Compactly Supported Tensor Product Complex Tight Framelets with Directionality

Abstract

Although tensor product real-valued wavelets have been successfully applied to many high-dimensional problems, they can capture well only edge singularities along the coordinate axis directions. As an alternative and improvement of tensor product real-valued wavelets and dual tree complex wavelet transform, recently tensor product complex tight framelets with increasing directionality have been introduced in [B. Han, Math. Model. Nat. Phenom., 8 (2013), pp. 18--47] and applied to image denoising in [B. Han and Z. Zhao, SIAM J. Imaging Sci., 7 (2014), pp. 997--1034]. Despite several desirable properties, those directional tensor product complex tight framelets are bandlimited and do not have compact support in the space/time domain. Since compactly supported wavelets and framelets are of great interest and importance in both theory and application, it remains as an unsolved problem whether there exist compactly supported tensor product complex tight framelets with directionality. In this paper, we shall answer this question by proving a theoretical result on directionality of tight framelets and by introducing an algorithm to construct compactly supported complex tight framelets with directionality. Our examples show that compactly supported complex tight framelets with directionality can be easily derived from any given eligible low-pass filters and refinable functions. Several examples of compactly supported tensor product complex tight framelets with directionality have been presented.