Directional tensor product complex tight framelets with low redundancy

Having the advantages of redundancy and flexibility, various types of tight frames have already shown impressive performance in applications such as image and video processing. For example, the undecimated wavelet transform, which is a particular case of tight frames, is known to have good performance for the denoising problem. Empirically, it is widely known that higher redundancy rate of a tight frame often leads to better performance in applications. The wavelet/framelet transform is often implemented in an undecimated fashion for the purpose of better performance in practice. Though high redundancy rate of a tight frame can improve performance in applications, as the dimension increases, it also makes the computational cost skyrocket and the storage of frame coefficients increase exponentially. This seriously restricts the usefulness of such tight frames for problems in moderately high dimensions such as video processing in dimension three. Inspired by the directional tensor product complex tight framelets TP-CTFmTP-CTFm with m≥3m≥3 in [15] and [20] and their impressive performance for image processing in [20] and [33], in this paper we introduce directional tensor product complex tight framelets TP-CTFm↓ (called reduced TP-CTFmTP-CTFm) with low redundancy. Such TP-CTFm↓ are particular examples of tight framelet filter banks with mixed sampling factors. In particular, we shall develop a directional tensor product complex tight framelet TP-CTF6↓ such that it performs nearly as well as the original TP-CTF6TP-CTF6 in [20] for image/video denoising/inpainting but it has significantly lower redundancy rates than TP-CTF6TP-CTF6 in every dimension. The TP-CTF6↓ in d   dimensions not only offers good directionality as the original TP-CTF6TP-CTF6 does but also has the low redundancy rate 3d−12d−1 (e.g., the redundancy rates are 2,223,357,513 and 72531 for dimension d=1,…,5d=1,…,5, respectively), in comparison with the redundancy rate 2d×3d−12d−1 of TP-CTF6TP-CTF6 in dimension d  . Moreover, our numerical experiments on image/video denoising and inpainting show that the performance using our proposed TP-CTF6↓ is often comparable with or sometimes better than several state-of-the-art frame-based methods which have much higher redundancy rates than that of TP-CTF6↓.