Exact Inversion of the Cone Transform Arising in an Application of a Compton Camera Consisting of Line Detectors

Abstract

Abstract.A Compton camera has been suggested for use in single photon emission computed tomography because a conventional gamma camera has low efficiency. Here we consider a cone transform brought about by a Compton camera with line detectors. A cone transform takes a given function on the 3-dimensional space and assigns to it the surface integral of the function over cones determined by the 1-dimensional vertex space, the 1-dimensional central axis, and the 1-dimensional opening angle. We generalize this cone transform to n-dimensional space and provide an inversion formula. Also, numerical simulations are presented to demonstrate our suggested algorithm in three dimensions.Key words. Compton camera, SPECT, tomography, cone transform, inversion, gamma camera AMS subject classifications. 44A12, 65R10, 92C55DOI. 10.1137/15M10336171. Introduction. A Radon-type transform that assigns to a given function its surface integral over various sets of cones has been studied, since it is known that this kind of a transform relates to a Compton camera. A Compton camera, also called an electronically collimated γ-camera, was introduced for use in single photon emission computed tomography (SPECT) because of the low efficiency of a conventional γ-camera [26,32]. A Compton camera has very high sensitivity and flexibility of geometrical design, so it has attracted a lot of interest and applications in many areas including monitoring nuclear power plants and astronomy [1,3].A standard Compton camera consists of two planar detectors: a scatter detector and an absorption detector, positioned one behind the other. A photon emitted from a radioactive source toward the camera undergoes Compton scattering in the scatter detector, and is absorbed in the absorption detector positioned behind (see Figure 1). In each detector, the position of the hit and the energy of the photon are measured. The scattering angle or the

The Radon Transform over Cones with Vertices on the Sphere and Orthogonal Axes

“…Variants of the conical Radon transform in R 2 are known as V-line or broken-ray transforms. These transforms appear in emission tomography with one-dimensional Compton cameras [5,22], or in the recently developed single scattering optical tomography [16]. In this paper, we consider the conical Radon transform in general dimension and further include a radial weight, that can be adjusted to a particular application at hand.…”

The V-line transform with some generalizations and cone differentiation

“…Corollary 3 One can invert the conical Radon transform T by using formula (24) to generate F from T f and then applying Theorem 3 or Theorem 4.…”

“…Many researchers have obtained interesting results on the CRT and the VLT for such setups (e.g. see [15,16,18,23,24,27,28,33,34,35,36,37,41,43,45,46,47]). A nice survey of this field was recently published in [44].…”

Inversion of the attenuated conical Radon transform with a fixed opening angle

“…During the last decade, the interest towards such transforms was triggered by the connection between the conical Radon transform and the mathematical models of many novel imaging modalities. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] In all the above works, the attenuation phenomena was neglected. However, in many medical imaging techniques, ignoring the effect of the attenuation of photon can significantly degrade the quality of the reconstruction image.…”