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

3D Image Reconstruction from Compton camera data