Due to simple implementation and relatively easy construction of 1D
wavelets, real-valued tensor product wavelets have been commonly used for
high-dimensional problems. However, real-valued tensor product wavelets are
known to have some shortcomings, in particular, they lack directionality.
For example, for 2D data such as images, edge singularities are ubiquitous
and play a more fundamental role in image processing than point
singularities. As a consequence, real-valued tensor product wavelets in 2D
can only capture edge singularities along the horizontal and vertical
directions. In this talk, we present a comprehensive theory and
construction of directional complex tight framelets. While keeping the
simple tensor product structure, our approach has the advantages of
improved directionality and uses finitely supported filter banks. In
particular, we propose a family of directional tensor product complex tight
framelets with increasing directions. We shall show that such complex tight
framelets have superior performance for the problem of image denoising, in
comparison with the well-known dual tree complex wavelet transform and
other known wavelet-based image denoising methods.
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