Leonid Kunyansky

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Chapters

• Agranovsky, M., Kuchment, P., & Kunyansky, L. (2017).

  On Reconstruction Formulas and Algorithms for the Thermoacoustic Tomography

  . In Photoacoustic Imaging and Spectroscopy. doi:10.1201/9781420059922-11
• Kuchment, P., & Kunyansky, L. (2015). Mathematics of photoacoustic and thermoacoustic tomography. In Handbook of Mathematical Methods in Imaging. Springer New York. doi:10.1007/978-1-4939-0790-8_51
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  This is the manuscript of the chapter for a planned Handbook of Mathematical Methods in Imaging that surveys the mathematical models, problems, and algorithms of the Thermoacoustic (TAT) and Photoacoustic (PAT) Tomography. TAT and PAT represent probably the most developed of the several novel ``hybrid'' methods of medical imaging. These new modalities combine different physical types of waves (electromagnetic and acoustic in case of TAT and PAT) in such a way that the resolution and contrast of the resulting method are much higher than those achievable using only acoustic or electromagnetic measurements.
• Kuchment, P., & Kunyansky, L. (2012).

  Mathematics of Photoacoustic and Thermoacoustic Tomography

  . In Mathematics of Photoacoustic and Thermoacoustic Tomography. doi:10.1007/springerreference_186900
• Agranovsky, M., Kuchment, P., & Kunyansky, L. (2009).

  On Reconstruction Formulas and Algorithms for the Thermoacoustic Tomography

  . In Photoacoustic Imaging and Spectroscopy. doi:10.1201/9781420059922.ch8

Journals/Publications

• Kunyansky, L., & Agranovsky, M. (2023). On the exactness of the universal backprojection formula for the spherical means Radon transform. Inverse Problems. doi:10.1088/1361-6420/acb2ee
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  Abstract The spherical means Radon transform f(x, r) is defined by the integral of a function f in ℝn over the sphere S(x, r) of radius r centered at a x, normalized by the area of the sphere. The problem of reconstructing f from the data f(x, r) where x belongs to a hypersurface Γ⊆ℝn and r ∈ (0, ∞) has important applications in modern imaging modalities, such as photo- and thermoacoustic tomography. When Γ coincides with the boundary ∂Ω of a bounded (convex) domain Ω⊆ℝn, a function supported within Ω can be uniquely recovered from its spherical means known on Γ. We are interested in explicit inversion formulas for such a reconstruction. If Γ=∂Ω, such formulas are only known for the case when Γ is an ellipsoid (or one of its partial cases). This gives rise to the natural question: can explicit inversion formulas be found for other closed hypersurfaces Γ? In this article we prove, for the so-called "universal backprojection inversion formulas", that their extension to non-ellipsoidal domains Ω is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.
• Kunyansky, L., McDugald, E., & Shearer, B. (2023).

  Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography

  . Inverse Problems. doi:10.1088/1361-6420/acd07a
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  Abstract Currently, theory of ray transforms of vector and tensor fields is well developed, but the Radon transforms of such fields have not been fully analyzed. We thus consider linearly weighted and unweighted longitudinal and transversal Radon transforms of vector fields. As usual, we use the standard Helmholtz decomposition of smooth and fast decreasing vector fields over the whole space. We show that such a decomposition produces potential and solenoidal components decreasing at infinity fast enough to guarantee the existence of the unweighted longitudinal and transversal Radon transforms of these components. It is known that reconstruction of an arbitrary vector field from only longitudinal or only transversal transforms is impossible. However, for the cases when both linearly weighted and unweighted transforms of either one of the types are known, we derive explicit inversion formulas for the full reconstruction of the field. Our interest in the inversion of such transforms stems from a certain inverse problem arising in magnetoacoustoelectric tomography (MAET). The connection between the weighted Radon transforms and MAET is exhibited in the paper. Finally, we demonstrate performance and noise sensitivity of the new inversion formulas in numerical simulations.
• Agranovsky, M., & Kunyansky, L. (2022).

  On the exactness of the universal backprojection formula for the spherical means Radon transform

