May  2012, 6(2): 315-320. doi: 10.3934/ipi.2012.6.315

Photo-acoustic inversion in convex domains

1. 

University of Münster, Department of Mathematics and Computer Science, Einsteinstrasse 72, 48159 Münster

Received  April 2011 Revised  April 2012 Published  May 2012

In photo-acoustics one has to reconstruct a function from its averages over spheres around points on the measurement surface. For special surfaces inversion formulas are known. In this paper we derive a formula for surfaces that bound smooth convex domains. It reconstructs the function modulo a smoothing integral operator. For special surfaces the integral operator vanishes, providing exact reconstruction.
Citation: Frank Natterer. Photo-acoustic inversion in convex domains. Inverse Problems & Imaging, 2012, 6 (2) : 315-320. doi: 10.3934/ipi.2012.6.315
References:
[1]

D. Finch, S. K. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. An., 35 (2004), 1213-1240.  Google Scholar

[2]

L. Kunyansky, Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra, Inverse Problems, 27 (2011), 025012. doi: 10.1088/0266-5611/27/2/025012.  Google Scholar

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F. Natterer, "The Mathematics of Computerized Tomography," Wiley-Teubner 1986, reprinted as Classics in Applied Mathematics, 32, SIAM, 2001.  Google Scholar

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F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309.  Google Scholar

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V. Palamodov, "Reconstructive Integral Geometry," Birkhäuser, 2004.  Google Scholar

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V. Palamodov, A uniform reconstruction formula in integral geometry, Inverse Problems, 17 (2012), 065014. Google Scholar

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E. T. Quinto, A. Rieder and T. Schuster, Local inversion of the sonar transform regularized by the approximate inverse, Inverse Problems, 27 (2011), 035006. doi: 10.1088/0266-5611/27/3/035006.  Google Scholar

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show all references

References:
[1]

D. Finch, S. K. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. An., 35 (2004), 1213-1240.  Google Scholar

[2]

L. Kunyansky, Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra, Inverse Problems, 27 (2011), 025012. doi: 10.1088/0266-5611/27/2/025012.  Google Scholar

[3]

F. Natterer, "The Mathematics of Computerized Tomography," Wiley-Teubner 1986, reprinted as Classics in Applied Mathematics, 32, SIAM, 2001.  Google Scholar

[4]

F. Natterer, Inversion of the attenuated Radon transform, Inverse Problems, 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309.  Google Scholar

[5]

V. Palamodov, "Reconstructive Integral Geometry," Birkhäuser, 2004.  Google Scholar

[6]

V. Palamodov, A uniform reconstruction formula in integral geometry, Inverse Problems, 17 (2012), 065014. Google Scholar

[7]

E. T. Quinto, A. Rieder and T. Schuster, Local inversion of the sonar transform regularized by the approximate inverse, Inverse Problems, 27 (2011), 035006. doi: 10.1088/0266-5611/27/3/035006.  Google Scholar

[8]

M. Xu and L. V. Wang, Universal back-projection algorithm for photoacoustic computed tomography, Physical Review, E 71 (2005), 016706. doi: 10.1103/PhysRevE.71.016706.  Google Scholar

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