## ROI reconstruction from truncated cone-beam projections

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### Abstract

Region-of-Interest (ROI) tomography aims at reconstructing a region of interest CC inside a body using only x-ray projections intersecting CC 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 ΓΓ in R3R3 verifying the classical Tuy condition. In this generic situation, the *non*-*trucated* cone-beam transform of smooth density functions ff admits an explicit inverse ZZ as originally shown by Grangeat. However ZZ cannot directly reconstruct ff from ROI-truncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities ff in L∞(B)L∞(B) where BB is a bounded ball in R3R3, our method iterates an operator UU combining ROI-truncated projections, inversion by the operator ZZ and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI C⊂BC⊂B, given ε>0ε>0, we prove that if CC is sufficiently large our iterative reconstruction algorithm converges at exponential speed to an εε-accurate approximation of ff in L∞L∞. The accuracy depends on the regularity of ff quantified by its Sobolev norm in W5(B)W5(B). Our result guarantees the existence of a critical ROI radius ensuring the convergence of our ROI reconstruction algorithm to an εε-accurate approximation of ff. 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 BB.Region-of-Interest (ROI) tomography aims at reconstructing a region of interest CC inside a body using only x-ray projections intersecting CC 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 ΓΓ in R3R3 verifying the classical Tuy condition. In this generic situation, the *non*-*trucated* cone-beam transform of smooth density functions ff admits an explicit inverse ZZ as originally shown by Grangeat. However ZZ cannot directly reconstruct ff from ROI-truncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities ff in L∞(B)L∞(B) where BB is a bounded ball in R3R3, our method iterates an operator UU combining ROI-truncated projections, inversion by the operator ZZ and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI C⊂BC⊂B, given ε>0ε>0, we prove that if CC is sufficiently large our iterative reconstruction algorithm converges at exponential speed to an εε-accurate approximation of ff in L∞L∞. The accuracy depends on the regularity of ff quantified by its Sobolev norm in W5(B)W5(B). Our result guarantees the existence of a critical ROI radius ensuring the convergence of our ROI reconstruction algorithm to an εε-accurate approximation of ff. 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 BB.

### Publication metadata

**Author(s): **Azencott A, Bodmann BG, Chowdhury T, Labate D, Sen A, Vera A

**Publication type: **Article

**Publication status:** Published

**Journal: **Inverse Problems and Imaging

**Year: **2018

**Volume: **12

**Issue: **1

**Pages: **29-57

**Online publication date: **01/02/2018

**Acceptance date: **19/09/2017

**ISSN (print): **1930-8337

**ISSN (electronic): **1930-8345

**Publisher: **American Institute of Mathematical Sciences

**URL: **https://doi.org/10.3934/ipi.2018002

**DOI: **10.3934/ipi.2018002

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