  . arXiv (Cornell University). doi:10.48550/arxiv.2207.08262
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  The spherical means Radon transform $\mathcal{M}f(x,r)$ is defined by the integral of a function $f$ in $\mathbb{R}^{n}$ over the sphere $S(x,r)$ of radius $r$ centered at a $x$, normalized by the area of the sphere. The problem of reconstructing $f$ from the data $\mathcal{M}f(x,r)$ where $x$ belongs to a hypersurface $\Gamma\subset\mathbb{R}^{n}$ and $r \in(0,\infty)$ has important applications in modern imaging modalities, such as photo- and thermo- acoustic tomography. When $\Gamma$ coincides with the boundary $\partial\Omega$ of a bounded (convex) domain $\Omega\subset\mathbb{R}^{n}$, a function supported within $\Omega$ can be uniquely recovered from its spherical means known on $\Gamma$. We are interested in explicit inversion formulas for such a reconstruction. If $\Gamma=\partial\Omega$, such formulas are only known for the case when $\Gamma$ is an ellipsoid (or one of its partial cases). This gives rise to the natural question: can explicit inversion formulas be found for other closed hypersurfaces $\Gamma$? In this article we prove, for the so-called "universal backprojection inversion formulas", that their extension to non-ellipsoidal domains $\Omega$ is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.
• Witte, R. S., Kunyansky, L., Alvarez, A., Kang, J., Huang, C., Perkins, C., & O'Donnell, M. (2022). Current Source Density Imaging Using Regularized Inversion of Acoustoelectric Signals. IEEE Transactions on Medical Imaging, 1-1. doi:10.1109/tmi.2022.3215748
• Kunyansky, L., & Eller, M. (2021). Parametrix for the inverse source problem of thermoacoustic tomography with reduced data. Inverse Problems, 37(4), 045003. doi:10.1088/1361-6420/abde16
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  Abstract Our goal is to solve the inverse source problem of thermo- and photoacoustic tomography, with data registered on an open surface partially surrounding the source of acoustic waves. The proposed modified time reversal algorithm recovers the source term up to an infinitely smooth error term. Similarly to (Eller M et al 2020 Inverse Problems 36 085012), numerical simulations show that the error term is quite small in practical terms. Unlike the latter method, the present technique is applicable in the presence of a known variable speed of sound. It is also significantly more efficient from a computational standpoint. It can be implemented using either standard finite difference techniques or through methods based on separation of variables, that for special geometries yield extremely fast image reconstruction. We illustrate our results with numerical simulations in 2 and 3 spatial dimensions.
• Eller, M., Hoskins, P., & Kunyansky, L. (2020).

  Microlocally accurate solution of the inverse source problem of thermoacoustic tomography

  . arXiv: Analysis of PDEs. doi:10.48550/arxiv.2004.07364
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  We consider the inverse source problem of thermo- and photoacoustic tomography, with data registered on an open surface partially surrounding the source of acoustic waves. Under the assumption of constant speed of sound we develop an explicit non-iterative reconstruction procedure that recovers the Radon transform of the sought source, up to an infinitely smooth additive error term. The source then can be found by inverting the Radon transform. Our analysis is microlocal in nature and does not provide a norm estimate on the error in the so obtained image. However, numerical simulations show that this error is quite small in practical terms. We also present an asymptotically fast implementation of this procedure for the case when the data are given on a circular arc in 2D.
• Eller, M., Hoskins, P., & Kunyansky, L. (2020). Microlocally accurate solution of the inverse source problem of thermoacoustic tomography. Inverse Problems, 36(8), 094004.
• Kunyansky, L. (2020). Microlocally accurate solution of the inverse source problem of thermoacoustic tomography. Arxiv.
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  We consider the inverse source problem of thermo- and photoacoustictomography, with data registered on an open surface partially surrounding thesource of acoustic waves. Under the assumption of constant speed of sound wedevelop an explicit non-iterative reconstruction procedure that recovers theRadon transform of the sought source, up to an infinitely smooth additive errorterm. The source then can be found by inverting the Radon transform. Our analysis is microlocal in nature and does not provide a norm estimate onthe error in the so obtained image. However, numerical simulations show thatthis error is quite small in practical terms. We also present an asymptoticallyfast implementation of this procedure for the case when the data are given on acircular arc in 2D.[Journal_ref: ]
• Do, N., & Kunyansky, L. (2018). Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data. Inverse Problems, 34(9), 094004.
• Terzioglu, F., Kuchment, P., & Kunyansky, L. (2017). Compton Camera Imaging And The Cone Transform. A Brief Overview. Inverse Problems.
• Terzioglu, F., Kunyansky, L., & Kuchment, P. (2018). Compton camera imaging and the cone transform: A brief overview. Inverse Problems, 34(5), 054002. doi:10.1088/1361-6420/aab0ab
• Kunyansky, L., Ingram, C. P., & Witte, R. S. (2017).

  Rotational magneto-acousto-electric tomography (MAET): theory and experimental validation

  . Inverse Problems. doi:10.1088/1361-6560/aa6222
• Nguyen, L. V., & Kunyansky, L. (2016).

  A Dissipative Time Reversal Technique for Photoacoustic Tomography in a Cavity

  . SIAM J. Imaging Science. doi:10.1137/15m1049683
• Nguyen, L., & Kunyansky, L. (2016). A dissipative time reversal technique for photo-acoustic tomography in а cavity. SIAM J. Imaging Science, 9(2), 748–769.
• Kunyansky, L. (2015). Inversion of the spherical means transform in corner-like domains by reduction to the classical Radon transform. Inverse Problems, 31(9).
• Ambartsoumian, G., & Kunyansky, L. (2014). Exterior/interior problem for the circular means transform with applications to intravascular imaging. Inverse Problems and Imaging, 8(2), 339-359.
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  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.
• Ambartsoumian, G., & Kunyansky, L. (2013).

