Efficient Algorithms for Discrete Wavelet Transform : With Applications to Denoising and Fuzzy Inference Systems

Transforms are an important part of an engineer s toolkit for solving signal processing and polynomial computation problems. In contrast to the Fourier transform-based approaches where a fixed window is used uniformly for a range of frequencies, the wavelet transform uses short windows at high frequencies and long windows at low frequencies. This way, the characteristics of non-stationary disturbances can be more closely monitored. In other words, both time and frequency information can be obtained by wavelet transform. Instead of transforming a pure time description into a pure frequency description, the wavelet transform finds a good promise in a time-frequency description.

Due to its inherent time-scale locality characteristics, the discrete wavelet transform (DWT) has received considerable attention in digital signal processing (speech and image processing), communication, computer science and mathematics. Wavelet transforms are known to have excellent energy compaction characteristics and are able to provide perfect reconstruction. Therefore, they are ideal for signal/image processing. The shifting (or translation) and scaling (or dilation) are unique to wavelets. Orthogonality of wavelets with respect to dilations leads to multigrid representation.

The nature of wavelet computation forces us to carefully examine the implementation methodologies. As the computation of DWT involves filtering, an efficient filtering process is essential in DWT hardware implementation. In the multistage DWT, coefficients are calculated recursively, and in addition to the wavelet decomposition stage, extra space is required to store the intermediate coefficients. Hence, the overall performance depends significantly on the precision of the intermediate DWT coefficients.