An efficient mollifier method for three-dimensional vector tomography: convergence analysis and implementation

Abstract

We consider the problem of three-dimensional vector tomography, that means the reconstruction of vector fields and their curl from line integrals over certain components of the field. It is well known that only the solenoidal part of the field can be recovered from these data. In this paper the method of approximate inverse is modified for vector fields and applied to this problem, leading to an efficient solver of filtered backprojection type. We prove convergence of the reconstructed solution, if the number of data tends to infinity, which means the method is exact. Finally, numerical results are presented for a straight flow through a cylinder.

The Formula of Grangeat for Tensor Fields of Arbitrary Order in n Dimensions

“…Reconstruction kernels are necessary to cope the problem of tensor tomography with the method of approximate inverse ; see, for example, Louis [13], Schuster [3], Rieder and Schuster [14]. It is clear that $\text{boldDf}\left(a,\omega \right)\omega +{\alpha }_1\left(a,\omega ,{\omega }_1\right){\omega }_1+{\alpha }_2\left(a,\omega ,{\omega }_2\right){\omega }_2=true{\int }_0^{\infty }\text{boldf}\left(a+t\omega \right)\text{d}t$ for certain coefficients α 1 , α 2 , where { ω, ω 1 , ω 2 } forms an orthonormal basis of ℝ 3 .…”

“…For m = 1, the operator D is the longitudinal X-ray transform of vector fields. A lot of numerical algorithms have been developed in recent years to solve the inverse problem D f = g in case m = 0 and m = 1; see, for example, Louis [1], Katsevich [2], Schuster [3], Derevtsov and Kashina [4], Sparr et al [5] among others. But also for tensor fields of order m > 1, this transform is of interest in various applications such as photoelasticity and plasma physics.…”

Vector tomography for the coronal magnetic field

“…To which extent we can reconstruct the divergence-free part V in (9) depends on the geometry of the view directions available. Schuster (2001) has shown that V can be reconstructed from observations from all three space directions. In our case, we have to expect that the solvability is limited by the fact that we usually have observations only from the ecliptic plane and in addition even from the ecliptic plane certain directions onto a given coronal volume element are missing due to the occultation of the Sun.…”

The approximate inverse in action III: 3D-Doppler tomography

“…Moreover, we only require an invariance of A * . In this respect Lemma 2.3 is an abstract modification of Theorem 3.1 from [11]. As a practical consequence we only need to find one single pair (e, υ) ∈ X × Y fulfilling (2.9), see Sect.…”

“…Here we present a convenient parameterization of this mapping and recall some of its properties which we will need later. The material is taken from [10] and [11].…”