Adaptive directional Haar tight framelets on bounded domains for digraph signal representations

TL;DR

This paper introduces adaptive directional Haar tight framelets on bounded domains, enabling efficient and directional signal representations on complex structures like digraphs, with potential applications in data analysis.

Contribution

It develops a novel construction of Haar-type tight framelets on compact sets, incorporating adaptivity and directionality for digraph signal processing.

Findings

Constructed Haar-type tight framelets on compact sets.

Demonstrated the use of these framelets for digraph signal representations.

Provided examples illustrating the effectiveness of the approach.

Abstract

Based on hierarchical partitions, we provide the construction of Haar-type tight framelets on any compact set $K\subseteq \mathbb{R}^d$. In particular, on the unit block $[0,1]^d$, such tight framelets can be built to be with adaptivity and directionality. We show that the adaptive directional Haar tight framelet systems can be used for digraph signal representations. Some examples are provided to illustrate results in this paper.