Three-Dimensional Image Reconstruction from Compton Camera Data

In this paper, we address analytically and numerically the inversion of the integral transform (cone or Compton transform) that maps a function on R³ 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 three dimensions (the much simpler two-dimensional case was considered in a previous publication). An admissibility condition on detectors geometry is formulated, under which all these inversion techniques will work.