1-D seislet transform

Next: 2-D seislet transform Up: From wavelets to seislets Previous: From wavelets to seislets

1-D seislet transform

The prediction and update operators employed in the lifting scheme can be understood as digital filters. In the -transform notation, the Haar prediction filter from equation 3 is

$\begin{displaymath} P(Z) = Z \end{displaymath}$

(8)

(shifting each sample by one), and the linear interpolation filter from equation 4 is

$\begin{displaymath} P(Z) = 1/2\,(1/Z+Z)\;. \end{displaymath}$

(9)

These predictions are appropriate for smooth signals but may not be optimal for a sinusoidal signal. In comparison, the prediction

$\begin{displaymath} P(Z) = Z/Z_0\;, \end{displaymath}$

(10)

where $Z_0 = e^{i\,\omega_0 \Delta t}$ , perfectly characterizes a sinusoid with $\omega_0$ circular frequency sampled on a $\Delta t$ grid. In other words, if a constant signal ( $\omega_0=0$ ) is perfectly predicted by shifting each trace to its neighbor, a sinusoidal signal ( $\omega_0\ne0$ ) requires the shift to be modulated by an appropriate frequency. Likewise, the linear interpolation in equation 9 needs to be replaced by a filter tuned to a particular frequency in order to predict a sinusoidal signal with that frequency perfectly:

$\begin{displaymath} P(Z) = 1/2\,(Z_0/Z+Z/Z_0)\;. \end{displaymath}$

(11)

The analysis easily extends to higher-order filters.

Next: 2-D seislet transform Up: From wavelets to seislets Previous: From wavelets to seislets

2013-03-02