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Complex-valued wavelets, the dual tree, and Hilbert pairs: why these lead to shift invariance and directional multi-dimensional wavelets

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If you have a question about this talk, please contact Carola-Bibiane Schoenlieb.

This talk will discuss some of the interesting mathematical properties of the dual tree wavelet construction, when it is used to generate analytic complex-valued wavelets. We shall show how the requirement for analyticity (zero energy at negative frequencies) implies that the wavelet bases from the two parts of the dual tree should form a Hilbert transform pair, and how this leads to the half-sample delay condition between the lowpass (scaling function) filters of each tree. It will then be shown how this results in approximate shift invariance of the complex wavelet transform, and the elegant Q-shift filter solution to this condition will be given. We will then discuss how the analytic nature of the wavelets in 1-D allows separable wavelet filters in higher dimensional spaces to be highly directionally selective. Finally we will present a wide range of applications for the dual-tree wavelets, including denoising, regularisation, motion / displacement estimation of 2-D and 3-D datasets, and object recognition.

The slides of the talk are online at http://www.damtp.cam.ac.uk/user/cbs31/MI_Cambridge/MathAndInfo_Network_files/nickkingsburytalk.pdf

This talk is part of the Mathematics & Information in Cambridge series.

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