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-CTFm with m > 3 in [14, 18] and their impressive performance for image processing in [18, 30], in this paper we introduce a directional tensor product complex tight framelet TP-CTF↓6 (called reduced TP-CTF6) with low redundancy. Such TP-CTF ↓ 6 is a particular example of tight framelet filter banks with mixed sampling factors. The TP-CTF↓6 in d dimensions not only offers good directionality but also has the low redundancy rate 3 −1 2d−1 (e.g., the redundancy rates are 2, 2 2 3 , 3 5 7 , 5 1 3 and 7 25 31 for dimension d = 1, . . . , 5, respectively). Moreover, our numerical experiments on image/video denoising and inpainting show that the performance using our proposed TP-CTF↓6 is often comparable or sometimes better than several state-of-the-art frame-based methods which have much higher redundancy rates than that of TP-CTF↓6.