Professur für Angewandte Mathematik

Das Paper  "Stabilizing Tensor Voting for 3D Curvature Estimation" von  Victor Alfaro Pérez, Virginie Uhlmann, und Brigitte Forster-Heinlein ist in den Proceedings "Workshop on mathematical modeling and scientific computing" erschienen: 

https://easychair.org/publications/paper/ssD6

Abstract: Curvature plays an important role in the function of biological membranes, and is therefore a readout of interest in microscopy data. The PyCurv library established itself as a valuable tool for curvature estimation in 3D microscopy images. However, in noisy images, the method exhibits visible instabilities, which are not captured by the standard error measures. In this article, we investigate the source of these instabilities, provide adequate measures to detect them, and introduce a novel post-processing step which corrects the errors. We illustrate the robustness of our enhanced method over various noise regimes and demonstrate that with our orientation correcting post-processing step, the PyCurv library becomes a truly stable tool for curvature quantification.

Das Paper "Rebricking frames and bases" von Thomas Fink, Brigitte Forster und Florian Heinrich ist "online first" im Journal of Mathematical Analysis and Applications erschienen:

https://doi.org/10.1016/j.jmaa.2024.129051

Abstract:

In 1949, Denis Gabor introduced the ``complex signal'' (nowadays called ``analytic signal'') by combining a real function f with its Hilbert transform Hf to a complex function f+iHf. His aim was to extract phase information, an idea that has inspired techniques as the monogenic signal and the complex dual tree wavelet transform. In this manuscript, we consider two questions: When do two real-valued bases or frames {fn:n∈ℕ} and {gn:n∈ℕ} form a complex basis or frame of the form {fn+ign:n∈ℕ}? And for which bounded linear operators A forms {fn+iAfn:n∈ℕ} a complex-valued orthonormal basis, Riesz basis or frame, when {fn:n∈ℕ} is a real-valued orthonormal basis, Riesz basis or frame? We call this approach \emph{rebricking}. It is well-known that the analytic signals don't span the complex vector space L2(ℝ;ℂ), hence H is not a rebricking operator. We give a full characterization of rebricking operators for bases, in particular orthonormal and Riesz bases, Parseval frames, and frames in general. We also examine the special case of finite dimensional vector spaces and show that we can use any real, invertible matrix for rebricking if we allow for permutations in the imaginary part.

Viel Spass beim Lesen!