Project overview

The geodesic ray transform, where one integrates a function or a tensor field along geodesics of a Riemannian metric, appears in several geometric inverse problems. Formally, instead of measuring a function $f$ (or a tensor field) directly, one observes a family of integral operators $A(f)$. A classical example is the standard X-ray or Radon transform [2,7], where one integrates a function along straight lines — it is the basis of medical imaging techniques such as CT (computer tomography) and PET (positron emission tomography).

In some other applications (such as geophysical and ultrasound imaging, or tensor tomography), one measures instead integrals along general geodesics that are not necessarily straight lines, leading to inverse problems with a more sophisticated geometric structure. For instance the geodesic Doppler transform is used in ultrasound tomography to detect tumours using blood flow measurements [6]. In practice, measurements of inverse problems are almost always subject to statistical noise. Formally one can consider, in the simplest case, an additive noise model where instead of $A(f)$ one observes $Y = A(f) +\epsilon Z$ where $Z$ is a Gaussian white noise and $\epsilon>0$ is the noise level. Standard recovery methods for inverse problems fail in this observational setup, and instead one has to incorporate a denoising step, such as a Bayesian regularisation method or spectral cutoff techniques (e.g., [3,4,5]).

The purpose of this project is to explore the theory behind some basic geometric inverse problems in the statistical framework where observations are corrupted by noise. This could first focus on problems where integrals are along geodesics of homogeneous backgrounds, for instance when integrating against great circles on the unit sphere [1] or when integrating along geodesics of the hyperbolic plane. The ultimate goal would be to understand these problems in truly anisotropic backgrounds where the Riemannian metric has no symmetries and develop efficient computational algorithms for regularisation (see [8]).

[1] Guillemin, V., The Radon transform on Zoll surfaces. Advances in Math. 22 (1976) 85–119.

[2] Helgason, S., The Radon transform, Birkhäuser, 1999.

[3] Johnstone I., Silverman, B., Speed of estimation in positron emission tomography and related inverse problems. Annals of Statistics 18 (1990) 251–280.

[4] Cavalier, L., Nonparametric statistical inverse problems. Inverse Problems 24 (2008) (electronic).

[5] Stuart, A. M., Inverse problems: a Bayesian perspective. Acta Numerica 19 (2010) 451-559.

[6] Wells P N T, Halliwell M, Skidmore R, Webb A J and Woodcock J P,

Tumour detection by ultrasonic Doppler blood-flow signals. Ultrasonics 15 (1977) 231–232.

[7] Kuchment, P., The Radon transform and medical imaging, CBMS-NSF Regional conference series in applied mathematics, 2014.

[8] Monard, F., Numerical implementation of geodesic X-ray transforms and their inversion, arXiv:1309.6042