Exterior/interior problem for the circular means transform with applicationsto intravascular imaging

Gaik Ambartsoumian; Leonid Kunyansky

doi:10.3934/ipi.2014.8.339

Article Contents

2014, Volume 8, Issue 2: 339-359. Doi: 10.3934/ipi.2014.8.339

This issue Previous Article Next Article Minimal partitions and image classification using a gradient-free perimeter approximation

Exterior/interior problem for the circular means transform with applications to intravascular imaging

Gaik Ambartsoumian^1, and
Leonid Kunyansky^2,

1.
Department of Mathematics, University of Texas, Arlington, TX 76019
2.
Department of Mathematics, University of Arizona, 617 N Santa Rita Ave, Tucson, AZ 85721

Received: July 31, 2013

Revised: November 30, 2013

Published: April 2014

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Abstract

Exterior inverse problem for the circular means transform (CMT) arises in the intravascular photoacoustic imaging (IVPA), in the intravascular ultrasound imaging (IVUS), as well as in radar and sonar. The reduction of the IPVA to the CMT is quite straightforward. As shown in the paper, in IVUS the circular means can be recovered from measurements by solving a certain Volterra integral equation. Thus, a tomography reconstruction in both modalities requires solving the exterior problem for the CMT.
Numerical solution of this problem usually is not attempted due to the presence of "invisible" wavefronts, which results in severe instability of the reconstruction. The novel inversion algorithm proposed in this paper yields a stable partial reconstruction: it reproduces the "visible" part of the image and blurs the "invisible" part. If the image contains little or no invisible wavefronts (as frequently happens in the IVPA and IVUS) the reconstruction is quantitatively accurate. The presented numerical simulations demonstrate the feasibility of tomography-like reconstruction in these modalities.

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Mathematics Subject Classification: Primary: 44A12, 65M32; Secondary: 92C55.

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