Shearlet Coorbit Spaces, Shearlet Transforms and Applications in Imaging

Beschreibung

Directional multiscale representation of images to address curved singularities has received much attention in harmonic analysis in the last 25 years. In particular, shearlets and curvelets provide an optimally sparse approximation of cartoon-like images. Shearlets possess a uniform construction for both the continuous and the discrete setting. This and the underlying group structure let them gain attraction in various theoretical and applied fields. In this thesis we contribute to both the continuous and the discrete setting. Having the shearlet group we discover isomorphisms to other groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group has an isomorphic symplectic subgroup while this is not true for the full shearlet group. Shearlet coorbit spaces are canonical smoothness spaces designed by applying the general coorbit theory of Feichtinger and Gröchenig. We examine structural properties of these shearlet coorbit spaces. We show an embedding into Besov spaces and examine traces onto the coordinate planes and their embedding into lower dimensional Besov and shearlet coorbit spaces. In the discrete setting we describe the implementation details of a fast and finite shearlet transform based on the FFT. We further describe how the discrete shearlet transform can be incorporated into convex imaging functionals for segmentation, decomposition and inpainting. We introduce a novel quadrature operator called linearized Riesz transform that corresponds to the shear operator. Based on the linearized Riesz transform we introduce finite discrete quasi-monogenic shearlets. Numerical experiments show the alignment of the directional information obtained from the shearlets and the quasi-monogenic orientation.

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Discrete Shearlet Transform Shearlet Coorbit Spaces Shearlet Group Shearlets

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