Inversion of weighted divergent beam and cone transforms

In this paper, we investigate the relations between the Radon and weighted divergent beam and cone transforms. Novel inversion formulas are derived for the latter two. The weighted cone transform arises, for instance, in image reconstruction from the data obtained by Compton cameras, which have promising applications in various fields, including biomedical and homeland security imaging and gamma ray astronomy. The inversion formulas are applicable for a wide variety of detector geometries in any dimension. The results of numerical implementation of some of the formulas in dimensions two and three are also provided.

[1]  Alexander Katsevich,et al.  Theoretically Exact Filtered Backprojection-Type Inversion Algorithm for Spiral CT , 2002, SIAM J. Appl. Math..

[2]  Guido Kanschat,et al.  Detecting small low emission radiating sources , 2010, 1012.3373.

[3]  John C. Schotland,et al.  Inversion formulas for the broken-ray Radon transform , 2010, Inverse Problems.

[4]  Peter Kuchment,et al.  The Radon Transform and Medical Imaging , 2014, CBMS-NSF regional conference series in applied mathematics.

[5]  Markus Haltmeier,et al.  Analytic Inversion of a Conical Radon Transform Arising in Application of Compton Cameras on the Cylinder , 2016, SIAM J. Imaging Sci..

[6]  Chang-Yeol Jung,et al.  Exact Inversion of the Cone Transform Arising in an Application of a Compton Camera Consisting of Line Detectors , 2016, SIAM J. Imaging Sci..

[7]  Yulia Nekova Georgieva-Hristova.  Mathematical problems of thermoacoustic and Compton camera imaging , 2010 .

[8]  Markus Haltmeier.  Exact reconstruction formulas for a Radon transform over cones , 2014 .

[9]  Bruce D. Smith.  Image Reconstruction from Cone-Beam Projections: Necessary and Sufficient Conditions and Reconstruction Methods , 1985, IEEE Transactions on Medical Imaging.

[10]  G. Shilov,et al.  Generalized Functions, Volume 1: Properties and Operations , 1967 .

[11]  Sunghwan Moon,et al.  Inversion of the conical Radon transform with vertices on a surface of revolution arising in an application of a Compton camera , 2017 .

[12]  Fatma Terzioglu,et al.  Three-Dimensional Image Reconstruction from Compton Camera Data , 2016, SIAM J. Imaging Sci..

[13]  Philip J. Bones,et al.  Towards direct reconstruction from a gamma camera based on Compton scattering , 1994, IEEE Trans. Medical Imaging.

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

[15]  Voichita Maxim.  Filtered Backprojection Reconstruction and Redundancy in Compton Camera Imaging , 2014, IEEE Transactions on Image Processing.

[16]  Donald C. Solmon,et al.  A Characterization of the Range of the Divergent Beam x-Ray Transform , 1983 .

[17]  J. M. Nightingale,et al.  Gamma-radiation imaging system based on the Compton effect , 1977 .

[18]  S. Helgason.  Integral Geometry and Radon Transforms , 2010 .

[19]  P. Grangeat.  Mathematical framework of cone beam 3D reconstruction via the first derivative of the radon transform , 1991 .

[20]  Mai K. Nguyen,et al.  Radon transforms on a class of cones with fixed axis direction , 2005 .

[21]  Fatma Terzioglu,et al.  Some inversion formulas for the cone transform , 2015, 1504.00344.

[22]  Manbir Singh,et al.  An electronically collimated gamma camera for single photon emission computed tomography. Part I: Theoretical considerations and design criteria , 1983 .

[23]  Frank Natterer,et al.  Mathematical methods in image reconstruction , 2001, SIAM monographs on mathematical modeling and computation.

[24]  H. Tuy.  AN INVERSION FORMULA FOR CONE-BEAM RECONSTRUCTION* , 1983 .

[25]  Moritz Allmaras,et al.  Passive Detection of Small Low-Emission Sources: Two-Dimensional Numerical Case Studies , 2016 .

[26]  Gaik Ambartsoumian,et al.  Inversion of the V-line Radon transform in a disc and its applications in imaging , 2012, Comput. Math. Appl..

[27]  Sunghwan Moon,et al.  A series formula for inversion of the V-line Radon transform in a disc , 2013, Comput. Math. Appl..

[28]  Alexander Katsevich,et al.  An improved exact filtered backprojection algorithm for spiral computed tomography , 2004, Adv. Appl. Math..

[29]  Per-Olof Persson,et al.  A Simple Mesh Generator in MATLAB , 2004, SIAM Rev..

[30]  Bruce Smith,et al.  Reconstruction methods and completeness conditions for two Compton data models. , 2005, Journal of the Optical Society of America. A, Optics, image science, and vision.

[31]  G T Gullberg,et al.  Application of spherical harmonics to image reconstruction for the Compton camera. , 1998, Physics in medicine and biology.

[32]  S. G. Gindikin,et al.  Selected Topics in Integral Geometry , 2003 .

[33]  Bruce D. Smith.  Line-reconstruction from Compton cameras: data sets and a camera design , 2011 .

[34]  Mai K. Nguyen,et al.  New properties of the V-line Radon transform and their imaging applications , 2015 .

[35]  Kennan T. Smith,et al.  The divergent beam x-ray transform , 1980 .

[36]  Gaik Ambartsoumian,et al.  Exact inversion of the conical Radon transform with a fixed opening angle , 2013, 1309.6581.

[37]  R. Carroll,et al.  A Bayesian approach to the detection of small low emission sources , 2011, Inverse problems.

[38]  M. Singh,et al.  An electronically collimated gamma camera for single photon emission computed tomography. Part I: Theoretical considerations and design criteria. , 1983, Medical physics.

[39]  Markus Haltmeier,et al.  The Radon Transform over Cones with Vertices on the Sphere and Orthogonal Axes , 2016, SIAM J. Appl. Math..

[40]  I. Gel'fand,et al.  Generalized Functions, Volume 4 , 2016 .