Longitudinal ray transform

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

Let be the components of a symmetric rank-m tesnor 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 .[1] 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 [2]

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

References

edit
  1. ^ a b V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,ISBN 90-6764-165-0. Chapter 2.on-line version
  2. ^ Denisjuk, Alexander. "Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve." Inverse problems 22.2 (2006): 399.
  3. ^ Wensrich, Chris M., et al. "Direct inversion of the Longitudinal Ray Transform for 2D residual elastic strain fields." Inverse Problems 40.7 (2024): 075011.
  4. ^ T. Schuster, An efficient method for three-dimensional vector tomography: convergence and implementation, Inverse problems, 17 (2001), 739-766