Coiflet
Encyclopedia
Coiflets are discrete wavelet
s 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.
. 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 (ie. 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, ie 6 for C6.
Wavelet
A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have...
s designed by Ingrid Daubechies
Ingrid Daubechies
Ingrid Daubechies is a Belgian physicist and mathematician. She was between 2004 and 2011 the William R. Kenan Jr. Professor in the mathematics and applied mathematics departments at Princeton University. In January 2011 she moved to Duke University as a Professor in mathematics. She is the first...
, at the request of Ronald Coifman
Ronald Coifman
Ronald Raphael Coifman is the Phillips Professor of Mathematics at Yale University. Coifman graduated from the University of Geneva in 1965.Coifman is a member of the American Academy of Arts and Sciences, the Connecticut Academy of Science and Engineering, and the National Academy of Sciences...
, 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.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 (ie. C6 wavelet = {−0.022140543057, 0.102859456942, 0.544281086116, −1.205718913884, 0.477859456942, 0.102859456942}).Mathematically
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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, ie 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 | ||||