Gaik Ambartsoumian

Research Interests

  • Integral Geometry
  • Inverse Problems
  • Computerized Tomography
  • Mathematical Problems of Imaging

Research Profile Links


Broken Rays, Cones, And Stars In Tomography.

Generalized Radon Transforms and Imaging by Scattered Particles. Broken Rays, Cones, and Stars in Tomography. World Scientific, 2023.

A generalized Radon transform (GRT) maps a function to its weighted integrals along a family of curves or surfaces. Such operators appear in mathematical models of various imaging modalities. The GRTs integrating along smooth curves and surfaces (lines, planes, circles, spheres, amongst others) have been studied at great lengths for decades, but relatively little attention has been paid to transforms integrating along non-smooth trajectories. Recently, an interesting new class of GRTs emerged at the forefront of research in integral geometry. The two common features of these transforms are the presence of a "vertex" in their paths of integration (broken rays, cones, and stars) and their relation to imaging techniques based on physics of scattered particles (Compton camera imaging, single scattering tomography, etc).

The book covers the relevant imaging modalities, their mathematical models, and the related GRTs. The discussion of the latter comprises a thorough exploration of their known mathematical properties, including injectivity, inversion, range description and microlocal analysis. The mathematical background required for reading most of the book is at the level of an advanced undergraduate student, which should make its content attractive for a large audience of specialists interested in imaging. Mathematicians may appreciate certain parts of the theory that are particularly elegant with connections to functional analysis, PDEs and algebraic geometry.

Journal Publications

  1. Microlocal analysis of the generalized V-line transforms, (with V. Krishnan, and E. T. Quinto) in preparation.
  2. A simple range characterization for spherical mean transform in odd dimensions and its applications, (with D. Agrawal, V. Krishnan, and N. Singhal) submitted, preprint at arXiv:2310.20702.
  3. V-line tensor tomography: numerical results, (with R. K. Mishra, and I. Zamindar) preprint at arXiv:2405.03249.
  4. V-line 2-tensor tomography in the plane, (with R. K. Mishra, and I. Zamindar) Inverse Problems 40 (2024), no. 3, 035003.
  5. Numerical implementation of generalized V-line transforms on 2D vector fields and their inversions, (with M. J. Latifi-Jebelli, and R. K. Mishra) SIAM Journal on Imaging Sciences 17 (2024), no. 1, pp 595-631.
  6. Inversion and symmetries of the star transform, (with M. J. Latifi-Jebelli) The Journal of Geometric Analysis 31 (2021), pp 11270-11291.
  7. Generalized V-line transforms in 2D vector tomography, (with M. J. Latifi-Jebelli, and R. K. Mishra) Inverse Problems 36 (2020), no. 10, 104002.
  8. Generalized Radon transforms and applications in tomography, (with E. T. Quinto) Inverse Problems 36 (2019), no. 2, 020301.
  9. The V-line transform with some generalizations and cone differentiation, (with M. J. Latifi-Jebelli) Inverse Problems 35 (2019), no. 3, 034003.
  10. Singular FIOs in SAR imaging, II: transmitter and receiver at different speeds, (with R. Felea, V. Krishnan, C. Nolan, and E. T. Quinto) SIAM Journal on Mathematical Analysis, vol. 50 (1), 2018, pp 591-621.
  11. Image reconstruction from radially incomplete spherical Radon data, (with R. Gouia-Zarrad, V. Krishnan, and S. Roy) European Journal of Applied Mathematics, vol. 29 (2018), issue 3, pp 470-493.
  12. Numerical inversion of a broken ray transform arising in single scattering optical tomography, (with S. Roy) IEEE Transactions on Computational Imaging, vol. 2 (2016), issue 2, pp 166-173.
  13. Inversion of a class of circular and elliptical Radon transforms, (with V. Krishnan) Contemporary Mathematics, vol. 653 (2015), 13174.
  14. Exact inversion of the conical Radon transform with a fixed opening angle, (with R. Gouia-Zarrad) Inverse Problems 30 (2014), no. 4, 045007.
  15. Exterior/interior problem for the circular means transform with applications to intravascular imaging, (with L. Kunyansky) Inverse Problems and Imaging, 8, no. 2 (2014), pp 339-359.
  16. Microlocal analysis of an ultrasound transform with circular source and receiver trajectories , (with J. Boman, V. Krishnan, and E. T. Quinto) Contemporary Mathematics, vol. 598 (2013), pp 45-58.
  17. A Series formula for inversion of the V-line transform in a disc, (with S. Moon) Computers & Mathematics with Applications, vol. 66, Issue 9, November 2013, pp 1567-1572.
  18. A class of singular Fourier integral operators in synthetic aperture radar imaging , (with R. Felea, V. Krishnan, C. Nolan, and E. T. Quinto) Journal of Functional Analysis 264 (2013), pp 246-269.
  19. Invasion speed in cellular automaton models for T. cruzi vector migration , (with B. Crawford, and C. Kribs-Zaleta) Bulletin of Mathematical Biology , vol. 75, no. 7 (2013), pp 1051-1081.
  20. Inversion of the v-line Radon transform in a disc and its applications in imaging, Computers & Mathematics with Applications, vol. 64, issue 3, August 2012, pp 260-265.
  21. Inversion of the circular Radon transform on an annulus, (with R. Gouia-Zarrad, and M. Lewis) Inverse Problems 26 (2010), 105015.
  22. A range description for the planar circular Radon transform, (with P. Kuchment) SIAM Journal on Mathematical Analysis vol. 38, no. 2 (2006), pp 681-692.
  23. On the injectivity of the circular Radon transform, (with P. Kuchment) Inverse Problems 21 (2005), pp 473-485.
  24. Reconstructions in limited view thermoacoustic tomography, (with Y. Xu, L. Wang, and P. Kuchment) Medical Physics 31(4) April 2004, pp 724-733.
  25. Continuation of functions representable by exponentials of infinite multiplicity with alternating exponents, (with A. Burobin) Matematicheskie Zametki, Russian Academy of Sciences, 73 (2003), no. 2, pp 163-172. (in Russian)

