Image Processing with Complex Wavelets

We rst review how wavelets may be used for multi-resolution image processing, describing the lter-bank implementation of the discrete wavelet transform (dwt) and how it may be extended via separable ltering for processing images and other multi-dimensional signals. We then show that the condition for inversion of the dwt (perfect reconstruction) forces many commonly used wavelets to be similar in shape, and that this shape produces severe shift variance (variation of dwt coeecient energy at any given scale with shift of the input signal). It is also shown that separable ltering with the dwt prevents the transform from providing directionally selective lters for diagonal image features. Complex wavelets can provide both shift invariance and good directional se-lectivity, with only modest increases in signal redundancy and computation load. However development of a complex wavelet transform (cwt) with perfect reconstruction and good lter characteristics has proved diicult until recently. We now propose the dual-tree cwt as a solution to this problem, yielding a transform with attractive properties for a range of signal and image processing applications, including motion estimation, denoising, texture analysis and synthesis, and object segmentation.