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

  . arXiv: Analysis of PDEs. doi:10.48550/arxiv.1308.6016
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  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.
• Cox, B. T., Holman, B., & Kunyansky, L. (2013). Photoacoustic tomography in a reflecting cavity. Progress in Biomedical Optics and Imaging - Proceedings of SPIE, 8581.
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  Abstract: Almost all known photoacoustic image reconstruction algorithms are based on the assumption that the acoustic waves leave the object (the imaged region) after a finite time. This assumption is fulfilled if the measurements are made in free space and reflections from the detectors are negligible. However, when the object is surrounded by acoustically hard detectors arrays (and/or by additional acoustic mirrors), the acoustic waves will bounce around in such a reverberant cavity many times (in the absence of absorption, forever). This paper proposes fast reconstruction algorithms for the measurements made from the walls of a rectangular reverberant cavity. The algorithms are tested using numerical simulations. © 2013 Copyright SPIE.
• Kunyansky, L., & Holman, B. R. (2015). Gradual time reversal in thermo- and photo- acoustic tomography within a resonant cavity. Inverse Problems, 31, 035008.
• Kunyansky, L., Holman, B., & Cox, B. T. (2013). Photoacoustic tomography in a rectangular reflecting cavity. Inverse Problems, 29(12).
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  Abstract: Almost all known image reconstruction algorithms for photoacoustic and thermoacoustic tomography assume that the acoustic waves leave the region of interest after a finite time. This assumption is reasonable if the reflections from the detectors and surrounding surfaces can be neglected or filtered out (for example, by time-gating). However, when the object is surrounded by acoustically hard detector arrays, and/or by additional acoustic mirrors, the acoustic waves will undergo multiple reflections. (In the absence of absorption, they would bounce around in such a reverberant cavity forever.) This disallows the use of the existing free-space reconstruction techniques. This paper proposes a fast iterative reconstruction algorithm for measurements made at the walls of a rectangular reverberant cavity. We prove the convergence of the iterations under a certain sufficient condition, and demonstrate the effectiveness and efficiency of the algorithm in numerical simulations. © 2013 IOP Publishing Ltd.
• Agranovsky, M., Kuchment, P., Kunyansky, L., & Wang, L. (2012). On Reconstruction Formulas and Algorithms for the Thermoacoustic Tomography. PHOTOACCOUSTIC IMAGING AND SPECTROSCOPY, 144, 89-101.
• Kunyansky, L. (2012). A mathematical model and inversion procedure for magneto-acousto-electric tomography. Inverse Problems, 28(3).
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  Abstract: Magneto-acousto-electric tomography (MAET), also known as the Lorentz force or Hall effect tomography, is a novel hybrid modality designed to be a high-resolution alternative to the unstable electrical impedance tomography. In this paper, we analyze the existing mathematical models of this method, and propose a general procedure for solving the inverse problem associated with the MAET. It consists in applying to the data one of the algorithms of thermo-acoustic tomography, followed by solving the Neumann problem for the Laplace equation and the Poisson equation. For the particular case when the region of interest is a cube, we present an explicit series solution resulting in a fast reconstruction algorithm. As we show, both analytically and numerically, the MAET is a stable technique yielding high-resolution images even in the presence of significant noise in the data. © 2012 IOP Publishing Ltd.
• Kunyansky, L., & Kunyansky, L. (2012). Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems and Imaging, 6(1), 111-131.
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  Abstract: We propose three fast algorithms for solving the inverse problem of the thermoacoustic tomography corresponding to certain acquisition geometries. Two of these methods are designed to process the measurements done with point-like detectors placed on a circle (in 2D) or a sphere (in 3D) surrounding the object of interest. The third inversion algorithm works with the data measured by the integrating line detectors arranged in a cylindrical assembly rotating around the object. The number of operations required by these techniques is equal to O(n 3 log n) and O(n 3 log 2n) for the 3D techniques (assuming the reconstruction grid with n 3 nodes) and to O(n 2 log n) for the 2D problem with n × n discretizetion grid. Numerical simulations show that on large computational grids our methods are at least two orders of magnitude faster than the finite-difference time reversal techniques. The results of reconstructions from real measurements done by the integrating line detectors are also presented, to demonstrate the practicality of our algorithms. © 2012 American Institute of Mathematical Sciences.
• Kuchment, P., & Kunyansky, L. (2011). 2D and 3D reconstructions in acousto-electric tomography. Inverse Problems, 27(5).
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  Abstract: We propose and test stable algorithms for the reconstruction of the internal conductivity of a biological object using acousto-electric measurements. Namely, the conventional impedance tomography scheme is supplemented by scanning the object with acoustic waves that slightly perturb the conductivity and cause the change in the electric potential measured on the boundary of the object. These perturbations of the potential are then used as the data for the reconstruction of the conductivity. The present method does not rely on 'perfectly focused' acoustic beams. Instead, more realistic propagating spherical fronts are utilized, and then the measurements that would correspond to perfect focusing are synthesized. In other words, we use synthetic focusing. Numerical experiments with simulated data show that our techniques produce high-quality images, both in 2D and 3D, and that they remain accurate in the presence of high-level noise in the data. Local uniqueness and stability for the problem also hold. © 2011 IOP Publishing Ltd.
• Kunyansky, L. (2011).

  A mathematical model and inversion procedure for

  . Inverse Problems.
• Kunyansky, L. (2011).

  Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries

  . arXiv: Analysis of PDEs. doi:10.48550/arxiv.1102.1413
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  We propose three fast algorithms for solving the inverse problem of the thermoacoustic tomography corresponding to certain acquisition geometries. Two of these methods are designed to process the measurements done with point-like detectors placed on a circle (in 2D) or a sphere (in 3D) surrounding the object of interest. The third inversion algorithm works with the data measured by the integrating line detectors arranged in a cylindrical assembly rotating around the object. The number of operations required by these techniques is equal to O(n^3 log n) and O(n^3 log^2 n) for the 3D techniques (assuming the reconstruction grid with n^3 nodes) and to O(n^2 log n) for the 2D problem with n-by-n discretizetion grid. Numerical simulations show that our methods are at least two orders of magnitude faster than the existing algorithms, without any sacrifice in accuracy or stability. The results of reconstructions from real measurements done by the integrating line detectors are also presented, to demonstrate the practicality of our methods.
• Kunyansky, L. (2011). Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra. Inverse Problems, 27(2).
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  Abstract: We present explicit filtration/backprojection-type formulae for the inversion of the spherical (circular) mean transform with the centers lying on the boundary of some polyhedra (or polygons, in 2D). The formulae are derived using the double-layer potentials for the wave equation, for domains with certain symmetries. The formulae are valid for a rectangle and certain triangles in 2D, and for a cuboid, certain right prisms and a certain pyramid in 3D. All the present inversion formulae yield exact reconstruction within the domain surrounded by the acquisition surface even in the presence of exterior sources. © 2011 IOP Publishing Ltd.
• Kuchment, P., & Kunyansky, L. (2010).

  Reconstructions in 2D acousto-electric tomography

  . Inverse Problems.
• Kuchment, P., Kuchment, P., Kunyansky, L., & Kunyansky, L. (2010). Synthetic focusing in ultrasound modulated tomography. Inverse Problems and Imaging, 4(4), 665-673.
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  Abstract: Several hybrid tomographic methods utilizing ultrasound modulation have been introduced lately. Success of these methods hinges on the feasibility of focusing ultrasound waves at an arbitrary point of interest. Such focusing, however, is difficult to achieve in practice. We thus propose a way to avoid the use of focused waves through what we call synthetic focusing, i.e. by reconstructing the would-be response to the focused modulation from the measurements corresponding to realistic unfocused waves. Examples of reconstructions from simulated data are provided. This non-technical paper describes only the general concept, while technical details will appear elsewhere. © 2010 American Institute of Mathematical Sciences.
• Kunyansky, L. (2010). Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra. Inverse Problems. doi:10.1088/0266-5611/27/2/025012
• Kunyansky, L., & Kuchment, P. (2010). 2D and 3D reconstructions in acousto-electric tomography. Inverse Problems. doi:10.1088/0266-5611/27/5/055013
• Kuchment, P., & Kunyansky, L. (2008).

  Mathematics of thermoacoustic tomography

  . European Journal of Applied Mathematics. doi:10.1017/s0956792508007353
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  Core share and HTML view are not possible as this article does not have html content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. The article presents a survey of mathematical problems, techniques and challenges arising in thermoacoustic tomography and its sibling photoacoustic tomography.
• Kuchment, P., & Kunyansky, L. (2008). A survey in mathematics for industry: Mathematics of thermoacoustic tomography. European Journal of Applied Mathematics, 19(2), 191-224.
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  Abstract: The article presents a survey of mathematical problems, techniques and challenges arising in thermoacoustic tomography and its sibling photoacoustic tomography. © 2008 Cambridge University Press.
• Kunyansky, L. (2008).

  ASYMPTOTICALLY OPTIMAL HIGH-ORDER ACCURATE ALGORITHMS FOR THE SOLUTION OF CERTAIN ELLIPTIC PDEs

  . Arxiv ?. doi:10.2172/953765
• Kunyansky, L. A. (2008). Thermoacoustic tomography with detectors on an open curve: An efficient reconstruction algorithm. Inverse Problems, 24(5).
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  Abstract: Practical applications of thermoacoustic tomography require numerical inversion of the spherical mean Radon transform with the centers of integration spheres occupying an open surface. A solution of this problem is needed (both in 2D and in 3D) because frequently the region of interest cannot be completely surrounded by the detectors, as happens, for example, in breast imaging. We present an efficient numerical algorithm for solving this problem in 2D (similar methods are applicable in the 3D case). Our method is based on the numerical approximation of plane waves by certain single-layer potentials related to the acquisition geometry. After the densities of these potentials have been precomputed, each subsequent image reconstruction has the complexity of the regular filtration backprojection algorithm for the classical Radon transform. The performance of the method is demonstrated in several numerical examples: one can see that the algorithm produces very accurate reconstructions if the data are accurate and sufficiently well sampled; on the other hand, it is sufficiently stable with respect to noise in the data. © 2008 IOP Publishing Ltd.
• Kunyansky, L. (2007). A series solution and a fast algorithm for the inversion of the spherical mean Radon transform. Inverse Problems. doi:10.1088/0266-5611/23/6/s02
• Kunyansky, L. A. (2007). A series solution and a fast algorithm for the inversion of the spherical mean Radon transform. Inverse Problems, 23(6), S11-S20.
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  Abstract: An explicit series solution is proposed for the inversion of the spherical mean Radon transform. Such an inversion is required in problems of thermo- and photo-acoustic tomography. Closed-form inversion formulae are currently known only for the case when the centres of the integration spheres lie on a sphere surrounding the support of the unknown function, or on certain unbounded surfaces. Our approach results in an explicit series solution for any closed measuring surface surrounding a region for which the eigenfunctions of the Dirichlet Laplacian are explicitly known - such as, for example, cube, finite cylinder, half-sphere etc. In addition, we present a fast reconstruction algorithm applicable in the case when the detectors (the centres of the integration spheres) lie on a surface of a cube. This algorithm reconstructs 3D images thousands times faster than backprojection-type methods. © 2007 IOP Publishing Ltd.
• Kunyansky, L. A. (2007). Explicit inversion formulae for the spherical mean Radon transform. Inverse Problems, 23(1), 373-383.
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  Abstract: We derive explicit formulae for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulae are important for problems of thermo- and photo-acoustic tomography. A closed-form inversion formula of a filtration-backprojection type is found for the case when the centres of the integration spheres lie on a sphere in surrounding the support of the unknown function. © 2007 IOP Publishing Ltd.
• Kunyansky, L. (2006).

