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Details of Grant 

EPSRC Reference: EP/M016773/1
Title: The geodesic X-ray transform: well-posedness and practical inversion
Principal Investigator: Holman, Dr S
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics
Organisation: University of Manchester, The
Scheme: First Grant - Revised 2009
Starts: 01 March 2015 Ends: 28 February 2017 Value (£): 82,757
EPSRC Research Topic Classifications:
Algebra & Geometry Mathematical Analysis
Numerical Analysis
EPSRC Industrial Sector Classifications:
Healthcare Energy
Related Grants:
Panel History:
Panel DatePanel NameOutcome
26 Nov 2014 EPSRC Mathematics Prioritisation Panel November 2014 Announced
Summary on Grant Application Form
X-ray computerized tomography is by now a common and well understood imaging technique in which X-ray projections are combined to create a three dimensional image. Each point in an X-ray projection gives the integral of the attenuation along a line from an X-ray source to a receiver point, and thus the problem of constructing an image becomes the mathematical problem of finding a function based on its integrals along straight lines. This problem was solved in 1917 by Johann Radon. A natural generalization is to ask whether a function can be determined by its integrals along a family of curves rather than straight lines. When the curves are defined in a special way, as the geodesics of a Riemannian metric, we have the geodesic X-ray transform. This transform also arises independently in several applications.

The proposed research aims to investigate whether the geodesic X-ray transform of a function is sufficient to uniquely determine the function, and whether there are numerical methods to recover the function which are not overly sensitive to noise. Mathematically this corresponds to determining whether the transform is injective, and whether the inverse is continuous. Currently there are a number of cases in which the geodesic X-ray transform is known to be injective with stability estimates for the inversion including when the curvature is bounded above in an appropriate manner, when the manifold is simple and the metric is sufficiently close to a real analytic metric, and in dimension at least three when the manifold can be foliated by strictly convex hyper surfaces including the boundary. While these results include many cases, in fact examples which arise in practical situations, when considering for instance travel-time tomography, still are not covered. For example, it is unknown whether the existence of conjugate points (i.e. when geodesics beginning at a single point cross each other) implies that the transform is not injective, and generally the relationship between the geometry of the geodesics and the injectivity or non-injectivity of the transform is not well understood. Studying the transform in the case of complicated geometries including conjugate points is important because this is truly the generic situation; that is, conjugate points almost always exist in practical situations.

At the beginning of the project, the researcher will fully investigate the application of microlocal analysis to the geodesic X-ray transform. This will allow classification of the cases in which it is possible to construct a parametrix, or approximate inverse, for the transform thus showing that the inversion is a Fredholm problem. A corollary is thus that the kernel is at most finite dimensional and inversion is stable on a complement of the kernel.

At the same time the researcher will investigate the possibility of introducing alternate natural metrics on the unit sphere bundle in order to control portions of the Pestov identities arising in the study of the transform via energy methods. The original proof of injectivity in the case of simple manifolds used these identities, and the proposed research will investigate whether this can be pushed further by carefully analysing alternate geometries in the unit sphere bundle.

After completing the microlocal study explained above, the researcher will investigate the application of analytic microlocal analysis to the question of injectivity for the geodesic X-ray transform. This will require development of an analytic calculus of Fourier integral operators, and the proposed research will attempt to do this using methods from harmonic analysis.

The research will also look at numerical methods for inverting the geodesic X-ray transform. The initial work in this direction will take advantage of a characterisation of the inverse transform in terms of projections onto certain subspaces in L2 of the unit sphere bundle, but will proceed from there to investigate other methods.
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Organisation Website: http://www.man.ac.uk