The broken ray transform: additional properties and new inversion formula

Abstract

The significance of the broken ray transform (BRT) is due to its occurrence in a number of modalities spanning optical, x-ray, and nuclear imaging. When data are indexed by the scatter location, the BRT is both linear and shift invariant. Analyzing the BRT as a linear system provides a new perspective on the inverse problem. In this framework we contrast prior inversion formulas and identify numerical issues. This has practical benefits as well. We clarify the extent of data required for global reconstruction by decomposing the BRT as a linear combination of cone beam transforms. Additionally we leverage the two dimensional Fourier transform to derive new inversion formulas that are computationally efficient for arbitrary scatter angles. Results of numerical simulations are presented.

Iterative Algorithms for Joint Scatter and Attenuation Estimation From Broken Ray Transform Data

“…Even positron emission tomography can be described in this way [3]. Recent analysis of the related broken ray transform (BRT) has clarified the conditioning and requirements for joint image recovery under the single-scatter approximation [3]- [9]. X-ray scatter imaging has a long history [10] with many potential applications (see [11], [12] and references therein).…”

“…Background 1) Measurement Geometries: Three measurement geometries have received attention recently. We focus on the selected volume tomography (SVT) [1] which has been referred to as a translation-only measurement geometry [9] and the BRT measurement system [3]. This is distinguished from rotational measurement geometries [7], [16]- [19] (related to secondary radiation tomography [1]).…”

“…For uniformly scattering media, analytic inversion of the attenuation image is well-defined for images with bounded support. However, the BRT is poorly scaled [7], [9]. For nonuniform scattering media, scattering effects can be canceled analytically incorporating data from a second scatter angle [5], [7], [9], [18].…”

“…However, the BRT is poorly scaled [7], [9]. For nonuniform scattering media, scattering effects can be canceled analytically incorporating data from a second scatter angle [5], [7], [9], [18]. Combining data from multiple scatter angles is trivial for media scattering isotropically.…”

“…Combining data from multiple scatter angles is trivial for media scattering isotropically. This extends easily to x-ray fluorescence imaging, but requires additional conditions for Bragg and Compton scatter imaging [9], [20]. Incorporating three or more source-detector pairs facilitates local analytic reconstruction of the attenuation image with improved reconstruction quality [3].…”