# American Institute of Mathematical Sciences

May  2014, 8(2): 339-359. doi: 10.3934/ipi.2014.8.339

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

 1 Department of Mathematics, University of Texas, Arlington, TX 76019, United States 2 Department of Mathematics, University of Arizona, 617 N Santa Rita Ave, Tucson, AZ 85721

Received  August 2013 Revised  December 2013 Published  May 2014

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.
Citation: Gaik Ambartsoumian, Leonid Kunyansky. Exterior/interior problem for the circular means transform with applications to intravascular imaging. Inverse Problems & Imaging, 2014, 8 (2) : 339-359. doi: 10.3934/ipi.2014.8.339
##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,, National Bureau of Standards Applied Mathematics Series, (1964). doi: 10.1119/1.1972842. Google Scholar [2] G. Ambartsoumian, R. Gouia-Zarrad and M. Lewis, Inversion of the circular Radon transform on an annulus,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/10/105015. Google Scholar [3] G. Ambartsoumian and P. Kuchment, A range description for the planar circular Radon transform,, SIAM J. Math. Anal., 38 (2006), 681. doi: 10.1137/050637492. Google Scholar [4] P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier and G. Paltauf, Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors,, Inverse Problems, 23 (2007). doi: 10.1088/0266-5611/23/6/S06. Google Scholar [5] M. Cheney and B. Borden, Fundamentals of Radar Imaging,, Society for Industrial and Applied Mathematics (SIAM), (2009). doi: 10.1137/1.9780898719291. Google Scholar [6] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer-Verlag, (1992). Google Scholar [7] S. Emelianov, B. Wang, J. Su, A. Karpiouk, E. Yantsen, K. Sokolov, J. Amirian, R. Smalling and S. Sethuraman, Intravascular ultrasound and photoacoustic imaging,, Proceedings of the 30-th Annual International IEEE EMBS Conference, (2008), 2. doi: 10.1109/IEMBS.2008.4649075. Google Scholar [8] D. Finch and Rakesh, The spherical mean value operator with centers on a sphere,, Inverse Problems, 23 (2007). doi: 10.1088/0266-5611/23/6/S04. Google Scholar [9] M. Haltmeier, O. Scherzer, P. Burgholzer, R. Nuster and G. Paltauf, Thermoacoustic tomography and the circular Radon transform: Exact inversion formula,, Mathematical Models and Methods in Applied Sciences, 17 (2007), 635. doi: 10.1142/S0218202507002054. Google Scholar [10] P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic and photoacoustic tomography,, European J. Appl. Math., 19 (2008), 191. doi: 10.1017/S0956792508007353. Google Scholar [11] L. Kunyansky, Inversion of the 3D exponential parallel-beam transform and the Radon transform with angle-dependent attenuation,, Inverse Problems 20 (2004), 20 (2004), 1455. doi: 10.1088/0266-5611/20/5/008. Google Scholar [12] L. Kunyansky, Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries,, Inverse Problems and Imaging, 6 (2012), 111. doi: 10.3934/ipi.2012.6.111. Google Scholar [13] A. Louis and E. T. Quinto, Local tomographic methods in Sonar,, in Surveys on solution methods for inverse problems, (2000), 147. Google Scholar [14] L. V. Nguyen, On singularities and instability of reconstruction in thermoacoustic tomography,, in Tomography and inverse transport theory, 559 (2011), 163. doi: 10.1090/conm/559/11078. Google Scholar [15] S. J. Norton, Reconstruction of a two-dimensional reflecting medium over a circular domain: exact solution,, J. Acoust. Soc. Am., 67 (1980), 1266. doi: 10.1121/1.384168. Google Scholar [16] S. Sethuraman, S. R. Aglyamov, J. H. Amirian, R. W. Smalling and S. Y. Emelianov, Development of a combined intravascular ultrasound and photoacoustic imaging system,, Photon Plus Ultrasound: Imaging and Sensing 2006, 6086 (2006). Google Scholar [17] S. Sethuraman, J. H. Amirian, S. H. Litovsky, R. W. Smalling and S. Y. Emelianov, Spectroscopic intravascular photoacoustic imaging to differentiate atherosclerotic plaques,, Optics Express, 16 (2008), 3362. doi: 10.1364/OE.16.003362. Google Scholar [18] Sethuraman, S. Mallidi, S. R. Aglyamov, J. H. Amirian, S. Litovsky, R. W. Smalling and S. Y. Emelianov, Intravascular photoacoustic imaging of atherosclerotic plaques: ex vivo study using a rabbit model of atherosclerosis,, Photon Plus Ultrasound: Imaging and Sensing 2007, 66437 (2007). Google Scholar [19] V. S. Vladimirov, Equations of Mathematical Physics,, (Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey.) Pure and Applied Mathematics, 3 (1971). Google Scholar [20] B. Wang, J. L. Su, J. Amirian, S. H. Litovsky, R. Smalling and S. Emelianov, Detection of lipid in atherosclerotic vessels using ultrasound-guided spectroscopic intravascular photoacoustic imaging,, Optics Express, 18 (2010), 4889. doi: 10.1364/OE.18.004889. Google Scholar [21] Y. Xu, L.-H. Wang, G. Ambartsoumian and P. Kuchment, Limited view thermoacoustic tomography,, Photoacoustic imaging and spectroscopy, (2009), 61. doi: 10.1201/9781420059922.ch6. Google Scholar

