We study the solution of minimax problems min x max y G(x) + K(x), y − F * (y) in finite-dimensional Hilbert spaces. The functionals G and F * we assume to be convex, but the operator K we allow to be non-linear. We formulate a natural extension of the modified primal-dual hybrid gradient method (PDHGM), originally for linear K, due to Chambolle and Pock. We prove the local convergence of the method, provided various technical conditions are satisfied. These include in particular the Aubin property of the inverse of a monotone operator at the solution. Of particular interest to us is the case arising from Tikhonov type regularisation of inverse problems with non-linear forward operators. Mainly we are interested in total variation and second-order total generalised variation priors. For such problems, we show that our general local convergence result holds when the noise level of the data f is low, and the regularisation parameter α is correspondingly small. We verify the numerical performance of the method by applying it to problems from magnetic resonance imaging (MRI) in chemical engineering and medicine. The specific applications are in diffusion tensor imaging (DTI) and MR velocity imaging. These numerical studies show very promising performance.Mathematics subject classification: 49M29, 90C26, 92C55.

We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost based on a Huber-regularised TV seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a combined quasiNewton/semismooth Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between TGV 2 and ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.

We study the extension of total variation (TV), total deformation (TD), and (second-order) total generalized variation (TGV 2 ) to symmetric tensor fields. We show that for a suitable choice of finite-dimensional norm, these variational seminorms are rotation-invariant in a sense natural and well suited for application to diffusion tensor imaging (DTI). Combined with a positive definiteness constraint, we employ these novel seminorms as regularizers in Rudin-Osher-Fatemi (ROF) type denoising of medical in vivo brain images. For the numerical realization, we employ the ChambollePock algorithm, for which we develop a novel duality-based stopping criterion which guarantees error bounds with respect to the functional values. Our findings indicate that TD and TGV 2 , both of which employ the symmetrized differential, provide improved results compared to other evaluated approaches.

The D/D genotype of the alpha2B-adrenoceptor is a novel genetic risk factor for acute coronary events, but not for hypertension.

We propose the use of the Kantorovich-Rubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularisation model endowed with a Kantorovich-Rubinstein discrepancy term and total variation regularization in the context of image denoising and cartoon-texture decomposition. We point out connections of this approach to several other recently proposed methods such as total generalized variation and norms capturing oscillating patterns. We also show that the respective optimization problem can be turned into a convex-concave saddle point problem with simple constraints and hence, can be solved by standard tools. Numerical examples exhibit interesting features and favourable performance for denoising and cartoon-texture decomposition.