Inversion Formulas for a Cylindrical Radon Transform

In this paper we study the inversion of a generalized Radon transform that maps a function in three dimensional space to a family of cylindrical integrals. We derive local backprojection-type inversion formulas for this cylindrical Radon transform. Our inversion formulas can be implemented in a straightforward manner with $\mathcal{O}(\mathtt{N}^{4/3})$ floating point operations, where $\mathtt{N}$ is the number of spatial reconstruction points. Numerical results are presented which demonstrate the accuracy and validity of the derived algorithms.

[1]  Lihong V. Wang,et al.  Reconstructions in limited-view thermoacoustic tomography. , 2004, Medical physics.

[2]  Markus Haltmeier,et al.  Inversion of Spherical Means and the Wave Equation in Even Dimensions , 2007, SIAM J. Appl. Math..

[3]  Peter Kuchment,et al.  On the injectivity of the circular Radon transform , 2005 .

[4]  Otmar Scherzer,et al.  THERMOACOUSTIC TOMOGRAPHY AND THE CIRCULAR RADON TRANSFORM: EXACT INVERSION FORMULA , 2007 .

[5]  Avinash C. Kak,et al.  Principles of computerized tomographic imaging , 2001, Classics in applied mathematics.

[6]  Markus Haltmeier.  FREQUENCY DOMAIN RECONSTRUCTION FOR PHOTO- AND THERMOACOUSTIC TOMOGRAPHY WITH LINE DETECTORS , 2006 .

[7]  Otmar Scherzer,et al.  A Reconstruction Algorithm for Photoacoustic Imaging Based on the Nonuniform FFT , 2009, IEEE Transactions on Medical Imaging.

[8]  Rakesh,et al.  Spherical means with centers on a hyperplane in even dimensions , 2009, 0911.4582.

[9]  Lihong V. Wang,et al.  Universal back-projection algorithm for photoacoustic computed tomography. , 2005 .

[10]  Leonid Kunyansky.  A series solution and a fast algorithm for the inversion of the spherical mean Radon transform , 2007 .

[11]  Eric Todd Quinto,et al.  Injectivity Sets for the Radon Transform over Circles and Complete Systems of Radial Functions , 1996 .

[12]  Markus Haltmeier,et al.  Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors , 2007 .

[13]  P. Burgholzer,et al.  Thermoacoustic tomography with integrating area and line detectors , 2005, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[14]  Linh V. Nguyen,et al.  Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media , 2008 .

[15]  M. Haltmeier,et al.  Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors , 2007 .

[16]  Stephen J. Norton,et al.  Reconstruction of a two‐dimensional reflecting medium over a circular domain: Exact solution , 1980 .

[17]  M. Haltmeier,et al.  Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  E. T. Quinto,et al.  Local Tomographic Methods in Sonar , 2000 .

[19]  Rakesh,et al.  Determining a Function from Its Mean Values Over a Family of Spheres , 2004, SIAM J. Math. Anal..

[20]  P. Burgholzer,et al.  Photoacoustic tomography using a fiber based Fabry-Perot interferometer as an integrating line detector and image reconstruction by model-based time reversal method , 2007, European Conference on Biomedical Optics.

[21]  Günther Paltauf,et al.  Photoacoustic microtomography using optical interferometric detection. , 2010, Journal of biomedical optics.

[22]  Leonid Kunyansky,et al.  Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra , 2010, 1009.0288.

[23]  Peter Kuchment,et al.  Mathematics of thermoacoustic and photoacoustic tomography , 2007 .

[24]  S. Helgason.  The Radon Transform , 1980 .

[25]  John A. Fawcett,et al.  Inversion of N-dimensional spherical averages , 1985 .

[26]  Otmar Scherzer,et al.  Filtered backprojection for thermoacoustic computed tomography in spherical geometry , 2005, Mathematical Methods in the Applied Sciences.

[27]  Minghua Xu,et al.  Exact frequency-domain reconstruction for thermoacoustic tomography. II. Cylindrical geometry , 2002, IEEE Transactions on Medical Imaging.

[28]  Aleksei Beltukov,et al.  Inversion of the Spherical Mean Transform with Sources on a Hyperplane , 2009, 0910.1380.

[29]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[30]  Yuan Xu,et al.  Exact frequency-domain reconstruction for thermoacoustic tomography. I. Planar geometry , 2002, IEEE Transactions on Medical Imaging.

[31]  L. Andersson.  On the determination of a function from spherical averages , 1988 .

[32]  Leonid Kunyansky,et al.  Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries , 2011, 1102.1413.

[33]  Jens Klein.  Inverting the spherical Radon transform for physically meaningful functions , 2003 .

[34]  K. P. Köstli,et al.  Two-dimensional photoacoustic imaging by use of Fourier-transform image reconstruction and a detector with an anisotropic response. , 2003, Applied optics.

[35]  Xu Xiao.  Photoacoustic imaging in biomedicine , 2008 .

[36]  L. Kunyansky,et al.  Explicit inversion formulae for the spherical mean Radon transform , 2006, math/0609341.