The shift property of the nondownsampled discrete wavelet transform

The wavelet transform is widely used in signal processing with its localization properties across scales both in the frequency and time domain. Because the FIR filters used in the wavelet transform always have phase delay, this will cause the positions of the detected signal components such as signal edges or spikes shift to different positions across scales. We first demonstrate the shift phenomena of wavelet transform, and then introduce several alignment methods of FIR filter coefficients to make certain characteristic points of signal the same across scales. We also give some applications of the proposed algorithm in edge detection to show the effectiveness of the algorithm