3D Image Reconstruction from Compton camera data

In this paper, we address analytically and numerically the inversion of the integral transform (\emph{cone} or \emph{Compton} transform) that maps a function on ${\mathbb{R}}^3$ to its integrals over conical surfaces. It arises in a variety of imaging techniques, e.g. in astronomy, optical imaging, and homeland security imaging, especially when the so called Compton cameras are involved. Several inversion formulas are developed and implemented numerically in $3D$ (the much simpler $2D$ case was considered in a previous publication). An admissibility condition on detectors geometry is formulated, under which all these inversion techniques will work.