Motion compensated local tomography

Abstract

In this paper we develop local tomography (LT) for image reconstruction from motion contaminated data. It is assumed that motion is known. We propose a new LT function fΛ, which is related to an original object f via an operator : . Because of motion, may fail to be a pseudo-differential operator (PDO). We obtain the conditions that guarantee that is a PDO. Under these conditions, similarly to the classical LT in is a PDO of order 1. Computation of fΛ depends on a weight function Φ. We show that Φ can be chosen in such a way that the operator has principal symbol |ξ|. This result has an interesting corollary for conventional exact reconstruction. It suggests a novel frequency-split approach to finding f from motion contaminated data. In practice tomographic data are discrete, and derivatives are usually replaced by their mollified analogs. We consider how mollification affects the singularities of the LT function fΛ. Using this approach we develop an algorithm for finding values of jumps of f using LT. We also consider various aspects of numerical implementation of LT and show the results of numerical experiments.