Back-projection inversion of a conical Radon transform
Résumé
In an effort to deal with many ionizing radiation imaging mechanisms involving the Compton effect,
we study a Radon transform on circular cone surfaces having a fixed axis direction, which is called
here conical Radon transform (CRT). Concretely we seek to recover a density function $f(x,y,z)$
in $\mathbb{R}^{3}$ from its integrals over such circular cone surfaces or its conical
projections. Although the existence of the inverse CRT has been established, it is
the aim of this work to use this result to extent the concept of back-projection well known in
Computed Tomography (CT) to this type of cone surfaces. We discuss in some details the features
of back-projection in relation to the corresponding conical Radon transform adjoint operator as well as
the filters that arise naturally from the exact solution of the inversion problem.
This intuitive approach is attractive, lends itself to efficient computational algorithms and may
provide hints and guide for more general back-projection methods on other classes of cone surfaces, for example occurring in
Compton camera imaging. Comprehensive numerical simulations results are presented and discussed
to illustrate and validate this approach based on the concept of back-projection.
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