In this paper we study a first-order primal-dual algorithm for convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O(1/N) in finite dimensions, which is optimal for the complete class of non-smooth problems we are considering in this paper. We further show accelerations of the proposed algorithm to yield optimal rates on easier problems. In particular we show that we can achieve O(1/N 2) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O(1/e N) on problems where both are uniformly convex. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and image segmentation.

Variational network reconstructions preserve the natural appearance of MR images as well as pathologies that were not included in the training data set. Due to its high computational performance, that is, reconstruction time of 193 ms on a single graphics card, and the omission of parameter tuning once the network is trained, this new approach to image reconstruction can easily be integrated into clinical workflow. Magn Reson Med 79:3055-3071, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

The novel concept of total generalized variation of a function u is introduced, and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher-order derivatives of u. 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.

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Abstract-Image restoration is a long-standing problem in low-level computer vision with many interesting applications. We describe a flexible learning framework based on the concept of nonlinear reaction diffusion models for various image restoration problems. By embodying recent improvements in nonlinear diffusion models, we propose a dynamic nonlinear reaction diffusion model with time-dependent parameters (i.e., linear filters and influence functions). In contrast to previous nonlinear diffusion models, all the parameters, including the filters and the influence functions, are simultaneously learned from training data through a loss based approach. We call this approach TNRD -Trainable Nonlinear Reaction Diffusion. The TNRD approach is applicable for a variety of image restoration tasks by incorporating appropriate reaction force. We demonstrate its capabilities with three representative applications, Gaussian image denoising, single image super resolution and JPEG deblocking. Experiments show that our trained nonlinear diffusion models largely benefit from the training of the parameters and finally lead to the best reported performance on common test datasets for the tested applications. Our trained models preserve the structural simplicity of diffusion models and take only a small number of diffusion steps, thus are highly efficient. Moreover, they are also well-suited for parallel computation on GPUs, which makes the inference procedure extremely fast.

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. Total variation (TV) based strategies, which were originally designed for denoising of images (1) have recently gained wide interest for many MRI applications beyond denoising. Examples include regularization for parallel imaging (2,3), the elimination of truncation artifacts (4), inpainting of sensitivity maps (5), their use as a regularization method for undersampled imaging techniques within the compressed sensing framework (6) as well as in iterative reconstruction of undersampled radial data sets (7,8). TV models have the main benefit that they are very well suited to remove random noise, incoherent noise-like artifacts from random subsampling and streaking artifacts from undersampled radial sampling, while preserving the edges in the image. However, the assumption of TV is that the images consist of regions, which are piecewise constant. Due to inhomogeneities of the exciting B 1 field of high field systems with 3T and above and of the receive coils, this assumption is often violated in practical MRI examinations. Additionally, even in situations when this is no severe problem, due to the assumption of piecewise constancy, the use of TV often leads to staircasing artifacts and results in patchy, cartoon-like images which appear unnatural. This article introduces the new concept of total generalized variation (TGV) as a penalty term for MRI problems. This mathematical theory has recently been developed (9), and while it is equivalent to TV in terms of edge preservation and noise removal, it can also be applied in imaging situations where the assumption that the image is piecewise constant is not valid. As a result, the application of TGV in MR imaging is far less restrictive. It is shown in this work that TGV can be applied for image denoising and during iterative image reconstruction of undersampled radial data sets from phased array coils, and yields results that are superior to conventional TV. THEORY The Concept of Total Generalized VariationThe total generalized variation introduced in (9) is a functional which is capable to measure, in some sense, image characteristics up to a certain order of differentiation. In this section, we will restrict ourselves to give a short introduction which is not rigorous in the mathematical sense. The reader interested in the mathematical background may find more information in the Appendix.First, recall the definition of the total variation, which is, for a given image u, usually expres...

We revisit the proofs of convergence for a first order primal-dual algorithm for convex optimization which we have studied a few years ago. In particular, we prove rates of convergence for a more general version, with simpler proofs and more complete results. The new results can deal with explicit terms and nonlinear proximity operators in spaces with quite general norms. MSC Classification: 49M29 65K10 65Y20 90C25

A large number of imaging problems reduce to the optimization of a cost function, with typical structural properties. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. We place particular emphasis on optimal first-order schemes that can deal with typical non-smooth and large-scale objective functions used in imaging problems. We illustrate and compare the different algorithms using classical non-smooth problems in imaging, such as denoising and deblurring. Moreover, we present applications of the algorithms to more advanced problems, such as magnetic resonance imaging, multilabel image segmentation, optical flow estimation, stereo matching, and classification.