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
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