Coiflet
![](../I/m/Wavelet_Coif1.svg.png)
Coiflets are discrete wavelets designed by Ingrid Daubechies, at the request of Ronald Coifman, to have scaling functions with vanishing moments. The wavelet is near symmetric, their wavelet functions have vanishing moments and scaling functions , and has been used in many applications using Calderón-Zygmund Operators.[1][2]
Theory
Some theories about Coiflet :[3]
Theorem 1
For a wavelet system {}, the following three equations are equivalent:
and similar equivalence holds between and
Theorem 2
For a wavelet system {}, the following six equations are equivalent:
and similar equivalence holds between and
Theorem 3
For a biorthogonal wavelet system {}, if either or
possesses a degree L of vanishing moments, then the following two equations
are equivalent:
for any such that
Coiflet coefficients
Both the scaling function (low-pass filter) and the wavelet function (High-Pass Filter) must be normalised by a factor . Below are the coefficients for the scaling functions for C6-30. 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. C6 wavelet = {−0.022140543057, 0.102859456942, 0.544281086116, −1.205718913884, 0.477859456942, 0.102859456942}).
Mathematically, this looks like where k is the coefficient index, B is a wavelet coefficient and C a scaling function coefficient. N is the wavelet index, i.e. 6 for C6.
k | C6 | C12 | C18 | C24 | C30 |
---|---|---|---|---|---|
-10 | -0.0002999290456692 | ||||
-9 | 0.0005071055047161 | ||||
-8 | 0.0012619224228619 | 0.0030805734519904 | |||
-7 | -0.0023044502875399 | -0.0058821563280714 | |||
-6 | -0.0053648373418441 | -0.0103890503269406 | -0.0143282246988201 | ||
-5 | 0.0110062534156628 | 0.0227249229665297 | 0.0331043666129858 | ||
-4 | 0.0231751934774337 | 0.0331671209583407 | 0.0377344771391261 | 0.0398380343959686 | |
-3 | -0.0586402759669371 | -0.0930155289574539 | -0.1149284838038540 | -0.1299967565094460 | |
-2 | -0.1028594569415370 | -0.0952791806220162 | -0.0864415271204239 | -0.0793053059248983 | -0.0736051069489375 |
-1 | 0.4778594569415370 | 0.5460420930695330 | 0.5730066705472950 | 0.5873348100322010 | 0.5961918029174380 |
0 | 1.2057189138830700 | 1.1493647877137300 | 1.1225705137406600 | 1.1062529100791000 | 1.0950165427080700 |
1 | 0.5442810861169260 | 0.5897343873912380 | 0.6059671435456480 | 0.6143146193357710 | 0.6194005181568410 |
2 | -0.1028594569415370 | -0.1081712141834230 | -0.1015402815097780 | -0.0942254750477914 | -0.0877346296564723 |
3 | -0.0221405430584631 | -0.0840529609215432 | -0.1163925015231710 | -0.1360762293560410 | -0.1492888402656790 |
4 | 0.0334888203265590 | 0.0488681886423339 | 0.0556272739169390 | 0.0583893855505615 | |
5 | 0.0079357672259240 | 0.0224584819240757 | 0.0354716628454062 | 0.0462091445541337 | |
6 | -0.0025784067122813 | -0.0127392020220977 | -0.0215126323101745 | -0.0279425853727641 | |
7 | -0.0010190107982153 | -0.0036409178311325 | -0.0080020216899011 | -0.0129534995030117 | |
8 | 0.0015804102019152 | 0.0053053298270610 | 0.0095622335982613 | ||
9 | 0.0006593303475864 | 0.0017911878553906 | 0.0034387669687710 | ||
10 | -0.0001003855491065 | -0.0008330003901883 | -0.0023498958688271 | ||
11 | -0.0000489314685106 | -0.0003676592334273 | -0.0009016444801393 | ||
12 | 0.0000881604532320 | 0.0004268915950172 | |||
13 | 0.0000441656938246 | 0.0001984938227975 | |||
14 | -0.0000046098383254 | -0.0000582936877724 | |||
15 | -0.0000025243583600 | -0.0000300806359640 | |||
16 | 0.0000052336193200 | ||||
17 | 0.0000029150058427 | ||||
18 | -0.0000002296399300 | ||||
19 | -0.0000001358212135 | ||||
Matlab function
F = coifwavf(W) returns the scaling filter associated with the Coiflet wavelet specified by the string W where W = 'coifN'. Possible values for N are 1, 2, 3, 4, or 5. [4]
References
- ↑ G. Beylkin, R. Coifman, and V. Rokhlin (1991),Fast wavelet transforms and numerical algorithms, Comm. Pure Appl. Math., 44, pp. 141-183
- ↑ Ingrid Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992, ISBN 0-89871-274-2
- ↑ "COIFLET-TYPE WAVELETS: THEORY, DESIGN, AND APPLICATIONS" (PDF).
- ↑ "coifwavf". http://www.mathworks.com/. Retrieved 22 January 2015. External link in
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