Shift-Variance Analysis of the Q-shift Dual-Tree Complex Wavelet Transform

We analyze the shift-variance of the Dual-Tree Complex Wavelet Transform (DTCWT) in this thesis. In our study, the DT-CWT is treated as a generalized sampling process. The analysis of shift-variance is performed in terms of two quantities: the shift-variance level (SVL) and the shift-variance measure (SVM). The SVL describes the amount of distance between the system and the set of systems which are so-called  -shift-invariant (specifying how the system has to respond to the shifted input signal). The SVM is equal to the SVL divided by two times of the system norm. We obtain the SVL and the SVM for the DT-CWT and the scaling function for Kingsbury’s Q-shift filters of length 8, 12, 18, and 22. We observe that the shiftvariance varies as the Q-shift filter length changes. When the length of the filter increases, the SVM becomes closer to the zero. This means that DT-CWT is almost shift-invariance. For better illustrating the shift-invariance property, we consider an input signal and its shifted version to compare the outputs corresponding to the DTCWT and the scaling function. Keywords: Dual-tree complex wavelet transforms, scaling function, shift-variance level, shift-variance measure, Q-shift filters.