Inversion of the circular Radon transform on an annulus

Abstract

The representation of a function by its circular Radon transform (CRT) and various related problems arise in many areas of mathematics, physics and imaging science. There has been a substantial spike of interest towards these problems in the last decade mainly due to the connection between the CRT and mathematical models of several emerging medical imaging modalities. This paper contains some new results about the existence and uniqueness of the representation of a function by its circular Radon transform with partial data. A new inversion formula is presented in the case of the circular acquisition geometry for both interior and exterior problems when the Radon transform is known for only a part of all possible radii. The results are not only interesting as original mathematical discoveries, but can also be useful for applications, e.g. in medical imaging.

Instability of the linearized problem in multiwave tomography of recovery both the source and the speed

“…The problem of recovery of f , given c has received a lot of attention in the past years. We refer to [1,2,3,4,5,6,7,8,9,10,13,14,15,17,18,23,24,28,29] for some works in this direction. If c is constant, then the inversion of Λ 1 is actually an integral geometry problem as well.…”

Image Reconstruction in Quantitative Photoacoustic Tomography with the Simplified $P_2$ Approximation

“…From the measured data, we attempt to infer information on the optical properties of the underlying medium, for instance, the coefficients σ a , σ s and Ξ. This inverse problem has been extensively investigated in the past decade, from mathematical, computational as well as practical perspectives; see for instance [2,3,4,5,13,15,17,18,19,20,29,35,36,37,39,44,46,49,54,55,56,58,64,66,69,70,73,77,78,79] and references therein.…”

“…This is the same approach that has been recently used in [25,57,63]. An alternative, in fact more popular, approach for PAT reconstruction is a two-step strategy: (i) to reconstruct the initial pressure field H from measured ultrasound data; and then (ii) to reconstruct the optical coefficients from the reconstructed initial pressure field H. The first step involves only the acoustic model and is independent of the optical model, and reconstruction algorithms for this step have previously been developed in many scenarios [2,3,4,15,17,30,36,38,44,58,69,73,77]. The second step of the reconstruction have been developed for both the diffusion model (12) [10,12] and the radiative transport model (1) [9,51,65], but not the simplified P 2 model (9), to our best knowledge.…”

Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data

“…Using equations (3) and (5) one can relate the data g(t, y) to the integrals I(r, y) of f over spheres (circles) with the centers lying on S : g(t, y) = ∂ ∂t I(t, y) 4πt , n = 3, y ∈ S, (7) g(t, y) = ∂ ∂t t 0 I(r, y) 2π √ t 2 − r 2 dr, n = 2, y ∈ S,…”