Wavelet Transforms in Image Processing

This paper is designed to be partly tutorial in nature and partly a summary of recent work by the authors in applying wavelets to various image processing problems. The tutorial part describes the lter-bank implementation of the discrete wavelet transform (DWT) and shows that most wavelets which permit perfect reconstruction are similar in shape and scale. We then discuss an important drawback of these wavelet transforms, which is that the distribution of energy between coeecients at diierent scales is very sensitive to shifts in the input data. We propose the Complex Wavelet Transform (CWT) as a solution to this problem and show how it may be applied in two dimensions. Finally we give brief details of applications of the CWT to motion estimation and image de-noising. Wavelets have become a popular tool for image compression research, although they have yet to make a big impact on image compression standards, most of which still use the discrete cosine transform (DCT) as their basic energy compaction (or decorrelation) process. A good review of wavelets and their application to compression may be found in Rioul and Vetterli 12] and in-depth coverage is given in the book by Vetterli and Kovacevic 15]. A recent issue of the Proceedings of the IEEE 5] has also been devoted to wavelets and includes many very readable articles by leading experts. The conventional discrete wavelet transform (DWT) may be regarded as equivalent to ltering the input signal with a bank of bandpass lters, whose impulse responses are all approximately given by scaled versions of a mother wavelet. The scaling factor between adjacent lters is usually 2:1, leading to octave bandwidths and centre frequencies that are one octave apart. At the coarsest scale, a lowpass lter is also required to represent fully the lowest frequencies of the signal. The outputs of the lters are usually maximally decimated so that the number of DWT output samples equals the number of input samples and the transform is invertable. The octave-band DWT is most eeciently implemented by Mallat's dyadic wavelet decomposition tree 11], a cascade of 2-band perfect-reconstruction lter banks, shown in g 1a. The main advantages of wavelets over the DCT are the absence of blocking artefacts and the multiscale nature of the DWT which allows near-optimal compression of features with a variety of diierent scales or sizes. Although wavelets are now a very popular alternative to the DCT for image …