Coiflets are discrete wavelets designed by Ingrid Daubechies, at the request of Ronald Coifman, to have scaling functions with vanishing moments, and has been used in many applications using Calderón–Zygmund operators.

Theory

Some theorems about Coiflets:

Theorem 1

, the following three equations are equivalent:

Theorem 2

, the following six equations are equivalent:

Theorem 3

For a biorthogonal wavelet possesses a degree L of vanishing moments, then the following two equations are equivalent:

Coiflet coefficients

Both the scaling function (low-pass filter) and the wavelet function (high-pass filter. 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, 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.

References