Kristian Bredies

Total Generalized Variation

The novel concept of total generalized variation of a function

Second order total generalized variation (TGV) for MRI.

u

Iterated Hard Shrinkage for Minimization Problems with Sparsity Constraints

is introduced, and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher-order derivatives of

Regularization with non-convex separable constraints

u

Parallel imaging with nonlinear reconstruction using variational penalties.

. Numerical examples illustrate the high quality of this functional as a regularization term for mathematical imaging problems. In particular this functional selectively regularizes on different regularity levels and, as a side effect, does not lead to a staircasing effect.

Non-local Total Generalized Variation for Optical Flow Estimation

Total variation was recently introduced in many different magnetic resonance imaging applications. The assumption of total variation is that images consist of areas, which are piecewise constant. However, in many practical magnetic resonance imaging situations, this assumption is not valid due to the inhomogeneities of the exciting B1 field and the receive coils. This work introduces the new concept of total generalized variation for magnetic resonance imaging, a new mathematical framework, which is a generalization of the total variation theory and which eliminates these restrictions. Two important applications are considered in this article, image denoising and image reconstruction from undersampled radial data sets with multiple coils. Apart from simulations, experimental results from in vivo measurements are presented where total generalized variation yielded improved image quality over conventional total variation in all cases. Magn Reson Med, 2011.

Fast quantitative susceptibility mapping using 3D EPI and total generalized variation

A new iterative algorithm for the solution of minimization problems in infinite-dimensional Hilbert spaces which involve sparsity constraints in form of

Total Generalized Variation in Diffusion Tensor Imaging

\ell^{p}

A generalized conditional gradient method for nonlinear operator equations with sparsity constraints

-penalties is proposed. In contrast to the well-known algorithm considered by Daubechies, Defrise, and De Mol, it uses hard instead of soft shrinkage. It is shown that the hard shrinkage algorithm is a special case of the generalized conditional gradient method. Convergence properties of the generalized conditional gradient method with quadratic discrepancy term are analyzed. This leads to strong convergence of the iterates with convergence rates

Recovering Piecewise Smooth Multichannel Images by Minimization of Convex Functionals with Total Generalized Variation Penalty

\mathcal{O}(n^{-1/2})