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1.  Siemens AG, Corporate Technology, OttoHahnRing 6, 81739 Munich, Germany 
2.  The University of Texas Medical Branch, 301 UniversityBoulevard, Galveston, TX 77555, United States 
3.  Institute for Mathematics and its Applications, University of Minnesota, 400 Lind Hall, Minneapolis, MN 55455, United States 
4.  Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX 77843, United States, United States 
References:
[1] 
R. Basko, G. L. Zeng and G. T. Gullberg, Analytical reconstruction formula for onedimensional compton camera, IEEE Transactions on Nuclear Science, 44 (1997), 13421346. 
[2] 
R. Basko, G. L. Zeng and G. T. Gullberg, Application of spherical harmonics to image reconstruction for the compton camera, Physics in Medicine and Biology, 43 (1998), 887894. 
[3] 
R. R. Brechner and M. Singh, Iterative reconstruction of electronically collimated spect images, IEEE Transactions on Nuclear Science, 37 (1990), 13281332. 
[4] 
T. F. Budinger, G. T. Gullberg and R. H. Huseman, Emission computed tomography, in "Image Reconstruction from Projections'' (ed. G. Herman), Springer Verlag, (1979), 147246. 
[5]  
[6] 
N. H. Clinthorne, ChorYi Ng, ChiaHo Hua, J. E. Gormley, J. W. LeBlanc, S. J. Wilderman and W. L. Rogers, Theoretical performance comparison of a comptonscatter aperture and parallelhole collimator, in "Nuclear Science Symposium, Conference Record," 2, IEEE, (1996), 788792. 
[7] 
S. Coles, "An Introduction to Statistical Modeling of Extreme Values," Springer Series in Statistics, SpringerVerlag London, Ltd., London, 2001. 
[8] 
M. J. Cree and P. J. Bones, Towards direct reconstruction from a gamma camera based on compton scattering, IEEE Transactions on Medical Imaging, 13 (1994), 398407. 
[9] 
Y. F. Du, Z. He, G. F. Knoll, D. K. Wehe and W. Li, Evaluation of a compton scattering camera using 3D position sensitive CdZnTe detectors, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 457 (2001), 203211. 
[10] 
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. 
[11] 
A. Faridani, E. L. Ritman and K. T. Smith, Local tomography, SIAM Journal on Applied Mathematics, 52 (1992), 459484. doi: 10.1137/0152026. 
[12] 
T. Fawcett, An introduction to ROC analysis, Pattern Recognition Letters, 27 (2006), 861874. 
[13] 
G. T. Gullberg, G. L. Zeng and R. Basko, Image reconstruction from vprojections acquired by compton camera, US Patent 5841141, 1998. 
[14] 
D. L. Gunter, Filtered backprojection algorithm for compton telescopes, US Patent 7345283, 2008. 
[15] 
T. Hebert, R. Leahy and M. Singh, Threedimensional maximumlikelihood reconstruction for an electronically collimated singlephotonemission, Journal of the Optical Society of America A, 7 (1990), 13051313. 
[16] 
W. H. Hill and K. L. Matthews, Experimental verification of a hand held electronicallycollimated radiation detector, in "Nuclear Science Symposium Conference Record, 2007," IEEE, 5 (2007), 37923797. 
[17] 
M. Hirasawa and T. Tomitani, An analytical image reconstruction algorithm to compensate for scattering angle broadening in compton cameras, Physics in Medicine and Biology, 48 (2003), 10091026. 
[18] 
Y. Hristova, "Mathematical Problems of Thermoacoustic and Compton Camera Imaging," Ph.D thesis, Texas A&M University, 2010. 
[19] 
A. C. Kak and M. Slaney, "Principles of Computerized Tomographic Imaging," IEEE Press, New York, 1988. 
[20] 
P. Kuchment, Generalized transforms of radon type and their applications, in "The Radon Transform, Inverse Problems, and Tomography" (eds. G. Olafsson and E. T. Quinto), Proc. Sympos. Appl. Math., 63, American Mathematical Society, Providence, RI, (2006), 6791. 
