# 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
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##### 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.

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