# Directional Framelets with Low Redundancy and Directional Quasi-tight Framelets

ASCW01 - Challenges in optimal recovery and hyperbolic cross approximation

Edge singularities are ubiquitous and hold key information for many high-dimensional problems. Consequently, directional representation systems are required to effectively capture edge singularities for high-dimensional problems. However, the increased angular resolution often significantly increases the redundancy rates of a directional system. High redundancy rates lead to expensive computational costs and large storage requirement, which hinder the usefulness of such directional systems for problems in moderately high dimensions such as video processing. In this talk, we attack this problem by using directional tensor product complex tight framelets with mixed sampling factors. Such introduced directional system has good directionality with a very low redundancy rate $\frac{3d-1}{2d-1}$, e.g., the redundancy rates are $2$, $2\frac{2}{3}$, $3\frac{5}{7}$, $5\frac{1}{3}$ and $7\frac{25}{31}$ for dimension $d=1,\ldots,5$. Our numerical experiments on image/video denoising and inpainting show that the performance of our proposed directional system with low redundancy rate is comparable or better than several state-of-the-art methods which have much higher redundancy rates. In the second part, we shall discuss our recent developments of directional quasi-tight framelets in high dimensions. This is a joint work with Chenzhe Diao, Zhenpeng Zhao and Xiaosheng Zhuang.

This talk is part of the Isaac Newton Institute Seminar Series series.