inverse Problems, Mar 5, 2014
We study a new class of Radon transforms defined on circular cones called the conical Radon trans... more We study a new class of Radon transforms defined on circular cones called the conical Radon transform. In R 3 it maps a function to its surface integrals over circular cones, and in 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 R 2 and 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.
De Gruyter eBooks, Jun 17, 2019
Bulletin of Mathematical Biology, Jun 18, 2013
The parasite Trypanosoma cruzi, known for causing Chagas' disease, is spread via insect vectors f... more The parasite Trypanosoma cruzi, known for causing Chagas' disease, is spread via insect vectors from the triatomine family. T. cruzi is maintained in sylvatic vector-host transmission cycles in certain parts of the Americas. Communication between the cycles occurs mainly through movement (migration) of the insect vectors. In this study, we develop a cellular automaton (CA) model in order to study invasion of a hypothetical strain of T. cruzi through the region defined by the primary sylvatic cycles in northern Mexico and parts of the southeastern United States. The model given is a deterministic CA, which can be described as a large metapopulation model in the format of a dynamical system with 9,376 equations. The migration rates in the model, used as coupling parameters between cells in the CA, are estimated by summing up the proportion of vectors crossing patch boundaries (i.e., crossing from one cell to another). Specifically, we develop methods for estimating speed and direction of invasion as a function of vector migration rates, including preference for a particular direction of migration. We develop two methods for estimating invasion speed: via orthogonal local velocity components and by direct computation of magnitude and direction of an overall velocity vector given a front created by cells identified as being invaded by the epidemic. Results indicate that invasion speed is greatly affected by both the physical and the epidemiological landscapes through which the infection wave passes. A power-law fit suggests that invasion speed increases at slightly less than the square root of increases in migration rate. Keywords T. cruzi • Cellular automaton • Invasion speed • Vector migration 1 Introduction Chagas' disease, caused by the parasite Trypanosoma cruzi, is considered endemic in Central and South America. However, fewer than 10 autochthonous human cases
European Journal of Applied Mathematics, Sep 11, 2017
We study inversion of the spherical Radon transform with centers on a sphere (the data acquisitio... more 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 and demonstrate its accuracy and efficiency on several numerical examples.
Contemporary mathematics, 2015
The paper considers a class of elliptical and circular Radon transforms appearing in problems of ... more The paper considers a class of elliptical and circular Radon transforms appearing in problems of ultrasound imaging. These transforms put into correspondence to an unknown image function f in 2D its integrals Rf along a family of ellipses (or circles). From the imaging point of view, of particular interest is the circular geometry of data acquisition. Here the generalized Radon transform R integrates f along ellipses (circles) with their foci (centers) located on a fixed circle C. We prove that such transforms can be uniquely inverted from radially incomplete data to recover the image function in annular regions. Our results hold for cases when f is supported inside and/or outside of the data acquisition circle C.
Neurourology and Urodynamics, 2019
SIAM journal on imaging sciences, Mar 7, 2024
Inverse Problems, Jan 16, 2024
arXiv (Cornell University), May 5, 2020
The star transform is a generalized Radon transform mapping a function of two variables to its in... more 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.
arXiv (Cornell University), Feb 23, 2017
In this article, we consider two bistatic cases arising in synthetic aperture radar imaging: when... more 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 F as a Fourier integral operator with fold/blowdown singularities. Next we analyze the normal operator F * F in both cases (where F * is the L 2-adjoint of F). When the transmitter and receiver move in the same direction, we prove that F * F belongs to a class of operators associated to two cleanly intersecting Lagrangians, I p,l (∆, C 1). When they move in opposite directions, F * F 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.
arXiv (Cornell University), Jun 22, 2023
In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-t... more In this article, we introduce and study various V-line transforms (VLTs) defined on symmetric 2-tensor fields in 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.
Contemporary mathematics, Jul 1, 2013
This article is dedicated to Sigurdur Helgason on the occasion of his eighty fifth birthday. We t... more This article is dedicated to Sigurdur Helgason on the occasion of his eighty fifth birthday. We thank him for creating so much beautiful mathematics and for being a friend and mentor to so many people in the field.
arXiv (Cornell University), May 15, 2023
Inverse Problems, Dec 30, 2019
Proceedings of SPIE, Feb 8, 2007
ABSTRACT
arXiv (Cornell University), Dec 14, 2016
The paper studies various properties of the V-line transform (VLT) in the plane and the conical R... more The paper studies various properties of the V-line transform (VLT) in the plane and the conical Radon transform (CRT) in R n. The 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 the VLT and the CRT, as well as proving some previously known results using more intuitive geometric ideas. Using our inversion formula for the 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 the Cone Differentiation Theorem.
