Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms

Abstract

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.

The attenuated geodesic x-ray transform

“…In this case, injectivity and stability were proved in [19] and inversions were given in [11]. Such a transform can also be considered over vector fields (the so-called Doppler transform [8,6,18]), or higher-order tensor fields [17,12]. Once this simplicity condition is violated by the presence of conjugate points, stability is at stake and involves the interplay of a few factors, as explained below.…”

The Attenuated Geodesic Ray Transform on Tensors: Generic Injectivity and Stability

“…On simple surfaces, s-injectivity was proven in [2] (see Remark 7.5 therein) following [28,29]. Inversion formulas/procedure were given on Euclidean unit disc [26] and on simple surfaces [25]. Range characterization of I a was studied in Euclidean case [33] and on simple surfaces [3].…”

Partial inversion of the 2D attenuated $ X $-ray transform with data on an arc

“…The method of reconstruction stated in the result above is yet another application of the theory of A-analytic functions originally developed by Bukhgeim [8] to address the tomography problem from complete data, see [2] for the attenuated case. For different approaches to the inversion of the attenuated X-ray transform from complete data we refer to [23,24], and further developments in [20,6,3,14,17].…”