Design of Q-shift complex wavelets for image processing using frequency domain energy minimization

This paper proposes a new method of designing finite-support wavelet filters, based on minimization of energy in key parts of the frequency domain. In particular this technique is shown to be very effective for designing families of filters that are suitable for use in the shift-invariant dual-tree complex wavelet structure that has been developed by the author recently, and has been shown to be important for a range of image processing applications. The dual-tree structure requires most of the wavelet filters to have a well-controlled group delay, equivalent to one quarter of a sample period, in order to achieve optimal shift invariance. The proposed new design technique allows this requirement to be included along with the usual smoothness and perfect reconstruction properties to yield wavelet filters with a unique combination of features: linear phase, tight frame, compact spatial support, good frequency domain selectivity with low sidelobe levels, approximate shift invariance, and good directional selectivity in two or more dimensions.