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Posters (INVW07)

Joint Reconstruction of PET/MRI by Parallel Level Sets

Total Variation Regularisation for Functions with Values in a Manifold

Multiresolution tomography of the ionosphere using sparse regularisation

Exact solutions to the one dimensional TGV regularisation problem

Combined positron emission tomography and magnetic resonance imaging scanners allow us to simultaneously image structure and function of the human body. As function follows structure the two solutions of the two inverse problems are expected to show similar shapes. We will exploit this fact by modelling a joint prior which favours parallel level sets and hence similar structures. The results indicate that by utilizing the intrinsic shared structure both solutions can be significantly improved.


While the total variation is among the most popular regularizers for variational problems, its extension to functions with values in a manifold is an open problem. We propose an algorithm based on a convex relaxation method for solving such problems for arbitrary Riemannian manifolds. The framework can be adapted to different manifolds including spheres and three-dimensional rotations, and allows to obtain more accurate solutions than labelling methods. We show several applications including variational processing of chromaticity values, normal fields, and camera trajectories.


In geophysics the ionosphere has an important role due to its interaction with satellite signals. The ionosphere can be imaged through computerized tomography using observations from ground based receivers. However, the uneven and sparse distribution of limited-angle observations makes this inverse problem particularly challenging. To help solve this, a sparse regularization technique is applied to wavelet basis functions. Advantages over a more conventional approach using Tikhonov regularization with spherical harmonics are presented.


Total generalised variation (TGV) is a recently introduced, high quality regulariser which has successfully used in many applications of mathematical imaging. It tends to preserve not only edges but also smooth structures, thus avoiding the staircasing effect, a well known artifact of total variation regularisation. We provide exact solutions to the one dimensional TGV problem with L^{2} fidelity term, using simple piecewise affine functions as data terms. This gives a further insight into the behaviour of the solutions with respect to the TGV parameters and more generally into the regularising mechanisms of TGV.

Carola-Bibiane Schönlieb (University of Cambridge)
 Total variation regularisation in PET reconstruction
Andrew Fitzgibbon (Microsoft Research)
Jakob Sauer Joergensen (Danmarks Tekniske Universitet)
Empirical phase transitions in computed tomography

Sparse reconstruction methods such as total variation (TV) minimization have shown potential for dose-reduction in x-ray computed tomography. The fundamental question of how many measurements ensure unique recovery is open, as standard recovery guarantees from compressed sensing do not apply to the deterministic CT sampling matrices. Recent theoretical analysis take important steps forward for certain special sampling patterns but for conventional CT sampling patterns the question of sufficient sampling remains unanswered. We study the question empirically in the spirit of the Donoho-Tanner phase diagram, i.e, by investigating the fraction of recovered image instances as function of sparsity and sampling.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons