The Scaling Sequences of Lowest Approximation Order
Below are the coefficients for the scaling functions for D2-20. The wavelet coefficients are derived by reversing the order of the scaling function coefficients and then reversing the sign of every second one, (i.e., D4 wavelet = {-0.1830127, -0.3169873, 1.1830127, -0.6830127}). Mathematically, this looks like where k is the coefficient index, b is a coefficient of the wavelet sequence and a a coefficient of the scaling sequence. N is the wavelet index, i.e., 2 for D2.
| D2 (Haar) | D4 | D6 | D8 | D10 | D12 | D14 | D16 | D18 | D20 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 0.6830127 | 0.47046721 | 0.32580343 | 0.22641898 | 0.15774243 | 0.11009943 | 0.07695562 | 0.05385035 | 0.03771716 |
| 1 | 1.1830127 | 1.14111692 | 1.01094572 | 0.85394354 | 0.69950381 | 0.56079128 | 0.44246725 | 0.34483430 | 0.26612218 |
| 0.3169873 | 0.650365 | 0.8922014 | 1.02432694 | 1.06226376 | 1.03114849 | 0.95548615 | 0.85534906 | 0.74557507 | |
| -0.1830127 | -0.19093442 | -0.03957503 | 0.19576696 | 0.44583132 | 0.66437248 | 0.82781653 | 0.92954571 | 0.97362811 | |
| -0.12083221 | -0.26450717 | -0.34265671 | -0.31998660 | -0.20351382 | -0.02238574 | 0.18836955 | 0.39763774 | ||
| 0.0498175 | 0.0436163 | -0.04560113 | -0.18351806 | -0.31683501 | -0.40165863 | -0.41475176 | -0.35333620 | ||
| 0.0465036 | 0.10970265 | 0.13788809 | 0.1008467 | 6.68194092e-4 | -0.13695355 | -0.27710988 | |||
| -0.01498699 | -0.00882680 | 0.03892321 | 0.11400345 | 0.18207636 | 0.21006834 | 0.18012745 | |||
| -0.01779187 | -0.04466375 | -0.05378245 | -0.02456390 | 0.043452675 | 0.13160299 | ||||
| 4.71742793e-3 | 7.83251152e-4 | -0.02343994 | -0.06235021 | -0.09564726 | -0.10096657 | ||||
| 6.75606236e-3 | 0.01774979 | 0.01977216 | 3.54892813e-4 | -0.04165925 | |||||
| -1.52353381e-3 | 6.07514995e-4 | 0.01236884 | 0.03162417 | 0.04696981 | |||||
| -2.54790472e-3 | -6.88771926e-3 | -6.67962023e-3 | 5.10043697e-3 | ||||||
| 5.00226853e-4 | -5.54004549e-4 | -6.05496058e-3 | -0.01517900 | ||||||
| 9.55229711e-4 | 2.61296728e-3 | 1.97332536e-3 | |||||||
| -1.66137261e-4 | 3.25814671e-4 | 2.81768659e-3 | |||||||
| -3.56329759e-4 | -9.69947840e-4 | ||||||||
| 5.5645514e-5 | -1.64709006e-4 | ||||||||
| 1.32354367e-4 | |||||||||
| -1.875841e-5 |
Parts of the construction are also used to derive the biorthogonal Cohen-Daubechies-Feauveau wavelets (CDFs).
Read more about this topic: Daubechies Wavelet
Famous quotes containing the words lowest and/or order:
“Public administrators would get along better if they would restrain the impulse to butt in or be dragged into trouble. They should remain silent until an issue is reduced to its lowest terms, until it boils down into something like a moral issue.”
—Calvin Coolidge (1872–1933)
“Perchance not he but Nature ailed,
The world and not the infant failed.
It was not ripe yet to sustain
A genius of so fine a strain,
Who gazed upon the sun and moon
As if he came unto his own,
And, pregnant with his grander thought,
Brought the old order into doubt.”
—Ralph Waldo Emerson (1803–1882)