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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.
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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