### 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.

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

Journal | Inverse Problems |

Volume | 23 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 2007 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**A series solution and a fast algorithm for the inversion of the spherical mean Radon transform.** / Kunyansky, Leonid.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - A series solution and a fast algorithm for the inversion of the spherical mean Radon transform

AU - Kunyansky, Leonid

PY - 2007/12/1

Y1 - 2007/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=36749062069&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36749062069&partnerID=8YFLogxK

U2 - 10.1088/0266-5611/23/6/S02

DO - 10.1088/0266-5611/23/6/S02

M3 - Article

AN - SCOPUS:36749062069

VL - 23

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 6

ER -