[21] 
P. Kuchment, K. Lancaster and L. Mogilevskaya, On local tomography, Inverse Problems, 11 (1995), 571589. 
[22] 
A. W. Lackie, K. L. Matthews, B. M. Smith, W. Hill, WeiHsung Wang and M. L. Cherry, A directional algorithm for an electronicallycollimated gammaray detector, in "Nuclear Science Symposium Conference Record, 2006," 1, IEEE, (2006), 264269. 
[23] 
J. W. LeBlanc, N. H. Clinthorne, C.H. Hua, E. Nygard, W. L. Rogers, D. K. Wehe, P. Weilhammer and S. J. Wilderman, Csprint: A prototype compton camera system for low energy gamma ray imaging, in "Nuclear Science Symposium, 1997," 1, IEEE, (1997), 357361. 
[24] 
C. M. Marianno, D. R. Boyle, W. S. Charlton, G. M. Gaukler and A. Veditz, A guide for detector development and deployment, in "51st Annual Meeting of the Institute of Nuclear Materials Management 2010," Curran Associates, Inc., (2010), 39013909. 
[25] 
V. Maxim, M. Frandeş and R. Prost, Analytical inversion of the Compton transform using the full set of available projections, Inverse Problems, 25 (2009), 095001, 21 pp. doi: 10.1088/02665611/25/9/095001. 
[26] 
F. Natterer, "The Mathematics of Computerized Tomography," Society for Industrial and Applied Mathematics, 2001. doi: 10.1137/1.9780898719284. 
[27] 
F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," Reprint of the 1986 original, Classics in Applied Mathematics, 32, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. 
[28] 
M. K. Nguyen, T. T. Truong and P. Grangeat, Radon transforms on a class of cones with fixed axis direction, Journal of Physics A: Mathematical and General, 38 (2005), 80038015. doi: 10.1088/03054470/38/37/006. 
[29] 
L. C. Parra, Reconstruction of conebeam projections from compton scattered data, IEEE Transactions on Nuclear Science, 47 (2000), 15431550. 
[30] 
R. C. Rohe, M. M. Sharfi, K. A. Kecevar, J. D. Valentine and C. Bonnerave, The spatiallyvariant backprojection point kernel function of an energysubtraction compton scatter camera for medical imaging, IEEE Transactions on Nuclear Science, 44 (1997), 24772482. 
[31] 
G. J. Royle and R. D. Speller, A flexible geometry compton camera for industrial gamma ray imaging, in "Nuclear Science Symposium, 1996. Conference Record.," 2, IEEE, (1996), 821824. 
[32] 
G. J. Royle and R. D. Speller, Compton scatter imaging of a nuclear industry site, in "Nuclear Science Symposium, 1997," 1, IEEE, (1997), 365368. 
[33] 
A. C. Sauve, A. O. III Hero, W. L. Rogers, S. J. Wilderman and N. H. Clinthorne, 3D image reconstruction for a compton spect camera model, IEEE Transactions on Nuclear Science, 46 (1999), 20752084. 
[34] 
V. Schoenfelder, H. Aarts, K. Bennett, H. de Boer, J. Clear, W. Collmar, A. Connors, A. Deerenberg, R. Diehl, A. von Dordrecht, J. W. den Herder, W. Hermsen, M. Kippen, L. Kuiper, G. Lichti, J. Lockwood, J. Macri, M. McConnell, D. Morris, R. Much, J. Ryan, G. Simpson, M. Snelling, G. Stacy, H. Steinle, A. Strong, B. N. Swanenburg, B. Taylor, C. de Vries and C. Winkler, Instrument description and performance of the imaging gammaray telescope comptel aboard the compton gammaray observatory, Astrophysical Journal Supplement Series, 86 (1993), 657692. 
[35] 
M. Singh, An electronically collimated gamma camera for single photon emission computed tomography. Part I: Theoretical considerations and design criteria, Medical Physics, 10 (1983), 421427. 
