# 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 and Imaging, 2017, 11 (6) : 1071-1090. doi: 10.3934/ipi.2017049
##### References:

show all references

##### References:
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.
 [1] Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems and Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649 [2] Sean Holman, Plamen Stefanov. The weighted Doppler transform. Inverse Problems and Imaging, 2010, 4 (1) : 111-130. doi: 10.3934/ipi.2010.4.111 [3] Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029 [4] Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems and Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021 [5] C E Yarman, B Yazıcı. A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group. Inverse Problems and Imaging, 2007, 1 (3) : 457-479. doi: 10.3934/ipi.2007.1.457 [6] Jean-François Crouzet. 3D coded aperture imaging, ill-posedness and link with incomplete data radon transform. Inverse Problems and Imaging, 2011, 5 (2) : 341-353. doi: 10.3934/ipi.2011.5.341 [7] Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems and Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879 [8] Yang Zhang. Artifacts in the inversion of the broken ray transform in the plane. Inverse Problems and Imaging, 2020, 14 (1) : 1-26. doi: 10.3934/ipi.2019061 [9] Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems and Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023 [10] Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems and Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721 [11] Gaik Ambartsoumian, Leonid Kunyansky. Exterior/interior problem for the circular means transform with applications to intravascular imaging. Inverse Problems and Imaging, 2014, 8 (2) : 339-359. doi: 10.3934/ipi.2014.8.339 [12] Shanshan Wang, Yanxia Chen, Taohui Xiao, Lei Zhang, Xin Liu, Hairong Zheng. LANTERN: Learn analysis transform network for dynamic magnetic resonance imaging. Inverse Problems and Imaging, 2021, 15 (6) : 1363-1379. doi: 10.3934/ipi.2020051 [13] Mathieu Lutz. Application of Lie transform techniques for simulation of a charged particle beam. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 185-221. doi: 10.3934/dcdss.2015.8.185 [14] Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006 [15] James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems and Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013 [16] Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325 [17] Hiroshi Fujiwara, Kamran Sadiq, Alexandru Tamasan. Partial inversion of the 2D attenuated $X$-ray transform with data on an arc. Inverse Problems and Imaging, 2022, 16 (1) : 215-228. doi: 10.3934/ipi.2021047 [18] Sebastian Reich, Seoleun Shin. On the consistency of ensemble transform filter formulations. Journal of Computational Dynamics, 2014, 1 (1) : 177-189. doi: 10.3934/jcd.2014.1.177 [19] Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039 [20] Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems and Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27

2020 Impact Factor: 1.639

## Tools

Article outline

Figures and Tables