Solution of the Integral Geometry Problem for 2-Tensor Fields by the Singular Value Decomposition Method

We consider the integral geometry problem of finding a symmetric 2-tensor field in a unit disk provided that the ray transforms of this field are known. We construct singular value decompositions of the operators of longitudinal, transversal, and mixed ray transforms that are the integrals of projections of a field onto the line where they are computed. We essentially use the results on decomposition of tensor fields and their representation in terms of potentials. The singular value decompositions are constructive and can be used for creating an algorithm for recovering a tensor field from its known ray characteristics.

[1]  A K Louis,et al.  Incomplete data problems in x-ray computerized tomography , 1986 .

[2]  Andreas Rieder,et al.  Incomplete data problems in X-ray computerized tomography , 1989 .

[3]  Harvey Thomas Banks,et al.  Standard errors and confidence intervals in inverse problems: sensitivity and associated pitfalls , 2007 .

[4]  S. G. Kazantsev,et al.  Singular value decomposition for the 2D fan-beam Radon transform of tensor fields , 2004 .

[5]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[6]  E. Derevtsov.  An approach to direct reconstruction of a solenoidal part in vector and tensor tomography problems , 2005 .

[7]  M. E. Davison.  A singular value decomposition for the radon transform in n-dimensional euclidean space , 1981 .

[8]  Alfred K. Louis,et al.  Singular value decomposition and its application to numerical inversion for ray transforms in 2D vector tomography , 2011 .

[9]  Hermann Weyl,et al.  The method of orthogonal projection in potential theory , 1940 .

[10]  E. T. Quinto.  Singular value decompositions and inversion methods for the exterior Radon transform and a spherical transform , 1983 .

[11]  S. Deans.  The Radon Transform and Some of Its Applications , 1983 .

[12]  M. C. Delfour,et al.  Dynamical free boundary problem for an incompressible potential fluid flow in a time-varying domain , 2004 .

[13]  Alfred K. Louis,et al.  Feature reconstruction in inverse problems , 2011 .

[14]  Alfred K. Louis.  Orthogonal Function Series Expansions and the Null Space of the Radon Transform , 1984 .

[15]  Thomas Schuster,et al.  The Method of Approximate Inverse: Theory and Applications , 2007 .

[16]  S. G. Kazantsev,et al.  Polynomial bases for subspaces of vector fields in the unit ball. Method of ridge functions , 2007 .

[17]  Peter Maass,et al.  The x-ray transform: singular value decomposition and resolution , 1987 .