Directional Tensor Product Complex Tight Framelets with Low Redundancy

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 ${TP-CTF}_m$ with $m\ge 3$ 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 ${TP-CTF}_6$) 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 $\frac{3^d-1}{2^d-1}$ (e.g., the redundancy rates are $2, 2\mathord{\frac{2}{3}}, 3\mathord{\frac{5}{7}}, 5\mathord{\frac{1}{3}}$ and $7\mathord{\frac{25}{31}}$ 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 ${TPCTF}^!_6$.