Colloquium

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.