An Analytical Algorithm for Tensor Tomography From Projections Acquired About Three Axes

Abstract

Tensor fields are useful for modeling the structure of biological tissues. The challenge to measure tensor fields involves acquiring sufficient data of scalar measurements that are physically achievable and reconstructing tensors from as few projections as possible for efficient applications in medical imaging. In this paper, we present a filtered back-projection algorithm for the reconstruction of a symmetric second-rank tensor field from directional X-ray projections about three axes. The tensor field is decomposed into a solenoidal and irrotational component, each of three unknowns. Using the Fourier projection theorem, a filtered back-projection algorithm is derived to reconstruct the solenoidal and irrotational components from projections acquired around three axes. A simple illustrative phantom consisting of two spherical shells and a 3D digital cardiac diffusion image obtained from diffusion tensor MRI of an excised human heart are used to simulate directional X-ray projections. The simulations validate the mathematical derivations and demonstrate reasonable noise properties of the algorithm. The decomposition of the tensor field into solenoidal and irrotational components provides insight into the development of algorithms for reconstructing tensor fields with sufficient samples in terms of the type of directional projections and the necessary orbits for the acquisition of the projections of the tensor field. Index Terms-Filtered back-projection algorithm, solenoidal and irrotational components, tensor tomography, directional X-ray projections. I. INTRODUCTION T ENSOR tomography has found important applications in the physical sciences [1], [2], mathematics [3], and medicine [4]. Here we consider the tensor tomography problem as Manuscript

Iterative inversion of the tensor momentum x-ray transform

“…, X m f in definition 2.3 should be considered together. On other hand, equality (7) allows to express an integral momentum with the integral momentum of higher order. That means that all the information of the momentum x-ray transform is contained in X m f(a, b) considered on G (see [12], where the range of the transform for n ⩾ 3 is described with differential operator only on X m f).…”

“…Operator X k,0 coincides with the integral momentum of order k (4). The partial momentum x-ray transform satisfies relations analogous to (5)- (7).…”

“…One of such extensions is the momentum ray transform (definition 2.3). Alternative approaches use the transverse ray transform data (see for instance [6,7]).…”

“…The authors of [14] considered the attenuated ray transform of order k, which coincides with I k in equation ( 4) in case of zero attenuation. Specifically, they established differential equations analogous to (7) and prove the uniqueness theorem for corresponding boundary value problem and initial value boundary problem for the non-stationary case.…”

Singular value decomposition for longitudinal, transverse and mixed ray transforms of 2D tensor fields

“…In particular, this is the diffusion magnetic resonance imaging (MRI) tomography and the cross-polarization optical coherence tomography [15,31]. We note article [49] which provides an overview of important applications of tensor tomography in physical sciences and medicine.…”