  Explicit inversion formulas for the spherical mean Radon transform

  . arXiv: Analysis of PDEs.
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  We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo- and photo- acoustic tomography. A closed-form inversion formula of a filtration-backprojection type is found for the case when the centers of the integration spheres lie on a sphere in Rn surrounding the support of the unknown function. An explicit series solution is presented for the case when the centers of the integration spheres lie on a general closed surface.
• Kunyansky, L. A. (2004). Inversion of the 3D exponential parallel-beam transform and the Radon transform with angle-dependent attenuation. Inverse Problems, 20(5), 1455-1478.
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  Abstract: The inversion problem for the 3D parallel-beam exponential ray transform is solved through inversion of a set of the 2D exponential Radon transforms with complex-valued angle-dependent attenuation. An inversion formula for the latter 2D transform is derived; it generalizes the known Kuchment-Shneiberg formula valid for real angle-dependent attenuation. We derive an explicit theoretically exact solution of the 3D problem which is valid for arbitrary closed trajectory that does not intersect itself. A simple reconstruction algorithm is described, applicable for certain sets of trajectories satisfying Orlov's condition. In the latter case, our inversion technique is as stable as the Tretiak-Metz inversion formula. Possibilities of further reduction of noise sensitivity are briefly discussed in the paper. The work of our algorithm is illustrated by an example of image reconstruction from two circular orbits.
• Bruno, O. P., & Kunyansky, L. (2002).