[36] 
M. Singh and D. Doria, An electronically collimated gamma camera for single photon emission computed tomography. Part II: Image reconstruction and preliminary experimental measurements, Medical Physics, 10 (1983), 428435. 
[37] 
B. Smith, Reconstruction methods and completeness conditions for two compton data models, Journal of the Optical Society of America A, 22 (2005), 445459. 
[38] 
G. Spence, G. Ford, R. Vanderplas and W. Charlton, Directionally sensitive neutron detectors using boron loaded semiconductors, in "50th Annual Meeting of the Institute of Nuclear Materials Management 2009. INMM 50th Annual Meeting," Curran Associates, Inc., (2009), 224231. 
[39] 
R. W. Todd, J. M. Nightingale and D. B. Everett, A proposed gamma camera, Nature, 251 (1974), 132134. 
[40] 
T. Tomitani and M. Hirasawa, Image reconstruction from limited angle compton camera data, Physics in Medicine and Biology, 47 (2002), 21292145. 
[41] 
T. T. Truong, M. K. Nguyen and H. Zaidi, The mathematical foundations of 3D compton scatter emission imaging,, International Journal of Biomedical Imaging, 2007 (). 
[42] 
S. Watanabe, T. Tanaka, K. Nakazawa, T. Mitani, K. Oonuki, T. Takahashi, T. Takashima, H. Tajima, Y. Fukazawa, M. Nomachi, S. Kubo, M. Onishi and Y. Kuroda, A Si/CdTe semiconductor compton camera, IEEE Transactions on Nuclear Science, 52 (2005), 20452051. 
[43] 
S. J. Wilderman, W. L. Rogers, G. F. Knoll and J. C. Engdahl, Fast algorithm for list mode backprojection of compton scatter camera data, IEEE Transactions on Nuclear Science, 45 (1998), 957962. 
[44] 
X. Xun, B. Mallick, R. J. Carroll and P. Kuchment, A Bayesian approach to the detection of small low emission sources, Inverse Problems, 27 (2011), 115009, 11 pp. doi: 10.1088/02665611/27/11/115009. 
show all references
References:
[1] 
R. Basko, G. L. Zeng and G. T. Gullberg, Analytical reconstruction formula for onedimensional compton camera, IEEE Transactions on Nuclear Science, 44 (1997), 13421346. 
[2] 
R. Basko, G. L. Zeng and G. T. Gullberg, Application of spherical harmonics to image reconstruction for the compton camera, Physics in Medicine and Biology, 43 (1998), 887894. 
[3] 
R. R. Brechner and M. Singh, Iterative reconstruction of electronically collimated spect images, IEEE Transactions on Nuclear Science, 37 (1990), 13281332. 
[4] 
T. F. Budinger, G. T. Gullberg and R. H. Huseman, Emission computed tomography, in "Image Reconstruction from Projections'' (ed. G. Herman), Springer Verlag, (1979), 147246. 
[5]  
[6] 
N. H. Clinthorne, ChorYi Ng, ChiaHo Hua, J. E. Gormley, J. W. LeBlanc, S. J. Wilderman and W. L. Rogers, Theoretical performance comparison of a comptonscatter aperture and parallelhole collimator, in "Nuclear Science Symposium, Conference Record," 2, IEEE, (1996), 788792. 
[7] 
S. Coles, "An Introduction to Statistical Modeling of Extreme Values," Springer Series in Statistics, SpringerVerlag London, Ltd., London, 2001. 
[8] 
M. J. Cree and P. J. Bones, Towards direct reconstruction from a gamma camera based on compton scattering, IEEE Transactions on Medical Imaging, 13 (1994), 398407. 
[9] 
Y. F. Du, Z. He, G. F. Knoll, D. K. Wehe and W. Li, Evaluation of a compton scattering camera using 3D position sensitive CdZnTe detectors, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 457 (2001), 203211. 
