![Wavelet Coif1](https://upload.wikimedia.org/wikipedia/commons/d/da/Wavelet_Coif1.svg)
Coiflets are discrete
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
s designed by
Ingrid Daubechies
Baroness Ingrid Daubechies ( ; ; born 17 August 1954) is a Belgian physicist and mathematician. She is best known for her work with wavelets in image compression.
Daubechies is recognized for her study of the mathematical methods that enhance i ...
, at the request of
Ronald Coifman
Ronald Raphael Coifman is the Sterling professor of Mathematics at Yale University. Coifman earned a doctorate from the University of Geneva in 1965, supervised by Jovan Karamata.
Coifman is a member of the American Academy of Arts and Sciences, ...
, to have scaling functions with vanishing
moment
Moment or Moments may refer to:
* Present time
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* The Moments, American R&B vocal group Albums
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s. 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. 141–183][Ingrid 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
Wavelet#Scaling function, 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 = ).
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.
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.
[{{cite web, title=coifwavf, url=http://www.mathworks.com/help/wavelet/ref/coifwavf.html, website=www.mathworks.com/, accessdate=22 January 2015]
References
Orthogonal wavelets
Wavelets