A Discrete Dyadic Wavelet Transform for Multidimensional Feature Analysis

Discrete non-redundant wavelet transforms have been successfully applied previously in image compression applications [2], [6], [17]. However, the lack of translation invariance and aliasing present after the decomposition stage [3] may introduce undesirable artifacts for the analysis of medical signals and images, and can justify the use of a redundant wavelet representation. The discrete dyadic wavelet transform is one example of a redundant representation. As originally proposed, the wavelet was a rst derivative of a smoothing function, and was used as a multiscale edge detector to obtain a translation-invariant parsimonious representation consisting of edges [14]. A reconstruction algorithm to approximate an original signal from its multiscale edge coe cients alone was devised in [14], [13]. On the other hand, previous applications described in [11], [9], [10], [4] made no attempt to obtain a parsimonious representation from a discrete dyadic wavelet transform. Rather, the transform intentionally remained highly redundant. This redundancy was exploited for image enhancement by rst modifying transform coe cients in some non-linear fashion and reconstructing. Here, we continue this theme and present a discrete dyadic wavelet transform as a redundant representation which can be implemented e ciently and is well matched for quanti cation problems in the analysis of medical images. The discrete dyadic wavelet transform was originally proposed in one and two dimensions. However, in medical imaging, there is a more general need for signal processing in more than two dimensions. In this Chapter, we extend the discrete dyadic wavelet transform to multiple dimensions and describe an e cient implementation within a fast hierarchical digital ltering scheme. When digital ltering of a nite-duration discrete signal is performed via circular con-