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.G. Beylkin, R. Coifman, and V. Rokhlin (1991),Fast wavelet transforms and numerical algorithms, Comm. Pure Appl. Math., 44, pp. 141–183Ingrid Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992,
Theory Some theorems about Coiflets:
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.
+Coiflets coefficients (normalized to have sum 2)kC6C12C18C24C30 −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.
References