February  2013, 7(1): 47-79. doi: 10.3934/ipi.2013.7.47

Detecting small low emission radiating sources

1. 

Siemens AG, Corporate Technology, Otto-Hahn-Ring 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

Received  February 2012 Revised  May 2012 Published  February 2013

In order to prevent influx of highly enriched nuclear material throu-gh border checkpoints, new advanced detection schemes need to be developed. Typical issues faced in this context are sources with very low emission against a dominating natural background radiation. Sources are expected to be small and shielded and hence cannot be detected from measurements of radiation levels alone. We consider collimated and Compton-type measurements and propose a detection method that relies on the geometric singularity of small sources to distinguish them from the more uniform background. The method is characterized by high sensitivity and specificity and allows for assigning confidence probabilities of detection. The validity of our approach can be justified using properties of related techniques from medical imaging. Results of numerical simulations are presented for collimated and Compton-type measurements. The 2D case is considered in detail.
Citation: Moritz Allmaras, David Darrow, Yulia Hristova, Guido Kanschat, Peter Kuchment. Detecting small low emission radiating sources. Inverse Problems & Imaging, 2013, 7 (1) : 47-79. doi: 10.3934/ipi.2013.7.47
References:
[1]

R. Basko, G. L. Zeng and G. T. Gullberg, Analytical reconstruction formula for one-dimensional compton camera,, IEEE Transactions on Nuclear Science, 44 (1997), 1342. Google Scholar

[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), 887. Google Scholar

[3]

R. R. Brechner and M. Singh, Iterative reconstruction of electronically collimated spect images,, IEEE Transactions on Nuclear Science, 37 (1990), 1328. Google Scholar

[4]

T. F. Budinger, G. T. Gullberg and R. H. Huseman, Emission computed tomography,, in, (1979), 147. Google Scholar

[5]

W. Charlton and G. Spence, Private, communication, (2009). Google Scholar

[6]

N. H. Clinthorne, Chor-Yi Ng, Chia-Ho Hua, J. E. Gormley, J. W. LeBlanc, S. J. Wilderman and W. L. Rogers, Theoretical performance comparison of a compton-scatter aperture and parallel-hole collimator,, in, 2 (1996), 788. Google Scholar

[7]

S. Coles, "An Introduction to Statistical Modeling of Extreme Values,", Springer Series in Statistics, (2001). Google Scholar

[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), 398. Google Scholar

[9]

Y. F. Du, Z. He, G. F. Knoll, D. K. Wehe and W. Li, Evaluation of a compton scattering camera using 3-D position sensitive CdZnTe detectors,, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, 457 (2001), 203. Google Scholar

[10]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar

[11]

A. Faridani, E. L. Ritman and K. T. Smith, Local tomography,, SIAM Journal on Applied Mathematics, 52 (1992), 459. doi: 10.1137/0152026. Google Scholar

[12]

T. Fawcett, An introduction to ROC analysis,, Pattern Recognition Letters, 27 (2006), 861. Google Scholar

[13]

G. T. Gullberg, G. L. Zeng and R. Basko, Image reconstruction from v-projections acquired by compton camera,, US Patent 5841141, (5841). Google Scholar

[14]

D. L. Gunter, Filtered back-projection algorithm for compton telescopes,, US Patent 7345283, (7345). Google Scholar

[15]

T. Hebert, R. Leahy and M. Singh, Three-dimensional maximum-likelihood reconstruction for an electronically collimated single-photon-emission,, Journal of the Optical Society of America A, 7 (1990), 1305. Google Scholar

[16]

W. H. Hill and K. L. Matthews, Experimental verification of a hand held electronically-collimated radiation detector,, in, 5 (2007), 3792. Google Scholar

[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), 1009. Google Scholar

[18]

Y. Hristova, "Mathematical Problems of Thermoacoustic and Compton Camera Imaging,", Ph.D thesis, (2010). Google Scholar

[19]

A. C. Kak and M. Slaney, "Principles of Computerized Tomographic Imaging,", IEEE Press, (1988). Google Scholar

[20]

P. Kuchment, Generalized transforms of radon type and their applications,, in, 63 (2006), 67. Google Scholar

[21]

P. Kuchment, K. Lancaster and L. Mogilevskaya, On local tomography,, Inverse Problems, 11 (1995), 571. Google Scholar

[22]

A. W. Lackie, K. L. Matthews, B. M. Smith, W. Hill, Wei-Hsung Wang and M. L. Cherry, A directional algorithm for an electronically-collimated gamma-ray detector,, in, 1 (2006), 264. Google Scholar

[23]

