Generalized Tree-Based Wavelet Transform

By Idan Ram, Michael Elad, Israel Cohen. 2010

98 citations recorded by the source corpus.

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.

Subfields: Computer Vision and Pattern Recognition, Media Technology, Speech & Audio

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Citations in the corpus

Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering (2016, methodology): “In this context, the authors of [12, 8, 10] revisited the construction of wavelet operators on graphs and techniques to perform mutli-scale pyramid transforms on graphs were proposed in [32, 27].”
Wavelets on Graphs via Deep Learning (2013, methodology): “by modifying the underlying graph [19, 17]) which effectively renders those transforms non-linear.”
A class of Laplacian multiwavelets bases for high-dimensional data (2015, methodology): “[36] Ram, I.”
Learning Laplacian Matrix in Smooth Graph Signal Representations (2014, background): “We believe that our new framework can benefit numerous emerging real world applications, such as the analysis of transportation, biomedical, and social networks, where it is critical to infer hidden relationships between data entities.”
Spectral Graph Skeletons for 3D Action Recognition (2014, methodology): “Recent approaches for using graphs in machine learning include graph kernels [1,14,38], generalizations of signal processing frameworks to the graph domain [7, 24], and also graph wavelets [5,6,12,20,22], such as the spectral graph wavelet transform [12].”
Data-Driven Tree Transforms and Metrics (2017, methodology): “Ram, Elad and Cohen [26] also proposed a “generalized tree transform” where folders are connected to multiple parents in the level above, however their work differs in two aspects.”