# American Institute of Mathematical Sciences

December  2017, 11(6): 1071-1090. doi: 10.3934/ipi.2017049

## Inversion of weighted divergent beam and cone transforms

 Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA

* Corresponding author

Received  December 2016 Published  September 2017

Fund Project: This work was supported in part by NSF DMS grant 1211463.

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.

Citation: Peter Kuchment, Fatma Terzioglu. Inversion of weighted divergent beam and cone transforms. Inverse Problems & Imaging, 2017, 11 (6) : 1071-1090. doi: 10.3934/ipi.2017049
##### References:
 [1] M. Allmaras, D. P. Darrow, Y. Hristova, G. Kanschat and P. Kuchment, Detecting small low emission radiating sources, Inverse Probl. Imaging, 7 (2013), 47-79.  doi: 10.3934/ipi.2013.7.47.  Google Scholar [2] M. Allmaras, W. Charlton, A. Ciabatti, Y. Hristova, P. Kuchment, A. Olson and J. Ragusa, Passive detection of small low-emission sources: Two-dimensional numerical case studies, Nuclear Sci. and Eng., 184 (2016), 125-150.   Google Scholar [3] G. Ambartsoumian, Inversion of the V-line Radon transform in a disc and its applications in imaging, Comput. Math. Appl., 64 (2012), 260-265.  doi: 10.1016/j.camwa.2012.01.059.  Google Scholar [4] G. Ambartsoumian and S. Moon, A series formula for inversion of the V-line Radon transform in a disc, Comput. Math. Appl., 66 (2013), 1567-1572.  doi: 10.1016/j.camwa.2013.01.039.  Google Scholar [5] R. Basko, G. L. Zeng and G. T. Gullberg, Application of spherical harmonics to image reconstruction for the Compton camera, Phys. Med. Biol., 43 (1998), 887-894.   Google Scholar [6] M. J. Cree and P. J. Bones, Towards direct reconstruction from a gamma camera based on Compton scattering, IEEE Trans. Med. Imaging, 13 (1994), 398-409.   Google Scholar [7] D. B. Everett, J. S. Fleming, R. W. Todd and J. M. Nightingale, Gamma-radiation imaging system based on the Compton effect, Proc. IEE, 124 (1977), 995-1000.   Google Scholar [8] D. Finch and D. Solomon, A characterization of the range of the divergent beam X-ray transform, SIAM J. Math. Anal, 14 (1983), 767-771.  doi: 10.1137/0514057.  Google Scholar [9] L. Florescu, V. A. Markel and J. C. Schotland, Inversion formulas for the broken-ray Radon transform, Inverse Problems, 27 (2011), 025002, 13pp. doi: 10.1088/0266-5611/27/2/025002.  Google Scholar [10] I. M Gel'fand, S. Gindikin and M. Graev, Selected Topics in Integral Geometry, (Transl. Math. Monogr. v. 220, Amer. Math. Soc.), Providence RI, 2003.  Google Scholar [11] I. M. Gel'fand and G. E. Shilov, Generalized Functions: Volume I Properties and Operations, Academic, New York, 1964.  Google Scholar [12] I. M. Gel’fand and A. B. Goncharov, Recovery of a compactly supported function starting from its integrals over lines intersecting a given set of points in space, (Russian)Dokl. Akad. Nauk SSSR, 290 (1986), 1037–1040; English translation in Soviet Math. Dokl., 34 (1987), 373–376  Google Scholar [13] R. Gouia-Zarrad and G. Ambartsoumian, Exact inversion of the conical Radon transform with a fixed opening angle, Inverse Problems, 30 (2014), 045007, 12pp. doi: 10.1088/0266-5611/30/4/045007.  Google Scholar [14] P. Grangeat, Mathematical framework of cone-beam reconstruction via the first derivative of the Radon transform, Mathematical Methods in Tomography (Oberwolfach, 1990), Lecture Notes in Math., Springer, New York, 1497 (1991), 66–97. doi: 10.1007/BFb0084509.  Google Scholar [15] M. Haltmeier, Exact reconstruction formulas for a Radon transform over cones, Inverse Problems, 30 (2014), 035001, 8pp. doi: 10.1088/0266-5611/30/3/035001.  Google Scholar [16] C. Hamaker, K. T. Smith, D. C. Solmon and S. L. Wagner, The divergent beam X-ray transform, Rocky Mountain J. Math., 10 (1980), 253-283.  doi: 10.1216/RMJ-1980-10-1-253.  Google Scholar [17] S. Helgason, Integral Geometry and Radon Transforms, Springer, Berlin, 2011. doi: 10.1007/978-1-4419-6055-9.  Google Scholar [18] Y. Hristova, Mathematical Problems of Thermoacoustic and Compton Camera Imaging, Dissertation Texas A&M University, 2010.  