Generalized Tree-Based Wavelet Transform

Idan Ram (1) , Michael Elad (2) , Israel Cohen (1)

1 EE-Technion - Department of Electrical Engineering - Technion [Haïfa]
2 Department of Computer Science [Haifa]

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.

Mots clés: 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

Idan Ram, Michael Elad, Israel Cohen. Generalized Tree-Based Wavelet Transform. IEEE Transactions on Signal Processing, 2011, 59 (9), pp.4199 -4209. ⟨10.1109/TSP.2011.2158428⟩. ⟨hal-00705951⟩