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##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,, National Bureau of Standards Applied Mathematics Series, (1964). doi: 10.1119/1.1972842. Google Scholar [2] G. Ambartsoumian, R. Gouia-Zarrad and M. Lewis, Inversion of the circular Radon transform on an annulus,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/10/105015. Google Scholar [3] G. Ambartsoumian and P. Kuchment, A range description for the planar circular Radon transform,, SIAM J. Math. Anal., 38 (2006), 681. doi: 10.1137/050637492. Google Scholar [4] P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier and G. Paltauf, Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors,, Inverse Problems, 23 (2007). doi: 10.1088/0266-5611/23/6/S06. Google Scholar [5] M. Cheney and B. Borden, Fundamentals of Radar Imaging,, Society for Industrial and Applied Mathematics (SIAM), (2009). doi: 10.1137/1.9780898719291. Google Scholar [6] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer-Verlag, (1992). Google Scholar [7] S. Emelianov, B. Wang, J. Su, A. Karpiouk, E. Yantsen, K. Sokolov, J. Amirian, R. Smalling and S. Sethuraman, Intravascular ultrasound and photoacoustic imaging,, Proceedings of the 30-th Annual International IEEE EMBS Conference, (2008), 2. doi: 10.1109/IEMBS.2008.4649075. Google Scholar [8] D. Finch and Rakesh, The spherical mean value operator with centers on a sphere,, Inverse Problems, 23 (2007). doi: 10.1088/0266-5611/23/6/S04. Google Scholar [9] M. Haltmeier, O. Scherzer, P. Burgholzer, R. Nuster and G. Paltauf, Thermoacoustic tomography and the circular Radon transform: Exact inversion formula,, Mathematical Models and Methods in Applied Sciences, 17 (2007), 635. doi: 10.1142/S0218202507002054. Google Scholar [10] P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic and photoacoustic tomography,, European J. Appl. Math., 19 (2008), 191. doi: 10.1017/S0956792508007353. Google Scholar [11] L. Kunyansky, Inversion of the 3D exponential parallel-beam transform and the Radon transform with angle-dependent attenuation,, Inverse Problems 20 (2004), 20 (2004), 1455. doi: 10.1088/0266-5611/20/5/008. Google Scholar [12] L. Kunyansky, Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries,, Inverse Problems and Imaging, 6 (2012), 111. doi: 10.3934/ipi.2012.6.111. Google Scholar [13] A. Louis and E. T. Quinto, Local tomographic methods in Sonar,, in Surveys on solution methods for inverse problems, (2000), 147. Google Scholar [14] L. V. Nguyen, On singularities and instability of reconstruction in thermoacoustic tomography,, in Tomography and inverse transport theory, 559 (2011), 163. doi: 10.1090/conm/559/11078. Google Scholar [15] S. J. Norton, Reconstruction of a two-dimensional reflecting medium over a circular domain: exact solution,, J. Acoust. Soc. Am., 67 (1980), 1266. doi: 10.1121/1.384168. Google Scholar [16] S. Sethuraman, S. R. Aglyamov, J. H. Amirian, R. W. Smalling and S. Y. Emelianov, Development of a combined intravascular ultrasound and photoacoustic imaging system,, Photon Plus Ultrasound: Imaging and Sensing 2006, 6086 (2006). Google Scholar [17] S. Sethuraman, J. H. Amirian, S. H. Litovsky, R. W. Smalling and S. Y. Emelianov, Spectroscopic intravascular photoacoustic imaging to differentiate atherosclerotic plaques,, Optics Express, 16 (2008), 3362. doi: 10.1364/OE.16.003362. Google Scholar [18] Sethuraman, S. Mallidi, S. R. Aglyamov, J. H. Amirian, S. Litovsky, R. W. Smalling and S. Y. Emelianov, Intravascular photoacoustic imaging of atherosclerotic plaques: ex vivo study using a rabbit model of atherosclerosis,, Photon Plus Ultrasound: Imaging and Sensing 2007, 66437 (2007). Google Scholar [19] V. S. Vladimirov, Equations of Mathematical Physics,, (Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey.) Pure and Applied Mathematics, 3 (1971). Google Scholar [20] B. Wang, J. L. Su, J. Amirian, S. H. Litovsky, R. Smalling and S. Emelianov, Detection of lipid in atherosclerotic vessels using ultrasound-guided spectroscopic intravascular photoacoustic imaging,, Optics Express, 18 (2010), 4889. doi: 10.1364/OE.18.004889. Google Scholar [21] Y. Xu, L.-H. Wang, G. Ambartsoumian and P. Kuchment, Limited view thermoacoustic tomography,, Photoacoustic imaging and spectroscopy, (2009), 61. doi: 10.1201/9781420059922.ch6. Google Scholar
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