Abstract The problem of reconstructing a watertight surface from a finite set of oriented points has received much attention over the last decades. In this paper, we propose a general higher order framework for surface reconstruction. It is based on the idea that position and normal defined by each oriented point can be used to construct an implicit local description of the unknown surface. On the one hand, this allows us to systematically explain and relate several popular methods, for example implicit moving least squares, smooth signed distance surface reconstruction as well as (screened) Poisson surface reconstruction. On the other hand, it allows to derive and discuss a number of new approaches for reconstructing either the signed distance or the indicator function of the sought object. All of these approaches are able to achieve competitive results but one of them turns out to be especially promising. To improve reconstructions in difficult real world scenarios where point clouds have been estimated from colour images, we introduce a hull constraint that encourages the surface to stay within a given region. Our framework is implemented on the GPU using a recent cyclic scheme called Fast Jacobi, which combines low implementational effort with high efficiency.

Alternating direction methods are a common tool for general mathematical programming and optimization. These methods have become particularly important in the field of variational image processing, which frequently requires the minimization of nondifferentiable objectives. This paper considers accelerated (i.e., fast) variants of two common alternating direction methods: the alternating direction method of multipliers (ADMM) and the alternating minimization algorithm (AMA). The proposed acceleration is of the form first proposed by Nesterov for gradient descent methods. In the case that the objective function is strongly convex, global convergence bounds are provided for both classical and accelerated variants of the methods. Numerical examples are presented to demonstrate the superior performance of the fast methods for a wide variety of problems.

The (constrained) minimization of a ratio of set functions is a problem frequently occurring in clustering and community detection. As these optimization problems are typically NP-hard, one uses convex or spectral relaxations in practice. While these relaxations can be solved globally optimally, they are often too loose and thus lead to results far away from the optimum. In this paper we show that every constrained minimization problem of a ratio of non-negative set functions allows a tight relaxation into an unconstrained continuous optimization problem. This result leads to a flexible framework for solving constrained problems in network analysis. While a globally optimal solution for the resulting non-convex problem cannot be guaranteed, we outperform the loose convex or spectral relaxations by a large margin on constrained local clustering problems.

Hypergraphs allow to encode higher-order relationships in data and are thus a very flexible modeling tool. Current learning methods are either based on approximations of the hypergraphs via graphs or on tensor methods which are only applicable under special conditions. In this paper we present a new learning framework on hypergraphs which fully uses the hypergraph structure. The key element is a family of regularization functionals based on the total variation on hypergraphs.

Spectral clustering is based on the spectral relaxation of the normalized/ratio graph cut criterion. While the spectral relaxation is known to be loose, it has been shown recently that a non-linear eigenproblem yields a tight relaxation of the Cheeger cut. In this paper, we extend this result considerably by providing a characterization of all balanced graph cuts which allow for a tight relaxation. Although the resulting optimization problems are non-convex and non-smooth, we provide an efficient first-order scheme which scales to large graphs. Moreover, our approach comes with the quality guarantee that given any partition as initialization the algorithm either outputs a better partition or it stops immediately.

We present novel wavelet-based inpainting algorithms. Applying ideas from anisotropic regularization and diffusion, our models can better handle degraded pixels at edges. We interpret our algorithms within the framework of forward-backward splitting methods in convex analysis and prove that the conditions for ensuring their convergence are fulfilled. Numerical examples illustrate the good performance of our algorithms.