X-ray transform

Abstract: Introduction: the problem of determining a metric by its hodograph and a linearization of the problem the kinetic equation in a Riemannian manifold. Part 1 The ray transform of symmetric tensor fields on Euclidean space: the ray transform and its relationship to the Fourier transform description of the kernel of the ray transform in the smooth case equivalence of the first two statements of theorem 2.2.1 in the case n=2 proof of theorem 2.2.2. the ray transform of a field-distribution decomposition of a tensor field into potential and solenoidal parts a theorem on the tangent component a theorem on conjugate tensor fields on the sphere primality of the ideal ([x]2, ) description of the image of the ray transform integral moments of the function I f inversion formulas for the ray transform proof of theorem 2.12.1 inversion of the ray transform on the space of field-distributions the Plancherel formula for the ray transform applications of the ray transform to an inverse problem of photoelasticity further results. Part 2 Some questions of tensor analysis. Part 3 The ray transform on a Riemannian manifold. Part 4 The transverse ray transform. Part 5 The truncated transverse ray transform. Part 6 The mixed ray transform. Part 7 The exponential ray transform (Part contents)

Abstract: Given a function f, the author specifies the singularities of f that are visible in a stable way from limited X-ray tomographic data. This determines which singularities of f can be stably recovere...

Abstract: Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces the geodesic X-ray transform is globally injective. In addition we prove stability estimates and propose a layer stripping type algorithm for reconstruction.

Abstract: Computed tomography is a non-destructive testing technique based on X-ray absorption that permits the 3D-visualisation of materials at micron-range resolutions. In this article, computed tomography is used to investigate fibre orientation and fibre position in various fibre-reinforced materials such as ceramic matrix composites, glass fibre-reinforced plastics or reinforced concrete. The goal of this article is to determine the quantitative orientation of fibres in fibre-reinforced materials. For this purpose, a mathematical technique based on the structure tensor is used to determine the local orientation of fibres. The structure tensor is easy to implement and results in a fast algorithm relying solely on local properties of the given reconstruction. In addition, the local X-ray transform is used to denoise fibres and to segment them from the matrix.