J. W. LeBlanc, N. H. Clinthorne, C.-H. Hua, E. Nygard, W. L. Rogers, D. K. Wehe, P. Weilhammer and S. J. Wilderman, C-sprint: A prototype compton camera system for low energy gamma ray imaging,, in, 1 (1997), 357. Google Scholar

[24]

C. M. Marianno, D. R. Boyle, W. S. Charlton, G. M. Gaukler and A. Veditz, A guide for detector development and deployment,, in, (2010), 3901. Google Scholar

[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). doi: 10.1088/0266-5611/25/9/095001. Google Scholar

[26]

F. Natterer, "The Mathematics of Computerized Tomography,", Society for Industrial and Applied Mathematics, (2001). doi: 10.1137/1.9780898719284. Google Scholar

[27]

F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction,", Reprint of the 1986 original, 32 (1986). Google Scholar

[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), 8003. doi: 10.1088/0305-4470/38/37/006. Google Scholar

[29]

L. C. Parra, Reconstruction of cone-beam projections from compton scattered data,, IEEE Transactions on Nuclear Science, 47 (2000), 1543. Google Scholar

[30]

R. C. Rohe, M. M. Sharfi, K. A. Kecevar, J. D. Valentine and C. Bonnerave, The spatially-variant backprojection point kernel function of an energy-subtraction compton scatter camera for medical imaging,, IEEE Transactions on Nuclear Science, 44 (1997), 2477. Google Scholar

[31]

G. J. Royle and R. D. Speller, A flexible geometry compton camera for industrial gamma ray imaging,, in, 2 (1996), 821. Google Scholar

[32]

G. J. Royle and R. D. Speller, Compton scatter imaging of a nuclear industry site,, in, 1 (1997), 365. Google Scholar

[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), 2075. Google Scholar

[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 gamma-ray telescope comptel aboard the compton gamma-ray observatory,, Astrophysical Journal Supplement Series, 86 (1993), 657. Google Scholar

[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), 421. Google Scholar

[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), 428. Google Scholar

[37]

B. Smith, Reconstruction methods and completeness conditions for two compton data models,, Journal of the Optical Society of America A, 22 (2005), 445. Google Scholar

[38]

G. Spence, G. Ford, R. Vanderplas and W. Charlton, Directionally sensitive neutron detectors using boron loaded semiconductors,, in, (2009), 224. Google Scholar

[39]

R. W. Todd, J. M. Nightingale and D. B. Everett, A proposed gamma camera,, Nature, 251 (1974), 132. Google Scholar

[40]

T. Tomitani and M. Hirasawa, Image reconstruction from limited angle compton camera data,, Physics in Medicine and Biology, 47 (2002), 2129. Google Scholar

[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 (). Google Scholar

[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), 2045. Google Scholar

[43]

S. J. Wilderman, W. L. Rogers, G. F. Knoll and J. C. Engdahl, Fast algorithm for list mode back-projection of compton scatter camera data,, IEEE Transactions on Nuclear Science, 45 (1998), 957. Google Scholar

[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). doi: 10.1088/0266-5611/27/11/115009. Google Scholar

show all references

References:
[1]

R. Basko, G. L. Zeng and G. T. Gullberg, Analytical reconstruction formula for one-dimensional compton camera,, IEEE Transactions on Nuclear Science, 44 (1997), 1342. Google Scholar

[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), 887. Google Scholar

[3]

R. R. Brechner and M. Singh, Iterative reconstruction of electronically collimated spect images,, IEEE Transactions on Nuclear Science, 37 (1990), 1328. Google Scholar

[4]

T. F. Budinger, G. T. Gullberg and R. H. Huseman, Emission computed tomography,, in, (1979), 147. Google Scholar

[5]

W. Charlton and G. Spence, Private, communication, (2009). Google Scholar

[6]

N. H. Clinthorne, Chor-Yi Ng, Chia-Ho Hua, J. E. Gormley, J. W. LeBlanc, S. J. Wilderman and W. L. Rogers, Theoretical performance comparison of a compton-scatter aperture and parallel-hole collimator,, in, 2 (1996), 788. Google Scholar

[7]

S. Coles, "An Introduction to Statistical Modeling of Extreme Values,", Springer Series in Statistics, (2001). Google Scholar

[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), 398. Google Scholar

[9]

Y. F. Du, Z. He, G. F. Knoll, D. K. Wehe and W. Li, Evaluation of a compton scattering camera using 3-D position sensitive CdZnTe detectors,, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, 457 (2001), 203. Google Scholar

[10]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar

[11]

A. Faridani, E. L. Ritman and K. T. Smith, Local tomography,, SIAM Journal on Applied Mathematics, 52 (1992), 459. doi: 10.1137/0152026. Google Scholar

[12]

T. Fawcett, An introduction to ROC analysis,, Pattern Recognition Letters, 27 (2006), 861. Google Scholar

[13]

