Longitudinal ray transform

In mathematics the longitudinal ray transform (LRT) is a generalization of the X-ray transform to symmetric tensor fields ^[1]

Let ${\displaystyle f_{i_{1}...i_{m}}}$ be the components of a symmetric rank-m tensor field (${\displaystyle m\geq }$) on Euclidean space ${\displaystyle \mathbf {R} ^{n}}$ (${\displaystyle n\geq 2}$). For a unit vector ${\displaystyle \xi ,|\xi |=1}$ and a point ${\displaystyle x\in \mathbf {R} ^{n}}$ the longitudinal ray transform is defined as

${\displaystyle g(x,\xi ):=If(x,\xi )=\int \limits _{-\infty }^{\infty }f_{i_{1}...i_{m}}(x+s\xi )\xi _{i_{1}}\cdots \xi _{i_{m}}\,\mathrm {d} s}$

where summation over repeated indices is implied. The transform has a null-space, assuming the components are smooth and decay at infinity any ${\displaystyle f=dg}$, the symmetrized derivative of a rank m-1 tensor field ${\displaystyle g}$, satisfies ${\displaystyle If=0}$.^[1] More generally the Saint-Venant tensor ${\displaystyle Wf}$ can be recovered uniquely by an explicit formula. For lines that pass through a curve similar results can be obtained to the case of the complete data case of all lines ^[2]

Applications of the LRT include Bragg edge neutron tomography of strain,^[3] and Doppler tomography of velocity vector fields.^[4]