Image processing with complex wavelets

this paper we consider how wavelets may be used for image processing. To date, there has been considerable interest in wavelets for image compression, and they are now commonly used by researchers for this purpose, even though the main international standards still use the discrete cosine transform (dct). However for image processing tasks, other than compression, the take-up of wavelets has been less enthusiastic. Here we analyse possible reasons for this and present some new ways to use wavelets which offer significant advantages. A good review of wavelets and their application to compression may be found in Rioul Vetterli (1991) and in-depth coverage is given in the book by Vetterli Kovacevic (1995). An issue of the Proceedings of the IEEE (Kovacevic Daubechies 1996) has been devoted to wavelets and includes many very readable articles by leading experts. In x 2 of this paper we introduce the basic discrete wavelet filter tree and show how it may be used to decompose multi-dimensional signals. In x 3 we show some typical wavelets and illustrate the similar shapes of those which all satisfy the perfect reconstruction constraints. Unfortunately, as explained in x 4, discrete