In mathematics the longitudinal ray transform (LRT) is a generalization of the X-ray transform to symmetric tensor fields V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,. Chapter 2.on-line version
Let be the components of a symmetric rank-m tensor field () on Euclidean space (). For a unit vector and a point the longitudinal ray transform is defined as
where summation over repeated indices is implied. The transform has a null-space, assuming the components are smooth and decay at infinity any , the symmetrized derivative of a rank m-1 tensor field , satisfies . More generally the Saint-Venant tensor 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 Denisjuk, Alexander. "Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve." Inverse problems 22.2 (2006): 399.
Applications of the LRT include Bragg edge neutron tomography of strain,Wensrich, Chris M., et al. "Direct inversion of the Longitudinal Ray Transform for 2D residual elastic strain fields." Inverse Problems 40.7 (2024): 075011. and Doppler tomography of velocity vector fields.T. Schuster, An efficient method for three-dimensional vector tomography: convergence and implementation, Inverse problems, 17 (2001), 739-766
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