The geodesic X-ray transform: well-posedness and practical inversion

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.

This project is motivated both by the many potential applications for inversion of the geodesic X-ray transform, and a fundamental interest in the theoretical questions surrounding injectivity and stability of the transform in complicated geometric situations which have been open for at least 30 years. Results from this project would answer such longstanding questions, as well as open the way for future research.

The proposed research fits within the larger mathematical discipline of inverse problems, the area of mathematics concerned with constructing images of objects based on a limited selection of measurements. Well established examples of the application of inverse problems include the creation of three dimensional images from multiple X-ray projections in computerised tomography, from energy emitted by spin relaxations in magnetic resonance imaging, or at a completely different scale, the location of oil, natural gas, water, or geological structures appropriate for carbon sequestration deep underground from seismic data. As a discipline, inverse problems seek to provide a common mathematical framework for the analysis of all these problems and implementation of practical algorithms to pass from raw data to useful images, or other types of information.

The research questions proposed for study in this project have been open for at least 30 years, and continue to be of great interest in the inverse problems community. Answers to the proposed questions would shed light on the fundamental capabilities of several different imaging methods. The geodesic X-ray transform arises directly in many applications including ultra-sound transmission tomography, optical tomography with variable index of refraction, seismic travel time tomography, and atmospheric shortwave or acoustic tomography. In particular, the microlocal analysis portion of the research will contribute to the understanding of artefacts which appear in images created by methods due to complicated ray geometries, and potentially give insight into how to remove these artefacts. Similar methods may shed light on reflection tomography problems as well and give insight into what types of receiver and source geometries should be used in that context. Methods developed could also apply to analysis of other cases in which integral geometry problems arise such as photo-acoustic tomography, and synthetic aperture radar. Additionally, injectivity of the geodesic X-ray transform has implications for elliptic inverse problems such as the anisotropic Calderon problem in which identifiability is implied by injectivity of a corresponding attenuated geodesic X-ray transform. Further research on the topics included in this proposal, which will certainly be impacted by the successful completion of the current research, would involve inversion of the geodesic X-ray transform acting on tensor fields, and the weighted geodesic X-ray transform.

Broadly, the development of new imaging technology relies on contributions from many disciplines, including mathematics for insight into the fundamental capabilities of such new technology, and for numerical algorithms to create images in practice. This project will contribute to such understanding.