Purpose: The paper addresses exact inversion of the integral transform, called the Compton (or cone) transform, that maps a three-dimensional (3-D) function to its integrals over conical surfaces in R³. Compton transform arises in passive detection of gamma-ray sources with a Compton camera, which has promising applications in medical and industrial imaging as well as in homeland security imaging and astronomy.

Approach: A generalized identity relating the Compton and the Radon transforms was formulated. The proposed relation can be used to devise a method for converting the Compton transform data of a function into its Radon projections. The function can then be recovered using well-known inversion techniques for the Radon transform.

Results: We derived a two-step method that uses the full set of available projections to invert the Compton transform: first, the recovery of the Radon transform from the Compton transform, and then the Radon transform inversion. The proposed technique is independent of the geometry of detectors as long as a generous admissibility condition is met.

Conclusions: We proposed an exact inversion formula for the 3-D Compton transform. The stability of the inversion algorithm was demonstrated via numerical simulations.