G. T. Gullberg, G. L. Zeng and R. Basko, Image reconstruction from v-projections acquired by compton camera,, US Patent 5841141, (5841). Google Scholar

[14]

D. L. Gunter, Filtered back-projection algorithm for compton telescopes,, US Patent 7345283, (7345). Google Scholar

[15]

T. Hebert, R. Leahy and M. Singh, Three-dimensional maximum-likelihood reconstruction for an electronically collimated single-photon-emission,, Journal of the Optical Society of America A, 7 (1990), 1305. Google Scholar

[16]

W. H. Hill and K. L. Matthews, Experimental verification of a hand held electronically-collimated radiation detector,, in, 5 (2007), 3792. Google Scholar

[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), 1009. Google Scholar

[18]

Y. Hristova, "Mathematical Problems of Thermoacoustic and Compton Camera Imaging,", Ph.D thesis, (2010). Google Scholar

[19]

A. C. Kak and M. Slaney, "Principles of Computerized Tomographic Imaging,", IEEE Press, (1988). Google Scholar

[20]

P. Kuchment, Generalized transforms of radon type and their applications,, in, 63 (2006), 67. Google Scholar

[21]

P. Kuchment, K. Lancaster and L. Mogilevskaya, On local tomography,, Inverse Problems, 11 (1995), 571. Google Scholar

[22]

A. W. Lackie, K. L. Matthews, B. M. Smith, W. Hill, Wei-Hsung Wang and M. L. Cherry, A directional algorithm for an electronically-collimated gamma-ray detector,, in, 1 (2006), 264. Google Scholar

[23]

J. W. LeBlanc, N. H. Clinthorne, C.-H. Hua, E. Nygard, W. L. Rogers, D. K. Wehe, P. Weilhammer and S. J. Wilderman, C-sprint: A prototype compton camera system for low energy gamma ray imaging,, in, 1 (1997), 357. Google Scholar

[24]

C. M. Marianno, D. R. Boyle, W. S. Charlton, G. M. Gaukler and A. Veditz, A guide for detector development and deployment,, in, (2010), 3901. Google Scholar

[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). doi: 10.1088/0266-5611/25/9/095001. Google Scholar

[26]

F. Natterer, "The Mathematics of Computerized Tomography,", Society for Industrial and Applied Mathematics, (2001). doi: 10.1137/1.9780898719284. Google Scholar

[27]

F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction,", Reprint of the 1986 original, 32 (1986). Google Scholar

[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), 8003. doi: 10.1088/0305-4470/38/37/006. Google Scholar

[29]

L. C. Parra, Reconstruction of cone-beam projections from compton scattered data,, IEEE Transactions on Nuclear Science, 47 (2000), 1543. Google Scholar

[30]

R. C. Rohe, M. M. Sharfi, K. A. Kecevar, J. D. Valentine and C. Bonnerave, The spatially-variant backprojection point kernel function of an energy-subtraction compton scatter camera for medical imaging,, IEEE Transactions on Nuclear Science, 44 (1997), 2477. Google Scholar

[31]

G. J. Royle and R. D. Speller, A flexible geometry compton camera for industrial gamma ray imaging,, in, 2 (1996), 821. Google Scholar

[32]

G. J. Royle and R. D. Speller, Compton scatter imaging of a nuclear industry site,, in, 1 (1997), 365. Google Scholar

[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), 2075. Google Scholar

[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 gamma-ray telescope comptel aboard the compton gamma-ray observatory,, Astrophysical Journal Supplement Series, 86 (1993), 657. Google Scholar

[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), 421. Google Scholar

[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), 428. Google Scholar

[37]

B. Smith, Reconstruction methods and completeness conditions for two compton data models,, Journal of the Optical Society of America A, 22 (2005), 445. Google Scholar

[38]

G. Spence, G. Ford, R. Vanderplas and W. Charlton, Directionally sensitive neutron detectors using boron loaded semiconductors,, in, (2009), 224. Google Scholar

[39]

R. W. Todd, J. M. Nightingale and D. B. Everett, A proposed gamma camera,, Nature, 251 (1974), 132. Google Scholar

[40]

T. Tomitani and M. Hirasawa, Image reconstruction from limited angle compton camera data,, Physics in Medicine and Biology, 47 (2002), 2129. Google Scholar

[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 (). Google Scholar

[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), 2045. Google Scholar

[43]

S. J. Wilderman, W. L. Rogers, G. F. Knoll and J. C. Engdahl, Fast algorithm for list mode back-projection of compton scatter camera data,, IEEE Transactions on Nuclear Science, 45 (1998), 957. Google Scholar

[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). doi: 10.1088/0266-5611/27/11/115009. Google Scholar

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