The geodesic X-ray transform with a GL(n,ℂ)-connection

2016/10/29 by Monard, François, Paternain, Gabriel P.
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1610.09571

Abstract

We derive reconstruction formulas for a family of geodesic ray transforms with connection, defined on simple Riemannian surfaces. Such formulas provide injectivity of such all transforms in a neighbourhood of constant curvature metrics and non-unitary connections with curvature close to zero. If certain Fredholm equations are injective in the absence of connection, then for any smooth enough connection multiplied by a complex parameter, the corresponding transform is injective for all values of that parameter outside a discrete set. Range characterizations are also provided, as well as numerical illustrations.

Related

• Functional relations, sharp mapping properties and regularization of the X-ray transform on disks of constant curvature
• Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature
• Numerical implementation of geodesic X-ray transforms and their inversion
• Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces
• Reconstruction formulas for X-ray transforms in negative curvature