Generalized Tree-Based Wavelet Transform
Abstract
In this paper we propose a new wavelet transform applicable to functions defined on high dimensional data, weighted graphs and networks. The proposed method generalizes the Haar-like transform recently introduced by Gavish , and can also construct data adaptive orthonormal wavelets beyond Haar. It is defined via a hierarchical tree, which is assumed to capture the geometry and structure of the input data, and is applied to the data using a modified version of the common one-dimensional (1D) wavelet filtering and decimation scheme. The adaptivity of this wavelet scheme is obtained by permutations derived from the tree and applied to the approximation coefficients in each decomposition level, before they are filtered. We show that the proposed transform is more efficient than both the 1D and two-dimension 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.
Keywords
Haar-like transform
K-SVD algorithm
approximation coefficient
data adaptive orthonormal wavelet
decimation scheme
generalized tree-based wavelet transform
geometry
hierarchical tree
high dimensional data
image denoising
network
subimage averaging scheme
wavelet filtering
wavelet scheme
weighted graph
Haar transforms
filtering theory
trees (mathematics)
wavelet transforms