# American Institute of Mathematical Sciences

April  2020, 14(2): 291-316. doi: 10.3934/ipi.2020013

## Robust and stable region-of-interest tomographic reconstruction using a robust width prior

 1 imec-IPI-Ghent University, B9000 Gent, Belgium 2 Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA

Received  May 2019 Revised  September 2019 Published  February 2020

Fund Project: The second author is supported by NSF grants DMS 1720487 and 1720452 and the third author is supported by NSF grant DMS 1715735.

Region-of-interest computed tomography (ROI CT) aims at reconstructing a region within the field of view by using only ROI-focused projections. The solution of this inverse problem is challenging and methods of tomographic reconstruction that are designed to work with full projection data may perform poorly or fail when applied to this setting. In this work, we study the ROI CT problem in the presence of measurement noise and formulate the reconstruction problem by relaxing data fidelity and consistency requirements. Under the assumption of a robust width prior that provides a form of stability for data satisfying appropriate sparsity-inducing norms, we derive reconstruction performance guarantees and controllable error bounds. Based on this theoretical setting, we introduce a novel iterative reconstruction algorithm from ROI-focused projection data that is guaranteed to converge with controllable error while satisfying predetermined fidelity and consistency tolerances. Numerical tests on experimental data show that our algorithm for ROI CT is competitive with state-of-the-art methods especially when the ROI radius is small.

Citation: Bart Goossens, Demetrio Labate, Bernhard G Bodmann. Robust and stable region-of-interest tomographic reconstruction using a robust width prior. Inverse Problems & Imaging, 2020, 14 (2) : 291-316. doi: 10.3934/ipi.2020013
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##### References:
Illustrations of recoverable regions for truncated projection data: (a) initial DBP methods [25,7,35] require at least one projection view in which the complete object is covered, (b) interior reconstruction is possible given a known subregion [20,34,30,17] and (c) no assumptions are made other than that the shape of the ROI is convex and approximate sparsity within a ridgelet domain (this paper). The gray dashed line indicates the measured area on the detector array for one particular source angle
Illustration of ROI tomographic reconstruction problem. (a) In ROI tomography, only projections intersecting the Region-of-Interest $S$ are known. (b) In the Radon-transform domain, the region $P(S)$ corresponding to $S$ is a strip-like domain
Schematic illustration of the solution space for $y$, given a current estimate $f$, as intersection of the balls $\left\Vert My-y_{0}\right\Vert ^{2}\leq \alpha$ and $\left\Vert y-Wf\right\Vert ^{2}\leq \beta$, as well as solutions favored by the sparsity prior $\left\Vert {y}\right\Vert _{\sharp}$ (see Sec. 3). Data consistent solutions may have a non-zero data fidelity, while data fidelity solutions are in general not consistent. We control the reconstruction error by combining fidelity and consistency constraints with the additional sparsity assumption
Commutative diagram of the measurement operator $\Phi$ and the restricted measurement operator $\Phi '$ (see Theorem 3.5). The following relationship holds: $P_{\tilde{\mathcal{H}}'}\Phi = \Phi 'P_{\mathcal{H}'}$.
Full LSCG reconstruction of the fan-beam data sets: (a) Preclinical - lungs, (b) Preclinical - abdomen. Images are cropped for visualization purposes
PSNR results for 2D fan-beam ROI reconstruction with increasing radius using (a) lungs and (b) adbomen. The PSNR is calculated inside the ROI
ROI reconstruction results for a fixed radius of $64$ pixels (or 3.2 mm). (a), (b): SIRA with different values of $\alpha$, $\beta$ and $u = \Vert (I-\tilde{W}\tilde{W}^{+})y_{0}\Vert _{2}^{2}$; (c): LSCG; (d): CS-TV, (e): CS-ridgelet, (f) Full view LSCG reconstruction (used for computing PSNR)
(a)-(c): Reconstruction results for different ROI radii using SIRA with fixed parameters $\alpha = 0$, $\beta = 0.25\left\Vert \left(I-\tilde{W}\tilde{W}^{+}\right)y_{0}\right\Vert _{2}^{2}.$ (d)-(f): Reconstruction results using LSCG for same ROI radii
CT fan-beam acquisition geometries (X-O CT system from Gamma Medica-Ideas) used in the experiments for this paper
 Data set Geometry parameter Preclinical - lungs Preclinical - abdomen Distance source-detector 146.09 mm 145.60 mm Distance source-object 41.70 mm 57.92 mm Detector offset -15.00 mm 12.14 mm Detector elements 592 592 Projection angles 512 640 Pixel pitch 0.20 mm 0.20 mm
 Data set Geometry parameter Preclinical - lungs Preclinical - abdomen Distance source-detector 146.09 mm 145.60 mm Distance source-object 41.70 mm 57.92 mm Detector offset -15.00 mm 12.14 mm Detector elements 592 592 Projection angles 512 640 Pixel pitch 0.20 mm 0.20 mm
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