Motion compensated local tomography

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 f Λ , 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 f Λ 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 f Λ . 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.