[10] 
L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. 
[11] 
A. Faridani, E. L. Ritman and K. T. Smith, Local tomography, SIAM Journal on Applied Mathematics, 52 (1992), 459484. doi: 10.1137/0152026. 
[12] 
T. Fawcett, An introduction to ROC analysis, Pattern Recognition Letters, 27 (2006), 861874. 
[13] 
G. T. Gullberg, G. L. Zeng and R. Basko, Image reconstruction from vprojections acquired by compton camera, US Patent 5841141, 1998. 
[14] 
D. L. Gunter, Filtered backprojection algorithm for compton telescopes, US Patent 7345283, 2008. 
[15] 
T. Hebert, R. Leahy and M. Singh, Threedimensional maximumlikelihood reconstruction for an electronically collimated singlephotonemission, Journal of the Optical Society of America A, 7 (1990), 13051313. 
[16] 
W. H. Hill and K. L. Matthews, Experimental verification of a hand held electronicallycollimated radiation detector, in "Nuclear Science Symposium Conference Record, 2007," IEEE, 5 (2007), 37923797. 
[17] 
M. Hirasawa and T. Tomitani, An analytical image reconstruction algorithm to compensate for scattering angle broadening in compton cameras, Physics in Medicine and Biology, 48 (2003), 10091026. 
[18] 
Y. Hristova, "Mathematical Problems of Thermoacoustic and Compton Camera Imaging," Ph.D thesis, Texas A&M University, 2010. 
[19] 
A. C. Kak and M. Slaney, "Principles of Computerized Tomographic Imaging," IEEE Press, New York, 1988. 
[20] 
P. Kuchment, Generalized transforms of radon type and their applications, in "The Radon Transform, Inverse Problems, and Tomography" (eds. G. Olafsson and E. T. Quinto), Proc. Sympos. Appl. Math., 63, American Mathematical Society, Providence, RI, (2006), 6791. 
[21] 
P. Kuchment, K. Lancaster and L. Mogilevskaya, On local tomography, Inverse Problems, 11 (1995), 571589. 
[22] 
A. W. Lackie, K. L. Matthews, B. M. Smith, W. Hill, WeiHsung Wang and M. L. Cherry, A directional algorithm for an electronicallycollimated gammaray detector, in "Nuclear Science Symposium Conference Record, 2006," 1, IEEE, (2006), 264269. 
[23] 
J. W. LeBlanc, N. H. Clinthorne, C.H. Hua, E. Nygard, W. L. Rogers, D. K. Wehe, P. Weilhammer and S. J. Wilderman, Csprint: A prototype compton camera system for low energy gamma ray imaging, in "Nuclear Science Symposium, 1997," 1, IEEE, (1997), 357361. 
[24] 
C. M. Marianno, D. R. Boyle, W. S. Charlton, G. M. Gaukler and A. Veditz, A guide for detector development and deployment, in "51st Annual Meeting of the Institute of Nuclear Materials Management 2010," Curran Associates, Inc., (2010), 39013909. 
[25] 
V. Maxim, M. Frandeş and R. Prost, Analytical inversion of the Compton transform using the full set of available projections, Inverse Problems, 25 (2009), 095001, 21 pp. doi: 10.1088/02665611/25/9/095001. 
[26] 
F. Natterer, "The Mathematics of Computerized Tomography," Society for Industrial and Applied Mathematics, 2001. doi: 10.1137/1.9780898719284. 
[27] 
F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," Reprint of the 1986 original, Classics in Applied Mathematics, 32, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. 
[28] 
M. K. Nguyen, T. T. Truong and P. Grangeat, Radon transforms on a class of cones with fixed axis direction, Journal of Physics A: Mathematical and General, 38 (2005), 80038015. doi: 10.1088/03054470/38/37/006. 
[29] 
L. C. Parra, Reconstruction of conebeam projections from compton scattered data, IEEE Transactions on Nuclear Science, 47 (2000), 15431550. 