  Fast, high-order solution of surface scattering problems

  . Must be conference proceedings..... doi:10.1109/aps.2000.874851
• Guillement, J. -., Jauberteau, F., Kunyansky, L., Novikov, R., & Trebossen, R. (2002). On single-photon emission computed tomography imaging based on an exact formula for the nonuniform attenuation correction. Inverse Problems, 18(6), L11-L19.
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  Abstract: An exact formula approach to nonuniform attenuation correction in single-photon emission computed tomography (SPECT) was discussed. The formula admits a numerical implementation via a direct generalization of the filtered backprojection (FBP) algorithm. The formula can be used for fast computing of an efficient first approximation for more complicated SPECT reconstruction techniques.
• Kuchment, P., & Kunyansky, L. (2002). Differential operators on graphs and photonic crystals. Advances in Computational Mathematics, 16(2-3), 263-290.
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  Abstract: Studying classical wave propagation in periodic high contrast photonic and acoustic media naturally leads to the following spectral problem: -Δu = λεu, where ε(x) (the dielectric constant) is a periodic function that assumes a large value ε near a periodic graph Σ in ℝ2 and is equal to 1 otherwise. High contrast regimes lead to appearence of pseudo-differential operators of the Dirichlet-to-Neumann type on graphs. The paper contains a technique of approximating these pseudo-differential spectral problems by much simpler differential ones that can sometimes be resolved analytically. Numerical experiments show amazing agreement between the spectra of the pseudo-differential and differential problems.
• Bruno, O. P., & Kunyansky, L. A. (2001). A Fast, High-Order Algorithm for the Solution of Surface Scattering Problems: Basic Implementation, Tests, and Applications. Journal of Computational Physics, 169(1), 80-110.
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  Abstract: We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates scattered fields through fast, high-order solution of the corresponding boundary integral equation. The high-order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of a novel approach which, based on high-order "two-face" equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D fast Fourier transforms (FFTs). This approach is faster and substantially more accurate, and it runs on dramatically smaller memories than other FFT and k-space methods. The present algorithm computes one matrix-vector multiplication in O(N6/5logN) to O(N4/3logN) operations, where N is the number of surface discretization points. The latter estimate applies to smooth surfaces, for which our high-order algorithm provides accurate solutions with small values of N; the former, more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling. Further, our approach exhibits super-algebraic convergence; it can be applied to smooth and nonsmooth scatterers, and it does not suffer from accuracy breakdowns of any kind. In this paper we introduce the main algorithmic components in our approach, and we demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds. © 2001 Academic Press.
• Bruno, O. P., & Kunyansky, L. A. (2001). Fast, high-order solution of surface scattering problems. IEEE Antennas and Propagation Society, AP-S International Symposium (Digest), 2, 554-557.
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  Abstract: A new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space was presented. The proposed algorithm was found to be faster, more accurate, and ran on lower memories than other Fast Fourier transform (FFT) or k-space methods. It was shown that the algorithm could evaluate the personal computer scattering from bodies of acoustical sizes of several hundreds.
• Bruno, O. P., & Kunyansky, L. A. (2001). Surface scattering in three dimensions: An accelerated high-order solver. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 457(2016), 2921-2934.
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  Abstract: We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates scattered fields through fast, high-order, accurate solution of the corresponding boundary integral equation. The high-order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of high-order equivalent source approximations, which allow for fast evaluation of non-adjacent interactions by means of the three-dimensional fast Fourier transform (FFT). Our acceleration scheme has dramatically lower memory requirements and yields much higher accuracy than existing FFT-accelerated techniques. The present algorithm computes one matrix-vector multiply in O(N6/5 log N) to O(N4/3 log N) operations (depending on the geometric characteristics of the scattering surface), it exhibits super-algebraic convergence, and it does not suffer from accuracy breakdowns of any kind. We demonstrate the efficiency of our method through a variety of examples. In particular, we show that the present algorithm can evaluate accurately, on a personal computer, scattering from bodies of acoustical sizes (ka) of several hundreds.
• Kunyansky, L. A. (2001). A new SPECT reconstruction algorithm based on the Novikov explicit inversion formula. Inverse Problems, 17(2), 293-306.
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  Abstract: We present a new reconstruction algorithm for single-photon emission computed tomography. The algorithm is based on the Novikov explicit inversion formula for the attenuated Radon transform with non-uniform attenuation. Our reconstruction technique can be viewed as a generalization of both the filtered backprojection algorithm and the Tretiak-Metz algorithm. We test the performance of the present algorithm in a variety of numerical experiments. Our numerical examples show that the algorithm is capable of accurate image reconstruction even in the case of strongly non-uniform attenuation coefficient, similar to that occurring in a human thorax.
• Kunyansky, L. A. (2001). Analytic reconstruction algorithms in emission tomography with variable attenuation. Journal of Computational Methods in Sciences and Engineering, 1(2-3), 267-286. doi:10.3233/jcm-2001-12-308
• Kunyansky, L., & Bruno, O. P. (2001). Surface scattering in three dimensions: an accelerated high–order solver. Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences. doi:10.1098/rspa.2001.0882
• Bruno, O. P., & Kunyansky, L. A. (2000). Fast, high-order solution of surface scattering problems. IEEE Antennas and Propagation Society, AP-S International Symposium (Digest), 4, 1860-1863.
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  Abstract: A fast, high-order algorithm for the solution of problems of acoustic scattering from smooth surfaces in three dimensions is presented. Numerical experiments indicate that this algorithm performs exceptionally well.
• Bruno, O., & Kunyansky, L. (2000). A fast, high-order algorithm for the solution of surface scattering problems: Basic implementation, tests, and applications. JOURNAL OF COMPUTATIONAL PHYSICS, 169(1), 80-110.
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  We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates scattered fields through fast, high-order solution of the corresponding boundary integral equation. The high-order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of a novel approach which, based on high-order "two-face" equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D fast Fourier transforms (FFTs), This approach is faster and substantially more accurate, and it runs on dramatically smaller memories than other FFT and k-space methods. The present algorithm computes one matrix-vector multiplication in O(N-6/5 log N) to O(N-4/3 log N) operations, where N is the number of surface discretization points. The latter estimate applies to smooth surfaces, for which our high-order algorithm provides accurate solutions with small values of N; the former, more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling. Further, our approach exhibits super-algebraic convergence. it can be applied to smooth and nonsmooth scatterers, and it does not suffer from accuracy breakdowns of any kind. In this paper we introduce the main algorithmic components in our approach, and we demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes Of several hundreds. (C) 2001 Academic Press.
• Kuchment, P., & Kunyansky, L. A. (1999). Spectral properties of high contrast band-gap materials and operators on graphs. Experimental Mathematics, 8(1), 1-28.
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  Abstract: The theory of classical waves in periodic high contrast photonic and acoustic media leads to the spectral problem -Δu = λεu, where the dielectric constant ε(x) is a periodic function which assumes a large value ε near a periodic graph Σ in ℝ2 and is equal to 1 otherwise. Existence and locations of spectral gaps are of primary interest. The high contrast asymptotics naturally leads to pseudodifferential operators of the Dirichlet-to-Neumann type on graphs and on more general structures. Spectra of these operators are studied numerically and analytically. New spectral effects are discovered, among them the "almost discreteness" of the spectrum for a disconnected graph and the existence of "almost localized" waves in some connected purely periodic structures.
• Kunyansky, L., & Kuchment, P. (1999). Spectral Properties of High Contrast Band-Gap Materials and Operators on Graphs. Experimental Mathematics. doi:10.1080/10586458.1999.10504384
• Kunyansky, L., Kuchment, P., & Axmann, W. (1999). Asymptotic methods for thin high-contrast two-dimensional PBG materials. Journal of Lightwave Technology. doi:10.1109/50.802986
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  This paper surveys recent analytic and numerical results on asymptotic models of spectra of electromagnetic (EM) waves in two-dimensional (2D) thin high-contrast photonic bandgap (PBG) materials. These models lead to discovery of interesting phenomena, including extremely narrow bands that can be used for spontaneous emission enhancement, gaps in the long wave regions, and asymptotic periodicity of the spectrum. The asymptotic results provide unexpectedly good qualitative (and sometimes quantitative) description of spectral behavior for materials of finite contrast. In some cases, simple ordinary differential models can be derived that yield a good approximation of the spectra. In such situations, one can obtain approximate analytic formulas for the dispersion relations.
• Kunyansky, L. (1998).

