Exponential vector field tomography

Abstract

In the exponential Radon transform in 1~2, the integrals of a scalar function f over lines, with exponential weight functions, are determined. In this paper we demonstrate how to define two different kinds of exponential Radon transforms for vector fields in R 2 in a natural way. It is shown that having data from these transforms it is possible to reconstruct the vector field uniquely. The motivation to study this problem is ultrasound measurements of flows, from which velocity spectra along lines can be determined. The first moment of these can be interpreted by means of one of the exponential vectorial Radon transforms.

Inversion of the scalar and vector attenuated X-ray transforms in a unit disc

“…The similar transformation for vector fields was first introduced by K. Stråhlen in [45] for the constant attenuation and was called the exponential vectorial transform.…”

“…The inverse problem here consists in determining the unknown vector field a(x) from the measured sinogram f (β, ϕ), provided that the attenuation map µ(x) is a known realvalued function. It should be stated that in the unattenuated case (when µ(x) ≡ 0) we can restore only the solenoidal part of the vector field (see [9,23,42,45]). In another words, since for arbitrary square integrable vector field we have the Helmholtz decomposition into the potential and solenoidal parts, L 2 (Ω) = ∇H 1 0 (Ω) ⊕ H(Ω; div = 0) (see [15, p. 216]), then in the case µ(x) ≡ 0 we can only determine the component H(Ω; div = 0).…”

Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography

“…The present definitions of the unweighted longitudinal and transversal Radon transforms coincide with those given in [4,6] (where they are mentioned under the names of "probe" and "normal" transforms, respectively). Our definitions of the weighted transforms appear to be new; they naturally extend the notion of "moments ray transforms" [6,11] to the case of Radon transforms.…”

“…There is a significant body of work on ray transforms (that involve integration over straight lines) of vector and tensor fields [1][2][3][4][5]. In particular, exponential and attenuated ray transforms were studied in [6][7][8][9], and momentum ray transforms were investigated in [10,11]. However, when it comes to the Radon transforms of vector fields (with integration over hyperplanes), there are very few publications [12,13]; moreover, the consideration is usually restricted to unweighted transforms of potential fields with finitely supported potentials.…”

Attenuated Vector Tomography -- An Approach to Image Flow Vector Fields with Doppler Ultrasonic Imaging

“…The Doppler flow measurements obtained using a directional beam of a continuous ultrasound wave can be formulated as a directional projection of a flow vector field [1,2,3]. The continuous wave reflects from points at different distances from the ultrasound detector.…”