Inversion of circular means and the wave equation on convex planar domains

We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary \partial \Om of a smooth convex bounded domain \Om \subset \R^2. As a main result we establish back-projection type inversion formulas that recover any initial data with support in \Om modulo an explicitly computed smoothing integral operator \K_\Om. For circular and elliptical domains the operator \K_\Om is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtained for recovering a function from its mean values over circles with centers on \partial \Om. Both reconstruction problems are, amongst others, essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography.Comment: [14 pages, 2 figures

Multiview Hilbert transformation in full-ring transducer array-based photoacoustic computed tomography

Based on the photoacoustic (PA) effect, PA tomography directly measures specific optical absorption, i.e., absorbed optical energy per unit volume. We recently developed a full-ring ultrasonic transducer array-based photoacoustic computed tomography (PACT) system for small-animal whole-body imaging. The system has a full-view detection angle and high in-plane resolution (∼100  μm). However, due to the bandpass frequency response of the piezoelectric transducer elements and the limited elevational detection coverage of the full-ring transducer array, the reconstructed images present bipolar (i.e., both positive and negative) pixel values, which cause ambiguities in image interpretation for physicians and biologists. We propose a multiview Hilbert transformation method to recover the unipolar initial pressure for full-ring PACT. The effectiveness of the proposed algorithm was first validated by numerical simulations and then demonstrated with ex vivo mouse brain structural imaging and in vivo mouse whole-body imaging

Universal inversion formulas for recovering a function from spherical means

The problem of reconstruction a function from spherical means is at the heart of several modern imaging modalities and other applications. In this paper we derive universal back-projection type reconstruction formulas for recovering a function in arbitrary dimension from averages over spheres centered on the boundary an arbitrarily shaped smooth convex domain. Provided that the unknown function is supported inside that domain, the derived formulas recover the unknown function up to an explicitly computed smoothing integral operator. For elliptical domains the integral operator is shown to vanish and hence we establish exact inversion formulas for recovering a function from spherical means centered on the boundary of elliptical domains in arbitrary dimension.Comment: [20 pages, 2 figures] Compared to the previous versions I corrected some typo