Abstract. In this work, a method for obtaining pixelwise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal prediction in order to obtain coverage guarantees for the error bounds, without making any assumption on the underlying data distribution. It is generally applicable to Bayesian regularization approaches, independent, e.g., of the concrete choice of the prior. Furthermore, the coverage guarantees can also be obtained in case only approximate sampling from the posterior is possible. With this in particular, the proposed framework is able to incorporate any learned prior in a black-box manner. Guaranteed coverage without assumptions on the underlying distributions is only achievable since the magnitude of the error bounds is, in general, unknown in advance. Nevertheless, experiments with multiple regularization approaches presented in the paper confirm that, in practice, the obtained error bounds are rather tight. For realizing the numerical experiments, a novel primal-dual Langevin algorithm for sampling from nonsmooth distributions is also introduced in this work, showing promising results in practice. While a proof of convergence for this primal-dual algorithm is still open, the theoretical guarantees of the proposed method do not require a guaranteed convergence of the sampling algorithm.

In this paper we introduce the notion of second-order total generalized variation (TGV) regularization for manifold-valued data in a discrete setting. We provide an axiomatic approach to formalize reasonable generalizations of TGV to the manifold setting and present two possible concrete instances that fulfill the proposed axioms. We provide well-posedness results and present algorithms for a numerical realization of these generalizations to the manifold setup. Further, we provide experimental results for synthetic and real data to further underpin the proposed generalization numerically and show its potential for applications with manifold-valued data.

The DjVu file format and image compression techniques are widely used in the archival of digital documents. Its key ingredients are the separation of the document into fore- and background layers and a binary switching mask, followed by a lossy, transform-based compression of the former and a dictionary-based compression of the latter. The lossy compression of the layers is based on a wavelet decomposition and bit truncation, which leads, in particular at higher compression rates, to severe compression artifacts in the standard decompression of the layers. The aim of this paper is to break ground for the variational decompression of DjVu files. To this aim, we provide an in-depth analysis and discussion of the compression standard with a particular focus on modeling data constraints for decompression. This allows to carry out DjVu decompression as regularized inversion of the compression procedure. As particular example, we evaluate the performance of such a framework using total variation and total generalized variation regularization. Furthermore, we provide routines for obtaining the necessary data constraints from a compressed DjVu file and for the forward and adjoint transformation operator involved in DjVu compression.

In this article, we evaluate Parallel Level Sets (PLS) and Bowsher's method as segmentation-free anatomical priors for regularized brain positron emission tomography (PET) reconstruction. We derive the proximity operators for two PLS priors and use the EM-TV algorithm in combination with the first order primal-dual algorithm by Chambolle and Pock to solve the non-smooth optimization problem for PET reconstruction with PLS regularization. In addition, we compare the performance of two PLS versions against the symmetric and asymmetric Bowsher priors with quadratic and relative difference penalty function. For this aim, we first evaluate reconstructions of 30 noise realizations of simulated PET data derived from a real hybrid positron emission tomography/magnetic resonance imaging (PET/MR) acquisition in terms of regional bias and noise. Second, we evaluate reconstructions of a real brain PET/MR data set acquired on a GE Signa time-of-flight PET/MR in a similar way. The reconstructions of simulated and real 3D PET/MR data show that all priors were superior to post-smoothed maximum likelihood expectation maximization with ordered subsets (OSEM) in terms of bias-noise characteristics in different regions of interest where the PET uptake follows anatomical boundaries. Our implementation of the asymmetric Bowsher prior showed slightly superior performance compared with the two versions of PLS and the symmetric Bowsher prior. At very high regularization weights, all investigated anatomical priors suffer from the transfer of non-shared gradients.

While current state of the art MR-PET scanners enable simultaneous MR and PET measurements, the acquired data sets are still usually reconstructed separately. We propose a new multi-modality reconstruction framework using second order Total Generalized Variation (TGV) as a dedicated multi-channel regularization functional that jointly reconstructs images from both modalities. In this way, information about the underlying anatomy is shared during the image reconstruction process while unique differences are preserved. Results from numerical simulations and in-vivo experiments using a range of accelerated MR acquisitions and different MR image contrasts demonstrate improved PET image quality, resolution, and quantitative accuracy.

A variational model for image reconstruction is introduced and analyzed in function space. Specific to the model is the data fidelity, which is realized via a basis transformation with respect to a Riesz basis followed by interval constraints. This setting in particular covers the task of reconstructing images constrained to data obtained from JPEG or JPEG 2000 compressed files. As image prior, the total generalized variation (TGV) functional of arbitrary order is employed. The present paper, the first of two works that deal with both analytical and numerical aspects of the model, provides a comprehensive analysis in function space and defines concrete instances for particular applications. A new, noncoercive existence result and optimality conditions, including a characterization of the subdifferential of the TGV functional, are obtained in the general setting.

The present work is the second of two papers on a variational model for image reconstruction whose specific features are twofold: First, data fidelity is realized by interval constraints on the coefficients of a Riesz basis representation, and second, total generalized variation (TGV) of arbitrary order is employed as image prior. While Part I provides a comprehensive analysis of the model in a general function space setting as well as discusses applications such as artifact-free JPEG, JPEG 2000 decompression, and variational zooming, Part II deals with the discretized setting and presents globally convergent algorithms for these applications. In the case of JPEG and JPEG 2000 decompression, highly improved color image reconstructions are obtained from the encoded files. The numerical results are supplemented by duality based stopping criteria, a parallelized implementation, and a comparison of results obtained with TV and second and third order TGV regularization.

The infimal convolution of total (generalized) variation--type functionals and its application as regularization for video and image reconstruction is considered. The definition of this particular type of regularization functional is motivated by the need of suitably combining spatial and temporal regularity requirements for video processing. The proposed functional is defined in an infinite dimensional setting, and important analytical properties are established. As applications, the reconstruction of compressed video data and of noisy still images is considered. The resulting problem settings are posed in function space, and suitable numerical solution schemes are established. Experiments confirm a significant improvement compared to standard total variation--type methods, which originates from the introduction of spatio-temporal and spatial anisotropies.

We propose a variational model for artifact-free JPEG decompression. It is based on the minimization of the total variation over the convex set U of all possible source images associated with given JPEG data. The general case where U represents a pointwise restriction with respect to an $L^2$-orthonormal basis is considered. Analysis of the infinite dimensional model is presented, including the derivation of optimality conditions. A discretized version is solved using a primal-dual algorithm supplemented by a primal-dual gap-based stopping criterion. Experiments illustrate the effect of the model. Good reconstruction quality is obtained even for highly compressed images, while a graphics processing unit (GPU) implementation is shown to significantly reduce computation time, making the model suitable for real-time applications.