Generalized Tree-Based Wavelet Transform

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 i...

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Veröffentlicht in:IEEE transactions on signal processing Jg. 59; H. 9; S. 4199 - 4209
Hauptverfasser: Ram, I., Elad, M., Cohen, I.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York, NY IEEE 01.09.2011
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Applied sciences
Approximation
Approximation algorithms
Approximation methods
Binary trees
Detection, estimation, filtering, equalization, prediction
Efficient signal representation
Exact sciences and technology
Filtering
hierarchical trees
Image denoising
Image processing
Information, signal and communications theory
Signal and communications theory
Signal processing
Signal processing algorithms
Signal representation. Spectral analysis
Signal, noise
Studies
Telecommunications and information theory
Transforms
Trees
Two dimensional
Wavelet
wavelet transform
Wavelet transforms
Averaging method
Filtering
Image processing
Noise reduction
wavelet transform
Two dimensional model
Haar function
Signal representation
Efficient signal representation
Algorithm
hierarchical trees
Wavelet transformation
One dimensional model
Signal processing
Haar transforms
Decimation
Singular value decomposition
Image denoising
weighted graph
data adaptive orthonormal wavelet
wavelet transforms
wavelet scheme
K-SVD algorithm
trees (mathematics)
filtering theory
image denoising
Haar-like transform
wavelet filtering
network
hierarchical tree
high dimensional data
subimage averaging scheme
geometry
generalized tree-based wavelet transform
approximation coefficient
decimation scheme
ISSN:1053-587X, 1941-0476
Online-Zugang:Volltext
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Beschreibung
Zusammenfassung: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.
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ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2011.2158428