arXiv subjects

Melanie Melching

Publications and source records attributed to Melanie Melching.

3 recordsLinked to original sources

arXiv2021

Diffusion Tensor Regularization with Metric Double Integrals

In this paper we propose a variational regularization method for denoising and inpainting of diffusion tensor magnetic resonance images. We consider these images as manifold-valued Sobolev functions, i.e. in an infinite dimensional setting, which are defined appropriately. The regularization functionals are defined as double integrals, which are equivalent to Sobolev semi-norms in the Euclidean setting. We extend the analysis of Ciak, Melching and Scherzer "Regularization with Metric Double Integrals of Functions with Values in a Set of Vectors", in: Journal of Mathematical Imaging and Vision (2019) concerning stability and convergence of the variational regularization methods by a uniqueness result, apply them to diffusion tensor processing, and validate our model in numerical examples with synthetic and real data.

math.OC↗

arXiv2019

Regularization with Metric Double Integrals for Vector Tomography

We present a family of non-local variational regularization methods for solving tomographic problems, where the solutions are functions with range in a closed subset of the Euclidean space, for example if the solution only attains values in an embedded sub-manifold. Recently, in Ciak, Melching and Scherzer "Regularization with Metric Double Integrals of Functions with Values in a Set of Vectors", in: Journal of Mathematical Imaging and Vision (2019), such regularization methods have been investigated analytically and their efficiency has been tested for basic imaging tasks such as denoising and inpainting. In this paper we investigate solving complex vector tomography problems with non-local variational methods both analytically and numerically.

math.OC↗

arXiv2018

Regularization with Metric Double Integrals of Functions with Values in a Set of Vectors

We present an approach for variational regularization of inverse and imaging problems for recovering functions with values in a set of vectors. We introduce regularization functionals, which are derivative-free double integrals of such functions. These regularization functionals are motivated from double integrals, which approximate Sobolev semi-norms of intensity functions. These were introduced in Bourgain, Brézis and Mironescu, "Another Look at Sobolev Spaces". In: Optimal Control and Partial Differential Equations-Innovations and Applications, IOS press, Amsterdam, 2001. For the proposed regularization functionals we prove existence of minimizers as well as a stability and convergence result for functions with values in a set of vectors.

math.OC↗