ROI reconstruction from truncated cone-beam projections

Robert Azencott; Bernhard G. Bodmann; Tasadduk Chowdhury; Demetrio Labate; Anando Sen; Daniel Vera

doi:10.3934/ipi.2018002

Article Contents

2018, Volume 12, Issue 1: 29-57. Doi: 10.3934/ipi.2018002

This issue Previous Article Stability for a magnetic Schrödinger operator on a Riemann surface with boundary Next Article Generalized stability estimates in inverse transport theory

ROI reconstruction from truncated cone-beam projections

1.
Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA
2.
Department of Biomedical Informatics, Columbia University, New York, NY 10032, USA
3.
Matematicas, Instituto Tecnologico Autonomo de México, 01080 Ciudad de México, CDMX, México

* Corresponding author
* Corresponding author

Received: October 31, 2016

Revised: August 31, 2017

Published: January 2018

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Abstract

Region-of-Interest (ROI) tomography aims at reconstructing a region of interest $C$ inside a body using only x-ray projections intersecting $C$ and it is useful to reduce overall radiation exposure when only a small specific region of a body needs to be examined. We consider x-ray acquisition from sources located on a smooth curve $Γ$ in $\mathbb R^3$ verifying the classical Tuy condition. In this generic situation, the non-trucated cone-beam transform of smooth density functions $f$ admits an explicit inverse $Z$ as originally shown by Grangeat. However $Z$ cannot directly reconstruct $f$ from ROI-truncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities $f$ in $L^{∞}(B)$ where $B$ is a bounded ball in $\mathbb R^3$, our method iterates an operator $U$ combining ROI-truncated projections, inversion by the operator $Z$ and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI $C \subset B$, given $ε >0$, we prove that if $C$ is sufficiently large our iterative reconstruction algorithm converges at exponential speed to an $ε$-accurate approximation of $f$ in $L^{∞}$. The accuracy depends on the regularity of $f$ quantified by its Sobolev norm in $W^5(B)$. Our result guarantees the existence of a critical ROI radius ensuring the convergence of our ROI reconstruction algorithm to an $ε$-accurate approximation of $f$. We have numerically verified these theoretical results using simulated acquisition of ROI-truncated cone-beam projection data for multiple acquisition geometries. Numerical experiments indicate that the critical ROI radius is fairly small with respect to the support region $B$.

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Mathematics Subject Classification: Primary: 44A12, 65R10; Secondary: 92C55.

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Figure 1. ROI-truncated cone-beam acquisition: projections are restricted to half-rays intersecting the ROI, which is a ball $C$ included in the target ball $B$

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Figure 2. Visual comparison of ROI reconstruction for 3D Shepp-Logan phantom using simulated Twin Circles acquisition and truncation of projection data. ROI radius =45 voxels. Middles sections are shown from the $xy$, $yz$ and $xz$ planes. From left to right: inversion by one-step Grangeat formula; our iterative ROI reconstruction; ground truth. The last column shows intensity profiles corresponding to the middle row of the images. Green: one-step Grangeat formula; blue: our algorithm; red: ground truth

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Figure 3. Visual comparison of ROI reconstruction for Mouse Tissue data using simulated Twin Circles acquisition and truncation of projection data. ROI radius =45 voxels. Middles sections are shown from the $xy$, $yz$ and $xz$ planes. From left to right: inversion by one-step Grangeat's formula; our iterative ROI reconstruction; ground truth. The last column shows intensity profiles corresponding to the middle row of the images. Green: one-step Grangeat formula; blue: our algorithm; red: ground truth

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Figure 4. Visual comparison of ROI reconstruction for 3D Shepp-Logan phantom and mouse tissue using simulated spiral acquisition and truncation of projection data. A representative horizontal section from the 3D reconstructed volume is shown. From left to right: inversion by one-step Katsevich formula; our iterative ROI reconstruction; ground truth

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Figure 5. Visual comparison of ROI reconstruction for 3D Shepp-Logan phantom and mouse tissue using simulated C-arm acquisition and truncation of projection data. A representative horizontal section from the 3D reconstructed volume is shown. From left to right: inversion by one-step FDK algorithm; our iterative ROI reconstruction; ground truth

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Table 1. Relative $L^1$ error of ROI reconstruction

Density data	ROI radius	Sources locations
Density data	ROI radius	Spherical	Spiral	Circle	Twin circles
Shepp-Logan	45 vox	10.3%	10.9%	13.2%	14.8%
	60 vox	8.6%	9.1%	11.6%	14.7%
	75 vox	7.6%	8.3%	7.4%	8.9%
	90 vox	7.3%	8.0%	4.4%	4.8%
Mouse tissue	45 vox	10.8%	11.4%	11.6%	12.5%
	60 vox	8.8%	9.7%	11.1%	9.4%
	75 vox	7.9%	8.8%	8.4%	8.3%
	90 vox	7.5%	8.4%	7.1%	7.8%
Human jaw	45 vox	11.4%	11.9%	12.9%	15.0%
	60 vox	9.6%	10.8%	12.8%	13.3%
	75 vox	9.0%	9.7%	10.2%	10.2%
	90 vox	8.2%	8.5%	9.8%	9.8%

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Table 2. Critical radius of convergence (voxels)

Density data	Source locations
Density data	Spherical	Spiral	Circle	Twin circles
Shepp-Logan	52 vox	56 vox	67 vox	73 vox
Mouse tissue	52 vox	57 vox	66 vox	49 vox
Human jaw	57 vox	70 vox	82 vox	82 vox

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