Seislet transform and seislet frame |

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

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

where , perfectly characterizes a sinusoid with circular frequency sampled on a grid. In other words, if a constant signal () is perfectly predicted by shifting each trace to its neighbor, a sinusoidal signal () 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:

The analysis easily extends to higher-order filters.

Seislet transform and seislet frame |

2013-07-26