Invertibility and Stability for A Generic class of Radon Transforms with Applications to Dynamic Operators

TL;DR

This paper analyzes the invertibility and stability of a class of dynamic Radon transforms, revealing conditions under which singularities can be recovered and providing stability estimates, with applications to Fan beam geometry.

Contribution

It establishes generic conditions for invertibility and stability of dynamic Radon transforms, extending understanding to moving objects and specific geometries.

Findings

Not all singularities are recoverable when the object moves.

Stability and injectivity hold under visibility, no conjugate points, and Bolker conditions.

Results apply to Fan beam geometry.

Abstract

Let X be an open subset of R^2. We study the dynamic operator, A, integrating over a family of level curves in X when the object changes between the measurement. We use analytic microlocal analysis to determine which singularities can be recovered by the data-set. Our results show that not all singularities can be recovered, as the object moves with a speed lower than the X-ray source. We establish stability estimates and prove that the injectivity and stability are of a generic set if the dynamic operator satisfies the visibility, no conjugate points, and local Bolker conditions. We also show this results can be implemented to Fan beam geometry.