TY - JOUR

T1 - Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data

AU - Do, N.

AU - Kunyansky, L.

N1 - Funding Information:
The authors would like to thank the anonymous referees for the numerous suggestions that helped to improve the manuscript. The second author is grateful for the partial support by the NSF through the awards NSF/DMS-1211521 and NSF/DMS-1418772.
Publisher Copyright:
© 2018 IOP Publishing Ltd.

PY - 2018/7/16

Y1 - 2018/7/16

N2 - We investigate the inverse source problem for the wave equation, arising in photo- and thermoacoustic tomography. There exist quite a few theoretically exact inversion formulas explicitly expressing the solution of this problem in terms of the measured data, under the assumption of the constant and known speed of sound. However, almost all of these formulas require data to be measured either on an unbounded surface, or on a closed surface completely surrounding the object. This is too restrictive for practical applications. The alternative approach we present, under certain restriction on geometry, yields a theoretically exact reconstruction of the standard Radon projections of the source from the data measured on a finite open surface. In addition, this technique reduces the time interval where the data should be known. In general, our method requires a pre-computation of densities of certain single-layer potentials. However, in the case of a truncated circular or spherical acquisition surface, these densities are easily obtained analytically, which leads to fully explicit asymptotically fast algorithms. We test these algorithms in a series of numerical simulations.

AB - We investigate the inverse source problem for the wave equation, arising in photo- and thermoacoustic tomography. There exist quite a few theoretically exact inversion formulas explicitly expressing the solution of this problem in terms of the measured data, under the assumption of the constant and known speed of sound. However, almost all of these formulas require data to be measured either on an unbounded surface, or on a closed surface completely surrounding the object. This is too restrictive for practical applications. The alternative approach we present, under certain restriction on geometry, yields a theoretically exact reconstruction of the standard Radon projections of the source from the data measured on a finite open surface. In addition, this technique reduces the time interval where the data should be known. In general, our method requires a pre-computation of densities of certain single-layer potentials. However, in the case of a truncated circular or spherical acquisition surface, these densities are easily obtained analytically, which leads to fully explicit asymptotically fast algorithms. We test these algorithms in a series of numerical simulations.

KW - explicit inversion formula

KW - photoacoustic tomography

KW - reduced data

KW - spherical means

KW - thermoacoustic tomography

KW - wave equation

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U2 - 10.1088/1361-6420/aacfac

DO - 10.1088/1361-6420/aacfac

M3 - Article

AN - SCOPUS:85051200369

SN - 0266-5611

VL - 34

JO - Inverse Problems

JF - Inverse Problems

IS - 9

M1 - 094004

ER -