Google Scholar [19] Y. Hristova, Inversion of a V-line transform arising in emission tomography, Journal of Coupled Systems and Multiscale Dynamics, 3 (2015), 272-277.   Google Scholar [20] C.-Y. Jung and S. Moon, Exact Inversion of the cone transform arising in an application of a Compton camera consisting of line detectors, SIAM J. Imaging Sciences, 9 (2016), 520-536.  doi: 10.1137/15M1033617.  Google Scholar [21] A. Katsevich, Theoretically exact filtered backprojection-type inversion algorithm for spiral CT, SIAM J. Appl. Math., 62 (2002), 2012-2026.  doi: 10.1137/S0036139901387186.  Google Scholar [22] A. Katsevich, An improved exact filtered backprojection algorithm for spiral computed tomography, Advances in Applied Mathematics, 32 (2004), 681-697.  doi: 10.1016/S0196-8858(03)00099-X.  Google Scholar [23] P. Kuchment, The Radon Transform and Medical Imaging, (CBMS-NSF Regional Conf. Ser. in Appl. Math. 85), Society for Industrial and Applied Mathematics, Philadelphia, 2014.  Google Scholar [24] P. Kuchment and F. Terzioglu, Three-dimensional image reconstruction from Compton camera data, SIAM J. Imaging Sci., 9 (2016), 1708-1725.  doi: 10.1137/16M107476X.  Google Scholar [25] V. Maxim, Filtered back-projection reconstruction and redundancy in Compton camera imaging, IEEE Trans. Image Process., 23 (2014), 332-341.  doi: 10.1109/TIP.2013.2288143.  Google Scholar [26] S. Moon and M. Haltmeier, Analytic inversion of a conical Radon transform arising in application of Compton cameras on the cylinder, SIAM J. Imaging Sci., 10 (2017), 535-557.  doi: 10.1137/16M1083116.  Google Scholar [27] S. Moon, Inversion of the conical Radon transform with vertices on a surface of revolution arising in an application of a Compton camera, Inverse Problems, 33 (2017), 065002, 11pp. doi: 10.1088/1361-6420/aa69c9.  Google Scholar [28] F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, (Monogr. Math. Model. Comput. 5), Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898718324.  Google Scholar [29] F. Natterer, The Mathematics of Computerized Tomography, (Classics in Applied Mathematics), Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898719284.  Google Scholar [30] M. K. Nguyen, T. T. Truong and P. Grangeat, Radon transforms on a class of cones with fixed axis direction, J. Phys. A: Math. Gen., 38 (2005), 8003-8015.  doi: 10.1088/0305-4470/38/37/006.  Google Scholar [31] P.-O. Persson and G. Strang, A simple mesh generator in MATLAB, SIAM Rev., 46 (2004), 329-345.  doi: 10.1137/S0036144503429121.  Google Scholar [32] D. Schiefeneder and M. Haltmeier, The Radon Transform over Cones with Vertices on the Sphere and Orthogonal Axes, SIAM J. Appl. Math., 77 (2017), 1335–1351, arXiv: 1606.03486. doi: 10.1137/16M1079476.  Google Scholar [33] M. Singh, An electronically collimated gamma camera for single photon emission computed tomography: I. Theoretical considerations and design criteria, Med. Phys., 10 (1983), 421-427.   Google Scholar [34] B. D. Smith, Image reconstruction from cone-beam projections: Necessary and sufficient conditions and reconstruction methods, IEEE Trans. Med. Imag., 4 (1985), 14-25.   Google Scholar [35] B. Smith, Reconstruction methods and completeness conditions for two Compton data models, J. Opt. Soc. Am. A, 22 (2005), 445-459.   Google Scholar [36] B. Smith, Line reconstruction from Compton cameras: Data sets and a camera design, Opt. Eng., 50 (2011), 053204. Google Scholar [37] F. Terzioglu, Some inversion formulas for the cone transform, Inverse Problems, 31 (2015), 115010, 21pp. doi: 10.1088/0266-5611/31/11/115010.  Google Scholar [38] T. T. Truong and M. K. Nguyen, New properties of the V-line Radon transform and their imaging applications, J. Phys. A, 48 (2015), 405204, 28pp. doi: 10.1088/1751-8113/48/40/405204.  Google Scholar [39] H. K. Tuy, An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math., 43 (1983), 546-552.  doi: 10.1137/0143035.  Google Scholar [40] X. Xun, B. Mallick, R. Carroll and P. Kuchment, Bayesian approach to detection of small low emission sources, Inverse Problems, 27 (2011), 115009, 11pp. doi: 10.1088/0266-5611/27/11/115009.  Google Scholar

