### Abstract

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.

Original language | English (US) |
---|---|

Article number | 094004 |

Journal | Inverse Problems |

Volume | 34 |

Issue number | 9 |

DOIs | |

State | Published - Jul 16 2018 |

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### Keywords

- explicit inversion formula
- photoacoustic tomography
- reduced data
- spherical means
- thermoacoustic tomography
- wave equation

### ASJC Scopus subject areas

- Theoretical Computer Science
- Signal Processing
- Mathematical Physics
- Computer Science Applications
- Applied Mathematics

### Cite this

**Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data.** / Do, N.; Kunyansky, Leonid.

Research output: Contribution to journal › Article

*Inverse Problems*, vol. 34, no. 9, 094004. https://doi.org/10.1088/1361-6420/aacfac

}

TY - JOUR

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

AU - Do, N.

AU - Kunyansky, Leonid

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

UR - http://www.scopus.com/inward/record.url?scp=85051200369&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85051200369&partnerID=8YFLogxK

U2 - 10.1088/1361-6420/aacfac

DO - 10.1088/1361-6420/aacfac

M3 - Article

AN - SCOPUS:85051200369

VL - 34

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 9

M1 - 094004

ER -