  Numerical and analytic study of problems of photonic crystals theory

  . This is my PhD Thesis.
• Kunyansky, L. (1992). Generalized and attenuated radon transforms: restorative approach to the numerical inversion. Inverse Problems. doi:10.1088/0266-5611/8/5/008
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  The problem of the function reconstruction on its line integrals with known weight function is considered. The approach studied consists of treating the attenuated projections by the radon transform inversion formula and considering the result of the inversion as a distorted image. A helpful formula describing the distortion is obtained. The norm of the distortion operator is estimated and several iterative restoration algorithms based on the integral transfers are investigated. The results of the numerical inversion of the attenuated radon transform are presented to demonstrate the features of the algorithms.

Proceedings Publications

• Kunyansky, L. (2014, August). Inversion of the spherical means transform by reduction to the classical Radon transform. In Workshop "Mathematics and Algorithms in Tomography", 13-15.
• Bruno, O., Kunyansky, L., Bermudez, A., Gomez, D., Hazard, C., Joly, P., & Roberts, J. (2013). Fast, high-order solution of surface scattering problems. In FIFTH INTERNATIONAL CONFERENCE ON MATHEMATICAL AND NUMERICAL ASPECTS OF WAVE PROPAGATION, 465-470.
• Kunyansky, L., Holman, B., & Cox, B. T. (2013). Photoacoustic tomography in a reflecting cavity. In Proceedings Volume 8581, Photons Plus Ultrasound: Imaging and Sensing 2013; 85811D (2013) https://doi.org/10.1117/12.2003571.
• Bruno, O., Kunyansky, L., , ., , ., & , . (1999). Fast, high-order solution of surface scattering problems. In IEEE ANTENNAS AND PROPAGATION SOCIETY INTERNATIONAL SYMPOSIUM, VOLS 1-4, 1860-1863.