show all references

##### References:
 [1] M. Allmaras, D. P. Darrow, Y. Hristova, G. Kanschat and P. Kuchment, Detecting small low emission radiating sources, Inverse Probl. Imaging, 7 (2013), 47-79.  doi: 10.3934/ipi.2013.7.47.  Google Scholar [2] M. Allmaras, W. Charlton, A. Ciabatti, Y. Hristova, P. Kuchment, A. Olson and J. Ragusa, Passive detection of small low-emission sources: Two-dimensional numerical case studies, Nuclear Sci. and Eng., 184 (2016), 125-150.   Google Scholar [3] G. Ambartsoumian, Inversion of the V-line Radon transform in a disc and its applications in imaging, Comput. Math. Appl., 64 (2012), 260-265.  doi: 10.1016/j.camwa.2012.01.059.  Google Scholar [4] G. Ambartsoumian and S. Moon, A series formula for inversion of the V-line Radon transform in a disc, Comput. Math. Appl., 66 (2013), 1567-1572.  doi: 10.1016/j.camwa.2013.01.039.  Google Scholar [5] R. Basko, G. L. Zeng and G. T. Gullberg, Application of spherical harmonics to image reconstruction for the Compton camera, Phys. Med. Biol., 43 (1998), 887-894.   Google Scholar [6] M. J. Cree and P. J. Bones, Towards direct reconstruction from a gamma camera based on Compton scattering, IEEE Trans. Med. Imaging, 13 (1994), 398-409.   Google Scholar [7] D. B. Everett, J. S. Fleming, R. W. Todd and J. M. Nightingale, Gamma-radiation imaging system based on the Compton effect, Proc. IEE, 124 (1977), 995-1000.   Google Scholar [8] D. Finch and D. Solomon, A characterization of the range of the divergent beam X-ray transform, SIAM J. Math. Anal, 14 (1983), 767-771.  doi: 10.1137/0514057.  Google Scholar [9] L. Florescu, V. A. Markel and J. C. Schotland, Inversion formulas for the broken-ray Radon transform, Inverse Problems, 27 (2011), 025002, 13pp. doi: 10.1088/0266-5611/27/2/025002.  Google Scholar [10] I. M Gel'fand, S. Gindikin and M. Graev, Selected Topics in Integral Geometry, (Transl. Math. Monogr. v. 220, Amer. Math. Soc.), Providence RI, 2003.  Google Scholar [11] I. M. Gel'fand and G. E. Shilov, Generalized Functions: Volume I Properties and Operations, Academic, New York, 1964.  Google Scholar [12] I. M. Gel’fand and A. B. Goncharov, Recovery of a compactly supported function starting from its integrals over lines intersecting a given set of points in space, (Russian)Dokl. Akad. Nauk SSSR, 290 (1986), 1037–1040; English translation in Soviet Math. Dokl., 34 (1987), 373–376  Google Scholar [13] R. Gouia-Zarrad and G. Ambartsoumian, Exact inversion of the conical Radon transform with a fixed opening angle, Inverse Problems, 30 (2014), 045007, 12pp. doi: 10.1088/0266-5611/30/4/045007.  Google Scholar [14] P. Grangeat, Mathematical framework of cone-beam reconstruction via the first derivative of the Radon transform, Mathematical Methods in Tomography (Oberwolfach, 1990), Lecture Notes in Math., Springer, New York, 1497 (1991), 66–97. doi: 10.1007/BFb0084509.  Google Scholar [15] M. Haltmeier, Exact reconstruction formulas for a Radon transform over cones, Inverse Problems, 30 (2014), 035001, 8pp. doi: 10.1088/0266-5611/30/3/035001.  Google Scholar [16] C. Hamaker, K. T. Smith, D. C. Solmon and S. L. Wagner, The divergent beam X-ray transform, Rocky Mountain J. Math., 10 (1980), 253-283.  doi: 10.1216/RMJ-1980-10-1-253.  Google Scholar [17] S. Helgason, Integral Geometry and Radon Transforms, Springer, Berlin, 2011. doi: 10.1007/978-1-4419-6055-9.  Google Scholar [18] Y. Hristova, Mathematical Problems of Thermoacoustic and Compton Camera Imaging, Dissertation Texas A&M University, 2010.  Google Scholar [19] Y. Hristova, Inversion of a V-line transform arising in emission tomography, Journal of Coupled Systems and Multiscale Dynamics, 3 (2015), 272-277.   Google Scholar [20] C.-Y. Jung and S. Moon, Exact Inversion of the cone transform arising in an application of a Compton camera consisting of line detectors, SIAM J. Imaging Sciences, 9 (2016), 520-536.  doi: 10.1137/15M1033617.  