  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.
