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April 2018, 12(2): 433-460. doi: 10.3934/ipi.2018019
Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms
François Monard
University of California Santa Cruz, Department of Mathematics, 1156 High St. Santa Cruz, CA 95064, USA
Received May 2017 Revised October 2017 Published February 2018
Fund Project: The author is supported by NSF grant DMS-1712790.
Figure(1)
This article extends the author's past work [11] to attenuated X-ray transforms, where the attenuation is complex-valued and only depends on position. We give a positive and constructive answer to the attenuated tensor tomography problem on the Euclidean unit disc in fan-beam coordinates. For a tensor of arbitrary order, we propose an equivalent tensor of the same order which can be uniquely and stably reconstructed from its attenuated transform, as well as an explicit and efficient procedure to do so.
Keywords: Tensor tomography, fan-beam coordinates, attenuated X-ray transform, radon transform, Doppler transform, killing decomposition, invariant distributions.
Mathematics Subject Classification: Primary: 35R30, 44A12; Secondary: 30E20, 92C55, 58J32.
Citation: François Monard. Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms. Inverse Problems & Imaging, 2018, 12 (2) : 433-460. doi: 10.3934/ipi.2018019
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Figure 1. Fan-beam coordinates, scattering relation $\mathcal{S}$ and antipodal scattering relation ${{\mathcal{S}}_{A}}$
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