Google Scholar [21] A. Katsevich, Theoretically exact filtered backprojection-type inversion algorithm for spiral CT, SIAM J. Appl. Math., 62 (2002), 2012-2026.  doi: 10.1137/S0036139901387186.  Google Scholar [22] A. Katsevich, An improved exact filtered backprojection algorithm for spiral computed tomography, Advances in Applied Mathematics, 32 (2004), 681-697.  doi: 10.1016/S0196-8858(03)00099-X.  Google Scholar [23] P. Kuchment, The Radon Transform and Medical Imaging, (CBMS-NSF Regional Conf. Ser. in Appl. Math. 85), Society for Industrial and Applied Mathematics, Philadelphia, 2014.  Google Scholar [24] P. Kuchment and F. Terzioglu, Three-dimensional image reconstruction from Compton camera data, SIAM J. Imaging Sci., 9 (2016), 1708-1725.  doi: 10.1137/16M107476X.  Google Scholar [25] V. Maxim, Filtered back-projection reconstruction and redundancy in Compton camera imaging, IEEE Trans. Image Process., 23 (2014), 332-341.  doi: 10.1109/TIP.2013.2288143.  Google Scholar [26] S. Moon and M. Haltmeier, Analytic inversion of a conical Radon transform arising in application of Compton cameras on the cylinder, SIAM J. Imaging Sci., 10 (2017), 535-557.  doi: 10.1137/16M1083116.  Google Scholar [27] S. Moon, Inversion of the conical Radon transform with vertices on a surface of revolution arising in an application of a Compton camera, Inverse Problems, 33 (2017), 065002, 11pp. doi: 10.1088/1361-6420/aa69c9.  Google Scholar [28] F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, (Monogr. Math. Model. Comput. 5), Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898718324.  Google Scholar [29] F. Natterer, The Mathematics of Computerized Tomography, (Classics in Applied Mathematics), Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898719284.  Google Scholar [30] M. K. Nguyen, T. T. Truong and P. Grangeat, Radon transforms on a class of cones with fixed axis direction, J. Phys. A: Math. Gen., 38 (2005), 8003-8015.  doi: 10.1088/0305-4470/38/37/006.  Google Scholar [31] P.-O. Persson and G. Strang, A simple mesh generator in MATLAB, SIAM Rev., 46 (2004), 329-345.  doi: 10.1137/S0036144503429121.  Google Scholar [32] D. Schiefeneder and M. Haltmeier, The Radon Transform over Cones with Vertices on the Sphere and Orthogonal Axes, SIAM J. Appl. Math., 77 (2017), 1335–1351, arXiv: 1606.03486. doi: 10.1137/16M1079476.  Google Scholar [33] M. Singh, An electronically collimated gamma camera for single photon emission computed tomography: I. Theoretical considerations and design criteria, Med. Phys., 10 (1983), 421-427.   Google Scholar [34] B. D. Smith, Image reconstruction from cone-beam projections: Necessary and sufficient conditions and reconstruction methods, IEEE Trans. Med. Imag., 4 (1985), 14-25.   Google Scholar [35] B. Smith, Reconstruction methods and completeness conditions for two Compton data models, J. Opt. Soc. Am. A, 22 (2005), 445-459.   Google Scholar [36] B. Smith, Line reconstruction from Compton cameras: Data sets and a camera design, Opt. Eng., 50 (2011), 053204. Google Scholar [37] F. Terzioglu, Some inversion formulas for the cone transform, Inverse Problems, 31 (2015), 115010, 21pp. doi: 10.1088/0266-5611/31/11/115010.  Google Scholar [38] T. T. Truong and M. K. Nguyen, New properties of the V-line Radon transform and their imaging applications, J. Phys. A, 48 (2015), 405204, 28pp. doi: 10.1088/1751-8113/48/40/405204.  Google Scholar [39] H. K. Tuy, An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math., 43 (1983), 546-552.  doi: 10.1137/0143035.  Google Scholar [40] X. Xun, B. Mallick, R. Carroll and P. Kuchment, Bayesian approach to detection of small low emission sources, Inverse Problems, 27 (2011), 115009, 11pp. doi: 10.1088/0266-5611/27/11/115009.  Google Scholar
A cone with vertex $u \in {{\mathbb{R}}^{n}}$, central axis direction vector $\beta \in {{\mathbb{S}}^{n-1}}$ and opening angle $\psi \in (0,\pi)$.
The density plot (left) and surface plot (right) of the phantom $f$ that consists of two concentric disks centered at $(0,0.4)$ with radii 0.25 and 0.5, and densities 1 and -0.5 units, respectively.
