Abstract
We propose a new acquisition geometry for electron density reconstruction in three dimensional x-ray Compton imaging using a monochromatic source. This leads us to a new three dimensional inverse problem where we aim to reconstruct a real valued function f (the electron density) from its integrals over spindle tori. We prove injectivity of a generalized spindle torus transform on the set of smooth functions compactly supported on a hollow ball. This is obtained through the explicit inversion of a class of Volterra integral operators, whose solutions give us an expression for the harmonic coefficients of f. The polychromatic source case is later considered, and we prove injectivity of a new spindle interior transform, apple transform and apple interior transform on the set of smooth functions compactly supported on a hollow ball.A possible physical model is suggested for both source types. We also provide simulated density reconstructions with varying levels of added pseudo random noise and model the systematic error due to the attenuation of the incoming and scattered rays in our simulation.
Highlights
In this paper we lay the foundations for a new three dimensional imaging technique in X-ray Compton scattering tomography
Three dimensional Compton tomography is considered in the literature, where a gamma source is reconstructed from its integrals over cones with a fixed axis direction [5, 6, 7]
We show that our problem can be decomposed as a set of one dimensional inverse problems, which we solve to provide an explicit expression for the harmonic coefficients of f
Summary
In this paper we lay the foundations for a new three dimensional imaging technique in X-ray Compton scattering tomography. In [3], Nguyen and Truong present an acquisition geometry in two dimensions for a monochromatic source (e.g. a gamma ray source) and energy resolving detector pair, where the source and detector remain at a fixed distance opposite one another and are rotated about the origin on the curve S1 (the unit circle). A new apple and apple interior transform are introduced in section 2.2 for the monochromatic and polychromatic backscatter problem Their injectivity is proven on the domain of smooth functions compactly supported on the intersection of the exterior of the unit ball and x3 > 0. We simulate the added effects due to the attenuation of the incoming and scattered rays in our data and see how this systematic error effects the quality of our reconstruction
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
