A new directional, low-redundancy, complex-wavelet transform

Shift variance and poor directional selectivity, two major disadvantages of the discrete wavelet transform, have previously been circumvented either by using highly redundant, non-separable wavelet transforms or by using restrictive designs to obtain a pair of wavelet trees with a transform-domain redundancy of 4.0 in 2D. We demonstrate that excellent shift-invariance properties and directional selectivity may be obtained with a transform-domain redundancy of only 2.67 in 2D. We achieve this by projecting the wavelet coefficients from Selesnick's (Selesnick et al., 2000) almost-shift-invariant, double-density wavelet transform so as to separate approximately the positive and negative frequencies, thereby increasing directionality. Subsequent decimation and a novel inverse projection maintain the low redundancy while ensuring perfect reconstruction. Although our transform generates complex-valued coefficients allowing processing capabilities that are impossible with real-valued coefficients, it may be implemented with a fast algorithm that uses only real arithmetic. To demonstrate the efficacy of our new transform, we show that it achieves state-of-the-art performance in a seismic image-processing application.