An inversion of the conical Radon transform arising in the Compton camera with helical movement

Abstract

Since the Compton camera was first introduced, various types of conical Radon transforms have been examined. Here, we derive the inversion formula for the conical Radon transform, where the cone of integration moves along a curve in three-dimensional space such as a helix. Along this three-dimensional curve, a detailed inversion formula for helical movement will be treated for Compton imaging in this paper. The inversion formula includes Hilbert transform and Radon transform. For the inversion of Compton imaging with helical movement, it is necessary to invert Hilbert transform with respect to the inner product between the vertex and the central axis of the cone of the Compton camera. However, the inner product function is not monotone. Thus, we should replace the Hilbert transform by the Riemann-Stieltjes integral over a certain monotone function related with the inner product function. We represent the Riemann-Stieltjes integral as a conventional Riemann integral over a countable union of disjoint intervals, whose end points can be computed using the Newton method. For the inversion of Radon transform, three dimensional filtered backprojection is used. For the numerical implementation, we analytically compute the Hilbert transform and Radon transform of the characteristic function of finite balls. Numerical test is given, when the density function is given by a characteristic function of a ball or three overlapping balls.