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

TL;DR

This paper introduces a stable partial reconstruction algorithm for the exterior inverse problem of the circular means transform, crucial for intravascular imaging techniques, effectively handling the instability caused by invisible wavefronts.

Contribution

It presents a novel inversion method that provides stable partial reconstructions of the exterior problem for the CMT, addressing instability issues in intravascular imaging.

Findings

The algorithm reconstructs the visible part of the image accurately.

Numerical simulations confirm the feasibility of tomography-like reconstructions.

The method handles the instability caused by invisible wavefronts effectively.

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.