Fabian Altekrüger

Fabian Altekrüger, Alexander Denker, Paul Hagemann et al.

Learning neural networks using only few available information is an important ongoing research topic with tremendous potential for applications. In this paper, we introduce a powerful regularizer for the variational modeling of inverse problems in imaging. Our regularizer, called patch normalizing flow regularizer (patchNR), involves a normalizing flow learned on small patches of very few images. In particular, the training is independent of the considered inverse problem such that the same regularizer can be applied for different forward operators acting on the same class of images. By investigating the distribution of patches versus those of the whole image class, we prove that our model is indeed a MAP approach. Numerical examples for low-dose and limited-angle computed tomography (CT) as well as superresolution of material images demonstrate that our method provides very high quality results. The training set consists of just six images for CT and one image for superresolution. Finally, we combine our patchNR with ideas from internal learning for performing superresolution of natural images directly from the low-resolution observation without knowledge of any high-resolution image.

Fabian Altekrüger, Johannes Hertrich, Gabriele Steidl

Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals with non-smooth Riesz kernels show a rich structure as singular measures can become absolutely continuous ones and conversely. In this paper we contribute to the understanding of such flows. We propose to approximate the backward scheme of Jordan, Kinderlehrer and Otto for computing such Wasserstein gradient flows as well as a forward scheme for so-called Wasserstein steepest descent flows by neural networks (NNs). Since we cannot restrict ourselves to absolutely continuous measures, we have to deal with transport plans and velocity plans instead of usual transport maps and velocity fields. Indeed, we approximate the disintegration of both plans by generative NNs which are learned with respect to appropriate loss functions. In order to evaluate the quality of both neural schemes, we benchmark them on the interaction energy. Here we provide analytic formulas for Wasserstein schemes starting at a Dirac measure and show their convergence as the time step size tends to zero. Finally, we illustrate our neural MMD flows by numerical examples.

Fabian Altekrüger, Paul Hagemann, Gabriele Steidl

Conditional generative models became a very powerful tool to sample from Bayesian inverse problem posteriors. It is well-known in classical Bayesian literature that posterior measures are quite robust with respect to perturbations of both the prior measure and the negative log-likelihood, which includes perturbations of the observations. However, to the best of our knowledge, the robustness of conditional generative models with respect to perturbations of the observations has not been investigated yet. In this paper, we prove for the first time that appropriately learned conditional generative models provide robust results for single observations.

Paul Hagemann, Johannes Hertrich, Fabian Altekrüger et al.

We propose conditional flows of the maximum mean discrepancy (MMD) with the negative distance kernel for posterior sampling and conditional generative modeling. This MMD, which is also known as energy distance, has several advantageous properties like efficient computation via slicing and sorting. We approximate the joint distribution of the ground truth and the observations using discrete Wasserstein gradient flows and establish an error bound for the posterior distributions. Further, we prove that our particle flow is indeed a Wasserstein gradient flow of an appropriate functional. The power of our method is demonstrated by numerical examples including conditional image generation and inverse problems like superresolution, inpainting and computed tomography in low-dose and limited-angle settings.

Moritz Piening, Fabian Altekrüger, Johannes Hertrich et al.

The solution of inverse problems is of fundamental interest in medical and astronomical imaging, geophysics as well as engineering and life sciences. Recent advances were made by using methods from machine learning, in particular deep neural networks. Most of these methods require a huge amount of (paired) data and computer capacity to train the networks, which often may not be available. Our paper addresses the issue of learning from small data sets by taking patches of very few images into account. We focus on the combination of model-based and data-driven methods by approximating just the image prior, also known as regularizer in the variational model. We review two methodically different approaches, namely optimizing the maximum log-likelihood of the patch distribution, and penalizing Wasserstein-like discrepancies of whole empirical patch distributions. From the point of view of Bayesian inverse problems, we show how we can achieve uncertainty quantification by approximating the posterior using Langevin Monte Carlo methods. We demonstrate the power of the methods in computed tomography, image super-resolution, and inpainting. Indeed, the approach provides also high-quality results in zero-shot super-resolution, where only a low-resolution image is available. The paper is accompanied by a GitHub repository containing implementations of all methods as well as data examples so that the reader can get their own insight into the performance.

