Compactly Supported Tensor Product Complex Tight Framelets with Directionality

TL;DR

This paper introduces a method to construct compactly supported tensor product complex tight framelets with directionality, addressing a key limitation of existing bandlimited versions, and demonstrates their practical applicability.

Contribution

It provides a theoretical proof and an algorithm for creating compactly supported complex tight framelets with directionality, filling a significant gap in wavelet theory.

Findings

Successfully constructed compactly supported tensor product complex tight framelets with directionality.

Demonstrated the ease of deriving these framelets from low-pass filters and refinable functions.

Presented multiple examples illustrating the effectiveness of the proposed methods.

Abstract

Although tensor product real-valued wavelets have been successfully applied to many high-dimensional problems, they can only capture well 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 [8] and applied to image denoising in [13]. Despite several desirable properties, the directional tensor product complex tight framelets constructed in [8,13] 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 satisfactorily 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.