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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| Published in: | IEEE transactions on signal processing Vol. 59; no. 9; pp. 4199 - 4209 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York, NY
IEEE
01.09.2011
Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 1053-587X, 1941-0476 |
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| 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. |
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| AbstractList | 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 [et al], 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. 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. |
| Author | Cohen, I. Elad, M. Ram, I. |
| Author_xml | – sequence: 1 givenname: I. surname: Ram fullname: Ram, I. email: idanram@tx.technion.ac.il organization: Dept. of Electr. Eng., Technion - Israel Inst. of Technol., Haifa, Israel – sequence: 2 givenname: M. surname: Elad fullname: Elad, M. email: elad@cs.technion.ac.il organization: Dept. of Comput. Sci., Technion - Israel Inst. of Technol., Haifa, Israel – sequence: 3 givenname: I. surname: Cohen fullname: Cohen, I. email: icohen@ee.technion.ac.il organization: Dept. of Electr. Eng., Technion - Israel Inst. of Technol., Haifa, Israel |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | Generalized Tree-Based Wavelet Transform |
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