Image processing with complex wavelets

We first review how wavelets may be used for multi-resolution image process ing, describing the filter-bank implementation of the discrete wavelet tran sform (DWT) and how it may be extended via separable filtering for processi ng images and other multi-dimensional signals. We then show that the condit ion for inversion of the DWT (perfect reconstruction) forces many commonly used wavelets to be similar in shape, and that this shape produces severe s hift dependence (variation of DWT coefficient energy at any given scale wit h shift of the input signal). It is also shown that separable filtering wit h the DWT prevents the transform from providing directionally selective fil ters for diagonal image features. Complex wavelets can provide both shift invariance and good directional sel ectivity, with only modest increases in signal redundancy and computation l oad. However, development of a complex wavelet transform (CWT) with perfect reconstruction and good filter characteristics has proved difficult until recently. We now propose the dual-tree CWT as a solution to this problem, y ielding a transform with attractive properties for a range of signal and im age processing applications, including motion estimation, denoising, textur e analysis and synthesis, and object segmentation.