Recovering a function from its integrals over conical surfaces through relations with the Radon transform

Abstract

This paper addresses the overdetermined problem of inverting the n-dimensional cone (or Compton) transform that integrates a function over conical surfaces in . The study of the cone transform originates from Compton camera imaging, a nuclear imaging method for the passive detection of gamma-ray sources. We present a new identity relating the n-dimensional cone and Radon transforms through spherical convolutions with arbitrary weight functions. This relationship, which generalizes a previously obtained identity, leads to various inversion formulas in n-dimensions under a mild assumption on the geometry of detectors. We present two such formulas along with the results of their numerical implementation using synthetic phantoms. Compared to our previously discovered inversion techniques, the new formulas are more stable and simpler to implement numerically.