Conference Proceedings

  1. Improved interior tomography reconstruction based on prior knowledge, (with E. Bridges and L. Florescu) American Institute of Physics Conference Proceedings, vol. 2302, 060002 (2020).
  2. Inversion of the star transform, (with M. J. Latifi-Jebelli) in Tomographic Inverse Problems: Theory and Applications, edited by M. Burger, B. Hahn and E. T. Quinto, Oberwolfach Reports, EMS, 2019.
  3. On the V-line Radon transform and its applications in imaging, (with R Gouia-Zarrad, and S. Moon) in Mathematics and Algorithms in Tomography, edited by M. Burger, A. K. Louis, and E. T. Quinto, Oberwolfach Reports, Vol. 11, Issue 3, EMS, 2014.
  4. Effect of refraction on dose reconstruction in optical-CT gel dosimetry, (with L. Florescu and C. Wuu) Journal of Physics: Conference Series vol. 444 (2013), 012063.
  5. Approximate inversion algorithm of the elliptical Radon transform, (with R. Gouia-Zarrad) Proceedings of ISMA 2012, Biomedical Applications (2012), ISBN 978-1-4673-0862-5.
  6. Tomographic reconstruction of nodular images from incomplete data, (with M. Xie) AIP Conference Proceedings, vol. 1301, issue 1, December 2010, pp 167-174.
  7. Reconstruction algorithms for interior and exterior spherical Radon transform-based ultrasound imaging, (with R. Vaidyanathan, M. Lewis, and T. Aktosun) Proceedings of SPIE, vol. 7265, Medical Imaging 2009: Ultrasonic Imaging and Signal Processing, 2009, 72651I 1-8.
  8. Thermoacoustic tomography: numerical results, (with S. K. Patch) Proceedings of SPIE, vol. 6437, Progress in Biomedical Optics and Imaging, 8 (2007), no. 14, pp 6437-47, preprint at arXiv:math.NA/0510638.

Book Chapters

  1. V-line and conical Radon transforms with applications in imaging, Chapter 7 in The Radon Transform: The First 100 Years and Beyond, edited by R. Ramlau and O. Scherzer, Radon Series on Computational and Applied Mathematics, De Gruyter, 2019.
  2. Microlocal analysis and imaging, (with R. Felea, V. Krishnan, C. Nolan, and E. T. Quinto) in Mathematics of Planet Earth, edited by H. Kaper and C. Rousseau, SIAM, 2015.
  3. Integral geometry and mathematical problems of image reconstruction, A chapter in Mathematical Models, Methods and Applications, edited by A.H. Siddiqi, P. Manchanda, and R. Bhardwaj, Industrial and Applied Mathematics Series, Springer, 2015.
  4. Limited view thermoacoustic tomography, (with Y. Xu, L. Wang, and P. Kuchment) Chapter 9 in Photoacoustic imaging and spectroscopy, edited by L. H. Wang, CRC Press, 2009.

Media Coverage

  1. "New book focuses on recent developments in mathematics of medical imaging", UTA College of Science News, May 2023.
  2. "Kicking Off Science Week with UTA Math's Gaik Ambartsoumian and Theresa Jorgensen", Voices Live Podcast, April 2022.
  3. "New math theory for improvement of imaging technology", UTA College of Science News, Feb. 2017.
  4. "Sharper image", Maverick Science Magazine, 2013.

Grant Support

  1. Principal Investigator, $385,127. Subaward of the following grant: "Development of the human dynamic neurochemical connectome scanner", PI: Ciprian Catana, Massachusetts General Hospital & Harvard Medical School $4,682,920. NIBIB/NINDS U01-EB029826, BRAIN Initiative: Development of Next Generation Human Brain Imaging Tools & Technologies, National Institutes of Health (NIH), 09/2020-05/2024.
  2. Principal Investigator (with Co-PI Venkateswaran Krishnan), $197,628. "Conical Radon transforms and their applications in tomography", NSF DMS-1616564, National Science Foundation, Division of Mathematical Sciences, 09/2016-08/2020.
  3. Principal Investigator, $35,000. "Integral geometric problems in tomography", # 360357, Mathematics and Physical Sciences-Collaboration Grants for Mathematicians, Simons Foundation, 09/2015-08/2020, Stopped in 2016 due to the reception of the NSF DMS-1616564 grant.
  4. Principal Investigator (with Co-PI Venkateswaran Krishnan), $175,899. "Elliptical Radon transforms in image reconstruction", NSF DMS-1109417, National Science Foundation, Division of Mathematical Sciences, 08/2011-07/2015.
  5. Principal Investigator (with Co-PIs Wei Qian and Daniel Terreros), $150,000. "Image reconstruction problems in tomosynthesis", NHARP 003656-0109-2009, Norman Hackerman Advanced Research Program Consortium Grant, The Texas Higher Education Coordinating Board, 08/2010-05/2013.
  6. Co-PI (with PI Matthew Lewis and Co-PI Tuncay Aktosun), $770,818. "Acoustic inverse scattering for breast microcalcification detection", BC063989, Department of Defense Medical Research Program Synergistic Idea Award, 01/2008-08/2011.
  7. Co-PI (with PI Jianzhong Su, Co-PIs Xin Lu and Hua Shan), $25,000. "AIMS Seventh International Conference on Dynamical Systems and Differential Equations, NSF DMS-0738356, National Science Foundation, 04/2008-03/2009.

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