08-16-2017, 11:00 PM
This example shows how the dual-tree discrete wavelet transform (DWT) provides advantages over the critically sampled DWT for signal and image processing. The dual-tree DWT is implemented as two separate two-channel filter banks. To gain the advantages described in this example, you cannot arbitrarily choose the scaling and wavelet filters used in the two trees. The lowpass (scaling) and highpass (wavelet) filters of one tree, $\{h_0,h_1\}$, must generate a scaling function and wavelet that are approximate Hilbert transforms of the scaling function and wavelet generated by the lowpass and highpass filters of the other tree, $\{g_0,g_1\}$. Therefore, the complex-valued scaling function and wavelet formed from the two trees are approximately analytic.
As a result, the dual-tree (complex) DWT exhibits less shift variance and more directional selectivity than the critically sampled DWT with only a $2^d$ redundancy factor for $d$-dimensional data. The redundancy in the dual-tree DWT is significantly less than the redundancy in the undecimated (stationary) DWT.
This example illustrates the approximate shift invariance of the dual-tree DWT, the selective orientation of the dual-tree analyzing wavelets in 2-D, and the use of the dual-tree DWT in image denoising.