Recent Progress in Applied and Computational Harmonic Analysis

Wavelet theory has been extensively developed in the function space L2 and discrete wavelet transform has successful applications in many areas. However, to understand better the performance of different discrete wavelet transforms, it is important to investigate their underlying discrete wavelet systems in l2. Though some preliminary results have been found recently, despite the fact that stability is a key issue in mathematical foundation of wavelet theory, results on stability of discrete wavelet systems in l2 have been barely developed so far. In the meantime, in recent years, to better handle edge singularities, it is realized that redundant wavelet transform is often preferred by providing better directionality and flexibility. Many different directional systems such as curvelets, framelets, and shearlets have been proposed in the literature. The hard/soft thresholding is theoretically optimal for discrete wavelet transform using orthogonal wavelet filter banks. Because the associated redundant transform is no longer orthonormal, it is not known so far what is the theoretically optimal thresholding strategies for redundant transforms, in particular, for the problem of image denoising. Research teams at University of Calgary and University of Alberta are currently collaborating together to investigate the discrete wavelet system associated with complex tight framelet filter banks and are exploring the thresholding strategies for such redundant transforms using Gaussian scale mixture, which has the capability to cope with correlated noise with superior performance. The idea of using patches in the state-of-the-art denoising algorithm BM3D also takes the advantages of redundancy. We are currently also exploring the thresholding strategies for the patch-based approach in image denoising using higher order singular value decomposition.