Inversion of the broken ray transform in the case of energy-dependent attenuation

• Published in: Physics in medicine & biology Vol. 60; no. 11; pp. 4313 - 4334
• Main Authors: Krylov, R, Katsevich, A
• Format: Journal Article
• Language: English
• Published: England IOP Publishing 07.06.2015
• Subjects: Algorithms; Humans; imaging; Linear Energy Transfer; Photons; ray integrals; scattering; Scattering, Radiation; X-Rays
• ISSN: 0031-9155; 1361-6560; 1361-6560
• DOI: 10.1088/0031-9155/60/11/4313

Broken Ray transform (BRT) arises when one considers a narrow x-ray beam propagating through medium under the assumption of single scattering. Previous algorithms for inverting the BRT assumed that the medium is characterized by a single attenuation coefficient μ. However x-rays lose their energy after Compton scattering and the energy loss depends on the scattering angle. Since the attenuation coefficient depends on energy, the μ's before and after scattering are different. When there are three or more detectors one should distinguish not only between μ's that are 'seen' by x-rays before and after scattering, but also between μ's that are 'seen' by x-rays traveling towards different detectors. The main thrust of this paper is inversion of the BRT with N 3 detectors under the assumption that the attenuation coefficient can be accurately approximated by a linear function of energy within the window of relevant energies. When the number of detectors is four or greater, we derive a family of inversion formulas. If N > 4, we find the optimal formula, which provides the best stability with respect to noise in the data. If N = 4, the family collapses into a single formula and no optimization is possible. If μ is independent of energy, N = 3 is sufficient for inversion. We also develop iterative reconstruction algorithms that can use global and local data. The results of testing the algorithms are presented.