Friederike Laus

Friederike Laus, Fabien Pierre, Gabriele Steidl

The contribution of this paper is two-fold. First, we introduce a generalized myriad filter, which is a method to compute the joint maximum likelihood estimator of the location and the scale parameter of the Cauchy distribution. Estimating only the location parameter is known as myriad filter. We propose an efficient algorithm to compute the generalized myriad filter and prove its convergence. Special cases of this algorithm result in the classical myriad filtering, respective an algorithm for estimating only the scale parameter. Based on an asymptotic analysis, we develop a second, even faster generalized myriad filtering technique. Second, we use our new approaches within a nonlocal, fully unsupervised method to denoise images corrupted by Cauchy noise. Special attention is paid to the determination of similar patches in noisy images. Numerical examples demonstrate the excellent performance of our algorithms which have moreover the advantage to be robust with respect to the parameter choice.

Jan Henrik Fitschen, Friederike Laus, Gabriele Steidl

We propose two models for the interpolation between RGB images based on the dynamic optimal transport model of Benamou and Brenier [8]. While the application of dynamic optimal transport and its extensions to unbalanced transform were examined for gray-values images in various papers, this is the first attempt to generalize the idea to color images. The nontrivial task to incorporate color into the model is tackled by considering RGB images as three-dimensional arrays, where the transport in the RGB direction is performed in a periodic way. Following the approach of Papadakis et al. [35] for gray-value images we propose two discrete variational models, a constrained and a penalized one which can also handle unbalanced transport. We show that a minimizer of our discrete model exists, but it is not unique for some special initial/final images. For minimizing the resulting functionals we apply a primal-dual algorithm. One step of this algorithm requires the solution of a four-dimensional discretized Poisson equation with various boundary conditions in each dimension. For instance, for the penalized approach we have simultaneously zero, mirror and periodic boundary conditions. The solution can be computed efficiently using fast Sin-I, Cos-II and Fourier transforms. Numerical examples demonstrate the meaningfulness of our model.

Ronny Bergmann, Friederike Laus, Johannes Persch et al.

Modern signal and image acquisition systems are able to capture data that is no longer real-valued, but may take values on a manifold. However, whenever measurements are taken, no matter whether manifold-valued or not, there occur tiny inaccuracies, which result in noisy data. In this chapter, we review recent advances in denoising of manifold-valued signals and images, where we restrict our attention to variational models and appropriate minimization algorithms. The algorithms are either classical as the subgradient algorithm or generalizations of the half-quadratic minimization method, the cyclic proximal point algorithm, and the Douglas-Rachford algorithm to manifolds. An important aspect when dealing with real-world data is the practical implementation. Here several groups provide software and toolboxes as the Manifold Optimization (Manopt) package and the manifold-valued image restoration toolbox (MVIRT).

Friederike Laus, Mila Nikolova, Johannes Persch et al. · mila

Nonlocal patch-based methods, in particular the Bayes' approach of Lebrun, Buades and Morel (2013), are considered as state-of-the-art methods for denoising (color) images corrupted by white Gaussian noise of moderate variance. This paper is the first attempt to generalize this technique to manifold-valued images. Such images, for example images with phase or directional entries or with values in the manifold of symmetric positive definite matrices, are frequently encountered in real-world applications. Generalizing the normal law to manifolds is not canonical and different attempts have been considered. Here we focus on a straightforward intrinsic model and discuss the relation to other approaches for specific manifolds. We reinterpret the Bayesian approach of Lebrun et al. (2013) in terms of minimum mean squared error estimation, which motivates our definition of a corresponding estimator on the manifold. With this estimator at hand we present a nonlocal patch-based method for the restoration of manifold-valued images. Various proof of concept examples demonstrate the potential of the proposed algorithm.

Tobias Alt, Andrea Ibisch, Clemens Meiser et al.

Generative AI models are capable of performing a wide variety of tasks that have traditionally required creativity and human understanding. During training, they learn patterns from existing data and can subsequently generate new content such as texts, images, audio, and videos that align with these patterns. Due to their versatility and generally high-quality results, they represent, on the one hand, an opportunity for digitalisation. On the other hand, the use of generative AI models introduces novel IT security risks that must be considered as part of a comprehensive analysis of the IT security threat landscape. In response to this risk potential, companies or authorities intending to use generative AI should conduct an individual risk analysis before integrating it into their workflows. The same applies to developers and operators, as many risks associated with generative AI must be addressed during development or can only be influenced by the operating organisation. Based on this, existing security measures can be adapted, and additional measures implemented.

Jan Henrik Fitschen, Friederike Laus, Gabriele Steidl

Recently, Papadakis et al. proposed an efficient primal-dual algorithm for solving the dynamic optimal transport problem with quadratic ground cost and measures having densities with respect to the Lebesgue measure. It is based on the fluid mechanics formulation by Benamou and Brenier and proximal splitting schemes. In this paper we extend the framework to color image processing. We show how the transportation problem for RGB color images can be tackled by prescribing periodic boundary conditions in the color dimension. This requires the solution of a 4D Poisson equation with mixed Neumann and periodic boundary conditions in each iteration step of the algorithm. This 4D Poisson equation can be efficiently handled by fast Fourier and Cosine transforms. Furthermore, we sketch how the same idea can be used in a modified way to transport periodic 1D data such as the histogram of cyclic hue components of images. We discuss the existence and uniqueness of a minimizer of the associated energy functional. Numerical examples illustrate the meaningfulness of our approach.