TY - THES A1 - Fink, Thomas T1 - Curvature Detection by Integral Transforms N2 - In various fields of image analysis, determining the precise geometry of occurrent edges, e.g. the contour of an object, is a crucial task. Especially the curvature of an edge is of great practical relevance. In this thesis, we develop different methods to detect a variety of edge features, among them the curvature. We first examine the properties of the parabolic Radon transform and show that it can be used to detect the edge curvature, as the smoothness of the parabolic Radon transform changes when the parabola is tangential to an edge and also, when additionally the curvature of the parabola coincides with the edge curvature. By subsequently introducing a parabolic Fourier transform and establishing a precise relation between the smoothness of a certain class of functions and the decay of the Fourier transform, we show that the smoothness result for the parabolic Radon transform can be translated into a change of the decay rate of the parabolic Fourier transform. Furthermore, we introduce an extension of the continuous shearlet transform which additionally utilizes shears of higher order. This extension, called the Taylorlet transform, allows for a detection of the position and orientation, as well as the curvature and other higher order geometric information of edges. We introduce novel vanishing moment conditions which enable a more robust detection of the geometric edge features and examine two different constructions for Taylorlets. Lastly, we translate the results of the Taylorlet transform in R^2 into R^3 and thereby allow for the analysis of the geometry of object surfaces. KW - Curvature KW - Wavelet KW - Shearlet KW - Parabolic Radon transform KW - Edge classification KW - Krรผmmung KW - Wavelet-Transformation KW - Shearlet KW - Radon-Transformation KW - Konturfindung Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus4-7684 ER - TY - JOUR A1 - Fink, Thomas A1 - Forster, Brigitte A1 - Heinrich, Florian T1 - Gaborโ€™s โ€œcomplex signalโ€ revisited: Complexifying frames and bases JF - PAMM (Proceedings in Applied Mathematics and Mechanics) N2 - In 1946, Dennis Gabor introduced the analytic signal ๐‘“ + ๐‘–๐ป๐‘“ for real-valued signals ๐‘“. Here, ๐ป is the Hilbert transform. This complexification of functions allows for an analysis of their amplitude and phase information and has ever since given well-interpretable insight into the properties of the signals over time. The idea of complexification has been reconsidered with regard to many aspects: examples are the dual tree complex wavelet transform, or via the Riesz transform and the monogenic signal, that is, a multi-dimensional version of the Hilbert transform, which in combination with multi-resolution approaches leads to Riesz wavelets, and others. In this context, we ask two questions: - Which pairs of real orthonormal bases (ONBs), Riesz bases, frames and Parseval frames {๐‘“ ๐‘› } ๐‘›โˆˆโ„• and {๐‘” ๐‘› } ๐‘›โˆˆโ„• can be โ€œrebrickedโ€ to complex-valued ones {๐‘“๐‘› + ๐‘–๐‘” ๐‘› } ๐‘›โˆˆโ„•? - And which real operators A allow for rebricking via the ansatz {๐‘“๐‘› + ๐‘–๐ด๐‘“๐‘› } ๐‘›โˆˆโ„•? In this short note, we give answers to these questions with regard to a characterization which linear operators A are suitable for rebricking while maintaining the structure of the original real valued family. Surprisingly, the Hilbert transform is not among them. Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:101:1-2023091815015485698899 VL - 23 IS - 3 PB - Wiley CY - Hoboken ER - TY - JOUR A1 - Fink, Thomas A1 - Forster-Heinlein, Brigitte A1 - Heinrich, Florian T1 - Rebricking frames and bases JF - Journal of Mathematical Analysis and Applications N2 - In 1949, Denis Gabor introduced the โ€œcomplex signalโ€ (nowadays called โ€œanalytic signalโ€) by combining a real function fwith its Hilbert transform Hfto 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:nโˆˆN} and {gn:nโˆˆN} form a complex basis or frame of the form {fn+ign:nโˆˆN}? And for which bounded linear operators Adoes {fn+iAfn:nโˆˆN} form a complex-valued orthonormal basis, Riesz basis or frame, when {fn:n โˆˆN} is a real-valued orthonormal basis, Riesz basis or frame? We call this approach rebricking. It is well-known that the analytic signals donโ€™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. KW - Real and imaginary parts of bases KW - frames and operators KW - Orthogonal and self-adjoint operators KW - Point spectrum KW - Approximate spectrum Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus4-19227 SN - 1096-0813 VL - 2025 PB - Elsevier CY - Amsterdam ER -