Presentations

• Kunyansky, L. (2023). Weighted Radon transforms of vector fields, with applications to magnitoacoustoelectric tomography. Tomographic Inverse Problems: Mathematical Challenges and Novel Applications. MFO Oberwolfach, Germany.
• Kunyansky, L. (2023, August). Weighted Radon transforms of vector fields, with applications to magnitoacoustoelectric tomography. Workshop on Mathematical Trends in Medical Imaging. University of Chicago.
• Kunyansky, L. (2023, May). Weighted Radon transforms of vector fields, with applications to magnitoacoustoelectric tomography. Programme "Rich and Nonlinear Tomography", Worksho "New tomographic methods using particles". Isaac Newton Institute, Cambridge, UK: Simons Foundation.
• Kunyansky, L. (2023, October). Weighted Radon transforms of vector fields, with applications to magnitoacoustoelectric tomography. Spectral theory and Applications. Texas A&M University.
• Kunyansky, L. (2022, May). Sparsity-based techniques for hybrid imaging modalities with missing low frequencies. The Tenth International Conference "Inverse Problems: Modeling and Simulation". Malta.
• Kunyansky, L., & Eller, M. (2022, Fall). Parametrix for the inverse source problem of thermoacoustic tomography with reduced data. Special Semester on Tomography Across the Scales (Zoom presentation). J. Radon Institute for Computational and Applied Mathematics, Linz, Austria.
• Kunyansky, L., & Eller, M. (2022, Spring). Parametrix for the inverse source problem of thermoacoustic tomography. Conference on Mathematics of Wave Phenomena 2022. Karlsruhe Institute of Technology, Germany (talk given over Zoom).
• Kunyansky, L., & Eller, M. (2021). Parametrix for the inverse source problem of thermoacoustic tomography with reduced data. International Zoom Inverse Problems seminar. University of California, Irvine (over ZOOM).
• Do, N., & Kunyansky, L. (2019, Spring). Theoretically exact solution of the inverse source problem for the wave equation with spatially and temporally reduced data. Joint Mathematics Meetings. Baltimore, MD.
• Kunyansky, L., & Do, N. (2019, Fall). Sparsity-based techniques for hybrid imaging modalities with missing low frequencies. Novel Medical Imaging Workshop. Texas A&M University.
• Kunyansky, L., & Do, N. (2019, Fall). Sparsity-based techniques for hybrid imaging modalities with missing low frequencies. Sparsity in Imaging Seminar. University of Arizona.
• Kunyansky, L., & Do, N. (2019, Fall). Sparsity-based techniques for hybrid imaging modalities with missing low frequencies. Tripods RWG3 seminar. University of Arizona.
• Kunyansky, L., & Do, N. (2019, February). Theoretically exact solution of the inverse source problem for the wave equation with limited data. Colloquium. Missouri State University: Department of Mathematics.
• Kunyansky, L., & Do, N. (2019, January). Seeing the invisibility through tomographic lens. Colloquium. California State University, Dominguez Hills: Mathematics Department.
• Kunyansky, L., & Do, N. (2019, July). Rotational magnetoacoustoelectric tomography and sparsity techniques. Applied Inverse Problems Conference. Grenoble.
• Kunyansky, L., & Do, N. (2019, July). Theoretically exact solution of the inverse source problem with spatially and temporally reduced data. Applied Inverse Problems Conference. Grenoble.
• Kunyansky, L., & Do, N. (2019, Spring). Sparsity-based techniques for hybrid imaging modalities with band-limited measurements. Inverse Problems Seminar. Texas A&M University.
• Kunyansky, L., & Do, N. (2019, Summer). Magnetoacoustic and magnetoelectric hybrid imaging modalities. Modern Challenges in Imaging. Tufts University, Boston.
• Kunyansky, L. (2018, Fall). Inverse source problem for the wave equation with reduced data: an explicit solution. Inverse problems seminar. Texas A&M University.
• Kunyansky, L., & Do, N. (2018, February). Inverse source problem for the wave equation with reduced data: an explicit solution. Modeling and Computation Seminar. University of Arizona: Department of Mathematics.
• Kunyansky, L., & Do, N. (2018, May). Theoretically exact solution of the inverse source problem for the wave equation with spatially and temporally reduced data. Summer School Waves and Particles in Random Media : Theory and Applications. Colorado State University: NSF and Colorado State University.
• Kunyansky, L., & Do, N. (2018, Spring). Theoretically exact solution of the inverse source problem with spatially and temporally reduced data. Colloquium. Georgetown University.
• Kunyansky, L. (2017, February). Medical imaging with sound, magnets, and currents. Undergraduate Research Seminar. U of A.
• Kunyansky, L., Ingram, P. C., & Witte, R. S. (2017, Spring). Lorentz force impedance tomography in 2D: Theory and Experiments. Quantitative Biology Colloquium. U of A.
• Kunyansky, L., Ingram, P. C., & Witte, R. S. (2017, Spring). Rotational Magneto-Acousto-Electric Tomography: Theory and Experiments.. 100 years of the Radon transform. Johannes Keppler University, Linz, Austria.
• Kunyansky, L., Ingram, P. C., & Witte, R. S. (2017, Spring). Rotational Magneto-Acousto-Electric Tomography: Theory and Experiments.. Inverse Problems Seminar. University College London..
• Kunyansky, L., Ingram, P., & Witte, R. S. (2017, January). Lorentz force impedance tomography in 2D: Theory and Experiments. Inverse Problems Seminar.. Texas A&M University.
• Kunyansky, L. (2016, April). Inversion of the spherical means transform in corner-like domains by reduction to the classical Radon transform. Colloquium, Kent State U-ty. Kent State University.
• Kunyansky, L. (2016, June). Inversion of the spherical means transform by reduction to the classical Radon transform,. Seminar on differential equations, Weierstrass Institute, Berlin.
• Kunyansky, L. (2016, June). Lorentz force impedance tomography in 2D: Theory and Experiments. Workshop, University of Lubeck. Lubeck, Germany.
• Kunyansky, L. (2016, March). Magneto-Acousto-Electric Tomography: Theory and experiments. MR Research Seminar, U of A. Department of Medical Imaging.
• Kunyansky, L. (2016, May). Lorentz Force Tomography in 2D. SIAM conference on Imaging Science. Albuquerque, NM: SIAM.
• Kunyansky, L. (2016, October). Lorentz Force Impedance Tomography in 2D: Theory and Experiments. AIM seminar, U-ty of Michigan. University of Michigan, Ann Arbor.
• Kunyansky, L. (2015, November). Inward gaze: Medical imaging with sound, magnets, and currents. Science Cafe. , Borderlands brewery, Tucson, AZ.
• Kunyansky, L. (2015, Summer). Inversion of the spherical means transform by reduction to the classical Radon transform. Minisymposium "Recent developments in tomography: from theory to applications" , Applied Inverse Problems Conference (AIP) 2015. Helsinki, Finland.
• Kunyansky, L. (2015, Summer). Inversion of the spherical means transform in corner-like domains by reduction to the classical Radon transform. Workshop "Hybrid methods in imaging". Banff International Research Station, Canada.
• Kunyansky, L. (2015, Summer). Inversion of the spherical means transform with centers lying on corner-like surfaces by reduction to the classical Radon transform. Workshop "Computational and Analytical Aspects of Image Reconstruction". ICERM, Providence.
• Kunyansky, L. (2014, April). Mathematics of Thermoacoustic Tomography. Applied Mathematics 35th Anniversary Meeting. U of A.
• Kunyansky, L. (2014, August). Inversion of spherical means transform by reduction to the classical Radon transform. Workshop "Mathematics and Algorithms in Tomography". Mathematisches Forschungsinstitut Oberwolfach, Germany.
• Kunyansky, L. (2014, June). Thermo- and photo- acoustic tomography in the presence of reflecting boundaries. Special session on Hybrid Imaging Methods, 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications. Madrid, Spain.
• Kunyansky, L. (2014, March). Magneto-Acousto-Electric Tomography and other hybrid modalities. Modeling and Computation Seminar. U of A.
• Kunyansky, L. (2014, November). Thermo- and photo- acoustic tomography in the presence of reflecting boundaries. Workshop "Advances in Mathematical Image Reconstruction". University of Innsbruck, Austria.

Poster Presentations

• Do, N., & Kunyansky, L. (2018, Fall). Inverse source problem for the wave equation with reduced data: an explicit solution. Mathematical Modeling and Computational Methods for Multiscale Problems in Science Engineering Conference. University of Arizona.

Others

• Kunyansky, L. (2016, January). Medical imaging with sound, magnets, and currents. Arizona Daily Star.
• Kunyansky, L. (2014, December). Inversion of the spherical means transform in corner-like domains by reduction to the classical Radon transform. Arxiv.org. http://arxiv.org/abs/1502.02640