A Directional Extension for Multidimensional Wavelet Transforms

Directional information is an important and unique feature of multidimensional signals. As a result of a separable extension from one-dimensional (1-D) bases, multidimensional wavelet transforms have very limited directionality. Furthermore, different directions are mixed in certain wavelet subbands. In this paper, we propose a simple directional extension for wavelets (DEW) that fixes this subband mixing problem and improves the directionality. The building block of the DEW is a two-channel 2-D filter bank with a checkerboard-shaped frequency partition. The DEW works with both the critically-sampled wavelet transform as well as the undecimated wavelet transform. In the 2-D case, it further divides the three wavelet subbands (i.e. horizontal, vertical, and diagonal) at each scale into six finer directional subbands. The DEW itself is critically-sampled, and hence will not increase the redundancy of the overall transform. Though nonseparable in essence, the proposed DEW has an efficient implementation that only requires 1-D filtering. Meanwhile, the DEW can be easily generalized to higher dimensions. In a nutshell, the proposed directional extension provides an optional tool to efficiently enhance the directionality of multidimensional wavelet transforms. Numerical experiments show that certain wavelet-based image processing applications will benefit from this improved directionality. Index Terms Wavelet transform, directional information, checkerboard filter bank, filter design, multidimensional signal processing, image denoising, feature extraction. Y. Lu is with the Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana IL 61801 (e-mail: [email protected]; WWW: http://www.ifp.uiuc.edu/∼yuelu). M. N. Do is with the Department of Electrical and Computer Engineering, the Coordinated Science Laboratory, and the Beckman Institute, University of Illinois at Urbana-Champaign, Urbana IL 61801 (e-mail: [email protected]; WWW: http://www.ifp.uiuc.edu/∼minhdo). This work was supported by the National Science Foundation under Grant CCR-0237633 (CAREER).