Abstract
Tensor tomography is fundamentally based on the assumption of a both anisotropic and linear contrast mechanism. While the X-ray or neutron dark-field contrast obtained with Talbot(-Lau) interferometers features the required anisotropy, a preceding detailed study of dark-field signal origination however found its specific orientation dependence to be a non-linear function of the underlying anisotropic mass distribution and its orientation, especially challenging the common assumption that dark-field signals are describable by a function over the unit sphere. Here, two approximative linear tensor models with reduced orientation dependence are investigated in a simulation study with regard to their applicability to grating based X-ray or neutron dark-field tensor tomography. By systematically simulating and reconstructing a large sample of isolated volume elements covering the full range of feasible anisotropies and orientations, direct correspondences are drawn between the respective tensors characterizing the physically based dark-field model used for signal synthesization and the mathematically motivated simplified models used for reconstruction. The anisotropy of freely rotating volume elements is thereby confirmed to be, for practical reconstruction purposes, approximable both as a function of the optical axis’ orientation or as a function of the interferometer’s grating orientation. The eigenvalues of the surrogate models’ tensors are found to exhibit fuzzy, yet almost linear relations to those of the synthesization model. Dominant orientations are found to be recoverable with a margin of error on the order of magnitude of 1^{circ }. Although the input data must adequately address the full orientation dependence of dark-field anisotropy, the present results clearly support the general feasibility of quantitative X-ray dark-field tensor tomography within an inherent yet acceptable statistical margin of uncertainty.
Highlights
Tensor tomography is fundamentally based on the assumption of a both anisotropic and linear contrast mechanism
The origination and actual orientation dependence of dark-field contrast for anisotropic mass distributions has been investigated in detail (Graetz et al.11), yielding a minimal yet non-linear model capturing the central features of general darkfield anisotropy including in particular the effect of varying scattering cross section in addition to the characteristic dependence of dark-field contrast on a structure’s correlation lengths
I.e., general dark-field anisotropy is concluded to be a function of two orientations, which has not been taken into account so far and in particular challenges the general assumption in present literature that dark-field signals may be described as a function over the unit sphere
Summary
Tensor tomography is fundamentally based on the assumption of a both anisotropic and linear contrast mechanism. I.e., general dark-field anisotropy is concluded to be a function of two orientations (the optical axis and a perpendicular axis of interferometer sensitivity), which has not been taken into account so far and in particular challenges the general assumption in present literature that dark-field signals may be described as a function over the unit sphere While this additional complexity, and the non-linearity in particular, are highly undesirable with regard to tomographic reconstruction, the derived physically based model allows to synthesize large amounts of anisotropic dark-field signals that will here be used to systematically study the general applicability of approximative linear tensor models suited for classic tensor tomography. The analyses form the basis towards a more detailed and quantitative understanding of X-ray or neutron dark-field tensor tomography
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