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:
“Perhaps it is the lowest of the qualities of an orator, but it is, on so many occasions, of chief importance,—a certain robust and radiant physical health; or—shall I say?—great volumes of animal heat.”
—Ralph Waldo Emerson (1803–1882)
“If there hadn’t been women we’d still be squatting in a cave eating raw meat, because we made civilization in order to impress our girl friends. And they tolerated it and let us go ahead and play with our toys.”
—Orson Welles (1915–1984)