Generalized Tree-Based Wavelet Transform

TL;DR

This paper introduces a generalized wavelet transform based on hierarchical trees for functions on graphs and high-dimensional data, improving representation efficiency and denoising performance over traditional wavelet methods.

Contribution

It presents a novel tree-based wavelet transform that adapts to data geometry, with a new tree construction method and application to image denoising.

Findings

More efficient than 1D and 2D wavelet transforms in image representation

Achieves denoising results comparable to K-SVD

Effective for data on graphs and high-dimensional spaces

Abstract

In this paper we propose a new wavelet transform applicable to functions defined on graphs, high dimensional data and networks. The proposed method generalizes the Haar-like transform proposed in [1], and it is defined via a hierarchical tree, which is assumed to capture the geometry and structure of the input data. It is applied to the data using a modified version of the common one-dimensional (1D) wavelet filtering and decimation scheme, which can employ different wavelet filters. In each level of this wavelet decomposition scheme, a permutation derived from the tree is applied to the approximation coefficients, before they are filtered. We propose a tree construction method that results in an efficient representation of the input function in the transform domain. We show that the proposed transform is more efficient than both the 1D and two-dimensional (2D) separable wavelet transforms in representing images. We also explore the application of the proposed transform to image denoising, and show that combined with a subimage averaging scheme, it achieves denoising results which are similar to those obtained with the K-SVD algorithm.