Nucleation and Atmospheric Aerosols, 2020
Proceedings of SPIE, Feb 26, 2009
Reconstruction algorithms for interior and exterior spherical Radon transform-based ultrasound im... more Reconstruction algorithms for interior and exterior spherical Radon transform-based ultrasound imaging. [Proceedings of SPIE 7265, 72651I (2009)]. Ravi Shankar Vaidyanathan, Matthew A. Lewis, Gaik Ambartsoumian, Tuncay Aktosun. Abstract. ...
Computers & mathematics with applications, Aug 1, 2012
Several novel imaging modalities proposed during the last couple of years are based on a mathemat... more Several novel imaging modalities proposed during the last couple of years are based on a mathematical model, which uses the V-line Radon transform (VRT). This transform, sometimes called broken-ray Radon transform, integrates a function along V-shaped piecewise linear trajectories composed of two intervals in the plane with a common endpoint. Image reconstruction problems in these modalities require inversion of the VRT. While there are ample results about inversion of the regular Radon transform integrating along straight lines, very little is known for the case of the V-line Radon transform. In this paper, we derive an exact inversion formula for the VRT of functions supported in a disc of arbitrary radius. The formula uses a two-dimensional restriction of VRT data, namely the incident ray is normal to the boundary of the disc, and the breaking angle is fixed. Our method is based on the classical filtered back-projection inversion formula of the Radon transform, and has similar features in terms of stability, speed, and accuracy.
Inverse Problems, Mar 5, 2014
We study a new class of Radon transforms defined on circular cones called the conical Radon trans... more We study a new class of Radon transforms defined on circular cones called the conical Radon transform. In R 3 it maps a function to its surface integrals over circular cones, and in 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 R 2 and 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.
De Gruyter eBooks, Jun 17, 2019
Bulletin of Mathematical Biology, Jun 18, 2013
The parasite Trypanosoma cruzi, known for causing Chagas' disease, is spread via insect vectors f... more The parasite Trypanosoma cruzi, known for causing Chagas' disease, is spread via insect vectors from the triatomine family. T. cruzi is maintained in sylvatic vector-host transmission cycles in certain parts of the Americas. Communication between the cycles occurs mainly through movement (migration) of the insect vectors. In this study, we develop a cellular automaton (CA) model in order to study invasion of a hypothetical strain of T. cruzi through the region defined by the primary sylvatic cycles in northern Mexico and parts of the southeastern United States. The model given is a deterministic CA, which can be described as a large metapopulation model in the format of a dynamical system with 9,376 equations. The migration rates in the model, used as coupling parameters between cells in the CA, are estimated by summing up the proportion of vectors crossing patch boundaries (i.e., crossing from one cell to another). Specifically, we develop methods for estimating speed and direction of invasion as a function of vector migration rates, including preference for a particular direction of migration. We develop two methods for estimating invasion speed: via orthogonal local velocity components and by direct computation of magnitude and direction of an overall velocity vector given a front created by cells identified as being invaded by the epidemic. Results indicate that invasion speed is greatly affected by both the physical and the epidemiological landscapes through which the infection wave passes. A power-law fit suggests that invasion speed increases at slightly less than the square root of increases in migration rate. Keywords T. cruzi • Cellular automaton • Invasion speed • Vector migration 1 Introduction Chagas' disease, caused by the parasite Trypanosoma cruzi, is considered endemic in Central and South America. However, fewer than 10 autochthonous human cases
European Journal of Applied Mathematics, Sep 11, 2017
We study inversion of the spherical Radon transform with centers on a sphere (the data acquisitio... more 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 and demonstrate its accuracy and efficiency on several numerical examples.
Contemporary mathematics, 2015
The paper considers a class of elliptical and circular Radon transforms appearing in problems of ... more The paper considers a class of elliptical and circular Radon transforms appearing in problems of ultrasound imaging. These transforms put into correspondence to an unknown image function f in 2D its integrals Rf along a family of ellipses (or circles). From the imaging point of view, of particular interest is the circular geometry of data acquisition. Here the generalized Radon transform R integrates f along ellipses (circles) with their foci (centers) located on a fixed circle C. We prove that such transforms can be uniquely inverted from radially incomplete data to recover the image function in annular regions. Our results hold for cases when f is supported inside and/or outside of the data acquisition circle C.
Neurourology and Urodynamics, 2019