G. Kutyniok

G. Kutyniok, W. -Q Lim, X. Zhuang

Over the past years, various representation systems which sparsely approximate functions governed by anisotropic features such as edges in images have been proposed. We exemplarily mention the systems of contourlets, curvelets, and shearlets. Alongside the theoretical development of these systems, algorithmic realizations of the associated transforms were provided. However, one of the most common shortcomings of these frameworks is the lack of providing a unified treatment of the continuum and digital world, i.e., allowing a digital theory to be a natural digitization of the continuum theory. In fact, shearlet systems are the only systems so far which satisfy this property, yet still deliver optimally sparse approximations of cartoon-like images. In this chapter, we provide an introduction to digital shearlet theory with a particular focus on a unified treatment of the continuum and digital realm. In our survey we will present the implementations of two shearlet transforms, one based on band-limited shearlets and the other based on compactly supported shearlets. We will moreover discuss various quantitative measures, which allow an objective comparison with other directional transforms and an objective tuning of parameters. The codes for both presented transforms as well as the framework for quantifying performance are provided in the Matlab toolbox ShearLab.

W. Dahmen, G. Kutyniok, W. -Q Lim et al.

This paper builds on recent developments of adaptive methods for linear transport equations based on certain stable variational formulations of Petrov-Galerkin type. The variational formulations allow us to employ meshes with cells of arbitrary aspect ratios. We develop a refinement scheme generating highly anisotropic partitions that is inspired by shearlet systems. We establish approximation rates for N-term approximations from corresponding piecewise polynomials for certain compact cartoon classes of functions. In contrast to earlier results in a curvelet or shearlet context the cartoon classes are concisely defined through certain characteristic parameters and the dependence of the approximation rates on these parameters is made explicit here. The approximation rate results serve then as a benchmark for subsequent applications to adaptive Galerkin solvers for transport equations. In numerical experiments, the new algorithms track C^2-curved shear layers and discontinuities stably and accurately, and realize essentially optimal rates. Finally, we treat parameter dependent transport problems, which arise in kinetic models as well as in radiative transfer. In heterogeneous media these problems feature propagation of singularities along curved characteristics precluding, in particular, fast marching methods based on ray-tracing. Since now the solutions are functions of spatial variables and parameters one has to address the curse of dimensionality. We show computationally, for a model parametric transport problem in heterogeneous media in 2 + 1 dimension, that sparse tensorization of the presently proposed spatial directionally adaptive scheme with hierarchic collocation in ordinate space based on a stable variational formulation high-dimensional phase space, the curse of dimensionality can be removed when approximating averaged bulk quantities.

A. Goeßmann, M. Götte, I. Roth et al.

Given observations of a physical system, identifying the underlying non-linear governing equation is a fundamental task, necessary both for gaining understanding and generating deterministic future predictions. Of most practical relevance are automated approaches to theory building that scale efficiently for complex systems with many degrees of freedom. To date, available scalable methods aim at a data-driven interpolation, without exploiting or offering insight into fundamental underlying physical principles, such as locality of interactions. In this work, we show that various physical constraints can be captured via tensor network based parameterizations for the governing equation, which naturally ensures scalability. In addition to providing analytic results motivating the use of such models for realistic physical systems, we demonstrate that efficient rank-adaptive optimization algorithms can be used to learn optimal tensor network models without requiring a~priori knowledge of the exact tensor ranks. As such, we provide a physics-informed approach to recovering structured dynamical laws from data, which adaptively balances the need for expressivity and scalability.

T. A. Bubba, G. Kutyniok, M. Lassas et al.

The high complexity of various inverse problems poses a significant challenge to model-based reconstruction schemes, which in such situations often reach their limits. At the same time, we witness an exceptional success of data-based methodologies such as deep learning. However, in the context of inverse problems, deep neural networks mostly act as black box routines, used for instance for a somewhat unspecified removal of artifacts in classical image reconstructions. In this paper, we will focus on the severely ill-posed inverse problem of limited angle computed tomography, in which entire boundary sections are not captured in the measurements. We will develop a hybrid reconstruction framework that fuses model-based sparse regularization with data-driven deep learning. Our method is reliable in the sense that we only learn the part that can provably not be handled by model-based methods, while applying the theoretically controllable sparse regularization technique to the remaining parts. Such a decomposition into visible and invisible segments is achieved by means of the shearlet transform that allows to resolve wavefront sets in the phase space. Furthermore, this split enables us to assign the clear task of inferring unknown shearlet coefficients to the neural network and thereby offering an interpretation of its performance in the context of limited angle computed tomography. Our numerical experiments show that our algorithm significantly surpasses both pure model- and more data-based reconstruction methods.

H. Lakshman, W. -Q Lim, H. Schwarz et al.

This paper proposes an image interpolation algorithm exploiting sparse representation for natural images. It involves three main steps: (a) obtaining an initial estimate of the high resolution image using linear methods like FIR filtering, (b) promoting sparsity in a selected dictionary through iterative thresholding, and (c) extracting high frequency information from the approximation to refine the initial estimate. For the sparse modeling, a shearlet dictionary is chosen to yield a multiscale directional representation. The proposed algorithm is compared to several state-of-the-art methods to assess its objective as well as subjective performance. Compared to the cubic spline interpolation method, an average PSNR gain of around 0.8 dB is observed over a dataset of 200 images.