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.
In this paper, we address exact inversion of the integral transform, called Compton (or cone) transform, that maps a function on R3 to its integrals over conical surfaces. Compton transform arises in passive detection of gammaray sources with a Compton camera which has promising applications in medical and industrial imaging as well as in homeland security imaging and astronomy. We present a two-step method that uses the full set of available projections for inverting the Compton transform: first the recovery of the Radon transform from the Compton transform, and then the Radon transform inversion. The first step can be done in various ways by means of the generalization of a previously obtained result relating the Compton and Radon transforms. This leads to a variety of Compton inversion formulas that are independent of the geometry of detectors as long as a generous admissibility condition is met. We formulate one inversion formula that is simpler and performs well in the case of noisy data, which is demonstrated by a numerical simulation.
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