arXiv subjects

G. Steidl

Publications and source records attributed to G. Steidl.

5 recordsLinked to original sources

arXiv2025

On Coorbit Fr\'echet Spaces

This paper is concerned with a new approach to coorbit space theory. Usually, coorbit spaces are defined by collecting all distributions for which the voice transform associated with a square-integrable group representation possesses a certain decay, usually measured in a Banach space norm such as weighted $L_p$-norms. Unfortunately, in cases where the representation does not satisfy certain integrability conditions, one is faced with a bottleneck, namely that the discretization of the coorbit spaces is surprisingly difficult. It turns out that in these cases the construction of coorbit spaces as Fr\'echet spaces is much more convenient since then atomic decompositions can be established in a very natural way.

math.FA↗

arXiv2025

A Jump-Diffusion Framework for Irregular Time Series Generation

We propose a framework for generative modeling of continuous-time processes from irregularly and asynchronously recorded data. It is based on the matching of generators and accommodates discontinuous trajectories. Analytical formulas for diffusion and jump bridges yield a family of reference generators that a neural network is trained to match. The key ingredient is that, for our constructed jump bridge, a parametrization of the jump kernel densities by scaled Gaussians admits closed-form expressions for the Kullback-Leibler divergence, allowing simulation-free training.

math.NA↗

arXiv2020

Continuous Wavelet Frames on the Sphere: The Group-Theoretic Approach Revisited

In \cite{AV99}, Antoine and Vandergheynst propose a group-theoretic approach to continuous wavelet frames on the sphere. The frame is constructed from a single so-called admissible function by applying the unitary operators associated to a representation of the Lorentz group, which is square-integrable modulo the nilpotent factor of the Iwasawa decomposition. We prove necessary and sufficient conditions for functions on the sphere, which ensure that the corresponding system is a frame. We strengthen a similar result in \cite{AV99} by providing a complete and detailed proof.

math.FA↗

arXiv2012

Fast Finite Shearlet Transform

In recent years it has turned out that shearlets have the potential to retrieve directional information so that they became interesting for many applications. Moreover the continuous shearlet transform has the outstanding property to stem from a square integrable group representation. However, to use shearlets and the shearlet transform for reasonable applications one needs fast algorithms to compute a discrete shearlet transform. In this tutorial we present the steps towards an implementation of a fast and finite shearlet transform that is only based on the FFT. Using band-limited shearlets we construct a Parseval frame that provides a simple and straightforward inverse shearlet transform. We provide all proofs and discuss several aspects of our implementation.

math.NA↗

arXiv2011

Convex Multiclass Segmentation with Shearlet Regularization

Segmentation plays an important role in many preprocessing stages in image processing. Recently, convex relaxation methods for image multi-labeling were proposed in the literature. Often these models involve the total variation (TV) semi-norm as regularizing term. However, it is well-known that the TV functional is not optimal for the segmentation of textured regions. In recent years directional representation systems were proposed to cope with curved singularities in images. In particular, curvelets and shearlets provide an optimally sparse approximation in the class of piecewise smooth functions with $C^2$ singularity boundaries. In this paper, we demonstrate that the discrete shearlet transform is suited as regularizer for the segmentation of curved structures. Neither the shearlet nor the curvelet transform where used as regularizer in a segmentation model so far. To this end, we have implemented a translation invariant finite discrete shearlet transform based on the FFT. We describe how the shearlet transform can be incorporated within the multi-label segmentation model and show how to find a minimizer of the corresponding functional by applying an alternating direction method of multipliers. Here the Parseval frame property of our shearlets comes into play. We demonstrate by numerical examples that the shearlet regularized model can better segment curved textures than the TV regularized one and that the method can also cope with regularizers obtained from non-local means.

math.NA↗