The Construction of Nonseparable Wavelet Bi-Frames and Associated Approximation Schemes
Ehler, Martin
Wavelet analysis and its fast algorithms are widely used in many fields of applied mathematics such as in signal and image processing. In the present thesis, we circumvent the restrictions of orthogonal and biorthogonal wavelet bases by constructing wavelet frames. They still allow for a stable decomposition, and so-called wavelet bi-frames provide a series expansion very similar to those of pairs of biorthogonal wavelet bases. Contrary to biorthogonal bases, primal and dual wavelets are no longer supposed to satisfy any geometrical conditions, and the frame setting allows for redundancy. This provides more flexibility in their construction. Finally, we construct families of optimal wavelet bi-frames in arbitrary dimensions with arbitrarily high smoothness. Then we verify that the n-term approximation can be described by Besov spaces and we apply the theoretical findings to image denoising.
Philipps-Universität Marburg
Mathematics
opus:1818
https://doi.org/10.17192/z2007.0693
urn:nbn:de:hebis:04-z2007-06933
opus:1818
2007
Inequalities in approximation (Bernstein, Jackson, Nikolcprime skiui-type inequalities)
Publikationsserver der Universitätsbibliothek Marburg
Universitätsbibliothek Marburg
Mathematics
Mathematik
Philipps-Universität Marburg
Die Konstruktion Nichtseparabler Wavelet-Bi-Frames und Zugehörige Approximations-Schemata
https://doi.org/10.17192/z2007.0693
2007-11-29
doctoralThesis
Frames
190
application/pdf
Wavelet
Die Wavelet-Analyse mit ihren schnellen Algorithmen wird bereits in vielen Bereichen der angewandten Mathematik benutzt, beispielsweise in der Bild- und Signal-Verarbeitung. In der vorliegenden Arbeit überwinden wir die Beschränkungen von orthogonalen und biorthogonalen Wavelet-Basen in dem wir Wavelet-Frames konstruieren. Diese erlauben noch immer eine stabile Zerlegung und sogenannte Wavelet-Bi-Frames bieten eine Reihenentwicklung, sehr ähnlich zu Paaren biorthogonaler Wavelet-Basen. Im Gegensatz zu biorthogonalen Wavelets müssen primale und duale Wavelets keine geometrischen Bedingungen mehr erfüllen und das Frame-Konzept erlaubt Redundanzen. Dies bietet mehr Flexibilität, die in Konstruktionen genutzt werden können. Schließlich konstruieren wir eine ganze Familie optimaler Wavelet-Bi-Frames in beliebigen Dimensionen mit beliebig hoher Regularität. Dann beschreiben wir die n-Term Approximationsraten durch Besov-Regularität und wir wenden unsere Resultate auf das Entrauschen von Bildern an.
Wavelets
Multivariate
2007-10-19
Wavelets
Wavelets
https://archiv.ub.uni-marburg.de/diss/z2007/0693/cover.png
Wavelet analysis and its fast algorithms are widely used in many fields of applied mathematics such as in signal and image processing. In the present thesis, we circumvent the restrictions of orthogonal and biorthogonal wavelet bases by constructing wavelet frames. They still allow for a stable decomposition, and so-called wavelet bi-frames provide a series expansion very similar to those of pairs of biorthogonal wavelet bases. Contrary to biorthogonal bases, primal and dual wavelets are no longer supposed to satisfy any geometrical conditions, and the frame setting allows for redundancy. This provides more flexibility in their construction. Finally, we construct families of optimal wavelet bi-frames in arbitrary dimensions with arbitrarily high smoothness. Then we verify that the n-term approximation can be described by Besov spaces and we apply the theoretical findings to image denoising.
Ehler, Martin
Ehler
Martin
Wavelets
English
The Construction of Nonseparable Wavelet Bi-Frames and Associated Approximation Schemes
Frames
Beste Approximation
Mehrdimensional
monograph
Nonlinear approximation
Fachbereich Mathematik und Informatik
Best approximation, Chebyshev systems
ths
Prof. Dr.
Dahlke
Stephan
Dahlke, Stephan (Prof. Dr.)
urn:nbn:de:hebis:04-z2007-06933
Reine und Angewandte Mathematik
Nichtlineare Approximation
2011-08-10
PRESERVATION_MASTER
VIEW
Image
PRESERVATION_MASTER