Moritz Piening, Fabian Altekrüger, Gabriele Steidl et al.

Contrast enhancement by Gadolinium-based contrast agents (GBCAs) is a vital tool for tumor diagnosis in neuroradiology. Based on brain MRI scans of glioblastoma before and after Gadolinium administration, we address enhancement prediction by neural networks with two new contributions. Firstly, we study the potential of generative models, more precisely conditional diffusion and flow matching, for uncertainty quantification in virtual enhancement. Secondly, we examine the performance of T1 scans from quantitive MRI versus T1-weighted scans. In contrast to T1-weighted scans, these scans have the advantage of a physically meaningful and thereby comparable voxel range. To compare network prediction performance of these two modalities with incompatible gray-value scales, we propose to evaluate segmentations of contrast-enhanced regions of interest using Dice and Jaccard scores. Across models, we observe better segmentations with T1 scans than with T1-weighted scans.

Sebastian Neumayer, Fabian Altekrüger

Theoretical guarantees for the robust solution of inverse problems have important implications for applications. To achieve both guarantees and high reconstruction quality, we propose learning a pixel-based ridge regularizer with a data-dependent and spatially varying regularization strength. For this architecture, we establish the existence of solutions to the associated variational problem and the stability of its solution operator. Further, we prove that the reconstruction forms a maximum-a-posteriori approach. Simulations for biomedical imaging and material sciences demonstrate that the approach yields high-quality reconstructions even if only a small instance-specific training set is available.

Johannes Hertrich, Christian Wald, Fabian Altekrüger et al.

Maximum mean discrepancy (MMD) flows suffer from high computational costs in large scale computations. In this paper, we show that MMD flows with Riesz kernels $K(x,y) = - \|x-y\|^r$, $r \in (0,2)$ have exceptional properties which allow their efficient computation. We prove that the MMD of Riesz kernels, which is also known as energy distance, coincides with the MMD of their sliced version. As a consequence, the computation of gradients of MMDs can be performed in the one-dimensional setting. Here, for $r=1$, a simple sorting algorithm can be applied to reduce the complexity from $O(MN+N^2)$ to $O((M+N)\log(M+N))$ for two measures with $M$ and $N$ support points. As another interesting follow-up result, the MMD of compactly supported measures can be estimated from above and below by the Wasserstein-1 distance. For the implementations we approximate the gradient of the sliced MMD by using only a finite number $P$ of slices. We show that the resulting error has complexity $O(\sqrt{d/P})$, where $d$ is the data dimension. These results enable us to train generative models by approximating MMD gradient flows by neural networks even for image applications. We demonstrate the efficiency of our model by image generation on MNIST, FashionMNIST and CIFAR10.

Fabian Altekrüger, Johannes Hertrich

Exploiting image patches instead of whole images have proved to be a powerful approach to tackle various problems in image processing. Recently, Wasserstein patch priors (WPP), which are based on the comparison of the patch distributions of the unknown image and a reference image, were successfully used as data-driven regularizers in the variational formulation of superresolution. However, for each input image, this approach requires the solution of a non-convex minimization problem which is computationally costly. In this paper, we propose to learn two kind of neural networks in an unsupervised way based on WPP loss functions. First, we show how convolutional neural networks (CNNs) can be incorporated. Once the network, called WPPNet, is learned, it can be very efficiently applied to any input image. Second, we incorporate conditional normalizing flows to provide a tool for uncertainty quantification. Numerical examples demonstrate the very good performance of WPPNets for superresolution in various image classes even if the forward operator is known only approximately.