BackgroundΒΆ
Tensor tomography refers to a family of methods for probing tensor-valued quantities in a volume-resolved manner. In traditional tomography, a scalar value (such as the absorptivity of a material) is reconstructed within a volume, from two-dimensional depth-probed images (for simplicity referred to as projections, although the images are not necessarily projections in the mathematical sense) of that volume. In tensor tomography, a measurement occurs over several channels, _e.g._, with different diffraction gratings, polarizations or scattering directions [Liebi2015], [Schaff2015]. Thus, the problem of tensor tomography can be understood as the reconstruction of a tensor field from two-dimensional projections of that tensor field [Gao2019].
In its most general sense, tensor tomography therefore requires both a tomography component (which contains a mapping between projection space and three-dimensional space) as well as a tensor representation component, which contains some mapping from the measured channels to the reconstructed tensor. In the simpler cases, this can be written on a linear form [Nielsen2023],
where \(P\) is a projection matrix, \(X\) is a matrix containing the coefficients of the three-dimensional tensor field, and \(A\) is a linear mapping from tensor field space to measurement space. \(D\) is the measured data. The tensor tomography problem then consists in finding an \(X\) that approximately satisfies this equality, possibly subject to other constraints. More generally, the equation may be written [Liebi2018]
In this case, \(P\) is some kind of transfer function that creates an image from a field, and additional information may need to be provided, _e.g._, at which polarizations a visible light measurement was performed.
SAXSTT is an example of a problem which can be written in the simpler linear form, and is currently the main method which is implemented in mumott, solvable as a regularized minimization problem. Here, \(A\) is a mapping from a function on the unit sphere to the reciprocal space map measured on a detector, and \(D\) contains scanning SAXS measurements. An example of a potential problem of somewhat greater complexity would be WAXS (or x-ray diffraction) tensor tomography, where the projection would also need to take into account the effect of the scattering direction on the transmitted intensity from each point in three-dimensional space.