The density plot of $256 \times 256$ image reconstructed from the simulated cone data using 256 counts for vertices $u$ (represented by white dots on the unit circle), 400 counts for directions $\beta$ and 90 counts for opening angles $\psi$ (left), and the comparison of $y$-axis profiles of the phantom and the reconstruction (right).
The density plot of $256 \times 256$ image reconstructed from cone data contaminated with $5\%$ Gaussian noise (left), and the comparison of $y$-axis profiles of the phantom and the reconstruction (right). The dimensions of the cone data are taken as in Fig. 3.
The density plot of $256 \times 256$ image reconstructed from the simulated cone data using 256 counts for vertices $u$ (represented by white dots around the square), 400 counts for directions $\beta$ and 90 counts for opening angles $\psi$ (left), and the comparison of $y$-axis profiles of the phantom and the reconstruction (right).
The density plot of $256 \times 256$ image reconstructed from cone data contaminated with $5\%$ Gaussian noise (left), and the comparison of $y$-axis profiles of the phantom and the reconstruction (right). The dimensions of the cone data are taken as in Fig. 5.
Comparison of the profiles of the reconstruction along the diagonal of the square region for the circular (left) and square (right) locations of the vertices (detectors).
The 3D ball phantom with radius 0.5, center (0, 0, 0.25) and unit density (left), and $90 \times 90$ image reconstructed via (33) from weighted cone data simulated using 1800 counts for vertices $u$ on the unit sphere, 1800 counts for directions $\beta$ and 200 counts for opening angles $\psi$ (right). The cross sections by the planes $x=0, y=0$ and $z=0.25$ are shown.
Comparison of the $x$-axis (left), $y$-axis (center) and $z$-axis (right) profiles of the reconstruction and the phantom given in Fig. 8.
The 3D ball phantom with radius 0.5, center (0, 0, 0.25) and unit density (left), and $90 \times 90$ image reconstructed via (34) from weighted cone data simulated using 1800 counts for vertices $u$ on the unit sphere, 1800 counts for directions $\beta$ and 200 counts for opening angles $\psi$ (right). The cross sections by the planes $x=0, y=0$ and $z=0.25$ are shown.
Comparison of the $x$-axis (left), $y$-axis (center) and $z$-axis (right) profiles of the reconstruction and the phantom given in Fig. 10.
The 3D ball phantom with radius 0.5, center (0, 0, 0.25) and unit density (left), and $90 \times 90$ image reconstructed via (34) from weighted cone data contaminated with $5\%$ Gaussian white noise (right). The dimensions of the cone projections are taken as in Fig. 10. The cross sections by the planes $x=0, y=0$ and $z=0.25$ are shown.
Comparison of the $x$-axis (left), $y$-axis (center) and $z$-axis (right) profiles of the reconstruction and the phantom given in Fig. 12.
The 3D ball phantom with radius 0.5, center (0, 0, 0.25) and unit density (left), and $90 \times 90$ image reconstructed via (35) from weighted divergent beam data simulated using 1800 counts for sources $u$ on the unit sphere and 30K counts for unit directions $\sigma$ (right). The cross sections by the planes $x=0, y=0$ and $z=0.25$ are shown.
Comparison of the $x$-axis (left), $y$-axis (center) and $z$-axis (right) profiles of the phantom and the reconstruction given in Fig. 14.
The 3D ball phantom with radius 0.5, center (0, 0, 0.25) and unit density (left), and $90 \times 90$ image reconstructed via (36) from weighted divergent beam data simulated using 1800 counts for sources $u$ on the unit sphere and 30K counts for unit directions $\sigma$ (right). The cross sections by the planes $x=0, y=0$ and $z=0.25$ are shown.
Comparison of the $x$-axis (left), $y$-axis (center) and $z$-axis (right) profiles of the reconstruction and the phantom given in Fig. 16.
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