|
|
|
| Seislet transform and seislet frame | |
|
Next: 2-D seislet transform
Up: From wavelets to seislets
Previous: From wavelets to seislets
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
|
(8) |
(shifting each sample by one), and the linear interpolation filter
from equation 4 is
|
(9) |
These predictions are appropriate for smooth signals but may not be
optimal for a sinusoidal signal. In
comparison, the prediction
|
(10) |
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:
|
(11) |
The analysis easily extends to higher-order filters.
|
|
|
| Seislet transform and seislet frame | |
|
Next: 2-D seislet transform
Up: From wavelets to seislets
Previous: From wavelets to seislets
2013-07-26