[30] 
R. C. Rohe, M. M. Sharfi, K. A. Kecevar, J. D. Valentine and C. Bonnerave, The spatiallyvariant backprojection point kernel function of an energysubtraction compton scatter camera for medical imaging, IEEE Transactions on Nuclear Science, 44 (1997), 24772482. 
[31] 
G. J. Royle and R. D. Speller, A flexible geometry compton camera for industrial gamma ray imaging, in "Nuclear Science Symposium, 1996. Conference Record.," 2, IEEE, (1996), 821824. 
[32] 
G. J. Royle and R. D. Speller, Compton scatter imaging of a nuclear industry site, in "Nuclear Science Symposium, 1997," 1, IEEE, (1997), 365368. 
[33] 
A. C. Sauve, A. O. III Hero, W. L. Rogers, S. J. Wilderman and N. H. Clinthorne, 3D image reconstruction for a compton spect camera model, IEEE Transactions on Nuclear Science, 46 (1999), 20752084. 
[34] 
V. Schoenfelder, H. Aarts, K. Bennett, H. de Boer, J. Clear, W. Collmar, A. Connors, A. Deerenberg, R. Diehl, A. von Dordrecht, J. W. den Herder, W. Hermsen, M. Kippen, L. Kuiper, G. Lichti, J. Lockwood, J. Macri, M. McConnell, D. Morris, R. Much, J. Ryan, G. Simpson, M. Snelling, G. Stacy, H. Steinle, A. Strong, B. N. Swanenburg, B. Taylor, C. de Vries and C. Winkler, Instrument description and performance of the imaging gammaray telescope comptel aboard the compton gammaray observatory, Astrophysical Journal Supplement Series, 86 (1993), 657692. 
[35] 
M. Singh, An electronically collimated gamma camera for single photon emission computed tomography. Part I: Theoretical considerations and design criteria, Medical Physics, 10 (1983), 421427. 
[36] 
M. Singh and D. Doria, An electronically collimated gamma camera for single photon emission computed tomography. Part II: Image reconstruction and preliminary experimental measurements, Medical Physics, 10 (1983), 428435. 
[37] 
B. Smith, Reconstruction methods and completeness conditions for two compton data models, Journal of the Optical Society of America A, 22 (2005), 445459. 
[38] 
G. Spence, G. Ford, R. Vanderplas and W. Charlton, Directionally sensitive neutron detectors using boron loaded semiconductors, in "50th Annual Meeting of the Institute of Nuclear Materials Management 2009. INMM 50th Annual Meeting," Curran Associates, Inc., (2009), 224231. 
[39] 
R. W. Todd, J. M. Nightingale and D. B. Everett, A proposed gamma camera, Nature, 251 (1974), 132134. 
[40] 
T. Tomitani and M. Hirasawa, Image reconstruction from limited angle compton camera data, Physics in Medicine and Biology, 47 (2002), 21292145. 
[41] 
T. T. Truong, M. K. Nguyen and H. Zaidi, The mathematical foundations of 3D compton scatter emission imaging,, International Journal of Biomedical Imaging, 2007 (). 
[42] 
S. Watanabe, T. Tanaka, K. Nakazawa, T. Mitani, K. Oonuki, T. Takahashi, T. Takashima, H. Tajima, Y. Fukazawa, M. Nomachi, S. Kubo, M. Onishi and Y. Kuroda, A Si/CdTe semiconductor compton camera, IEEE Transactions on Nuclear Science, 52 (2005), 20452051. 
[43] 
S. J. Wilderman, W. L. Rogers, G. F. Knoll and J. C. Engdahl, Fast algorithm for list mode backprojection of compton scatter camera data, IEEE Transactions on Nuclear Science, 45 (1998), 957962. 
[44] 
X. Xun, B. Mallick, R. J. Carroll and P. Kuchment, A Bayesian approach to the detection of small low emission sources, Inverse Problems, 27 (2011), 115009, 11 pp. doi: 10.1088/02665611/27/11/115009. 
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