Daubechies wavelets are a family of orthogonal wavelets named after Belgian physicist and mathematician Ingrid Daubechies. They are used in discrete wavelet transform.
Definition
Scale function coefficients (low pass filter in orthogonal filter banks) must satisfy following conditions ( is length of filter).
Normalization
or (then coefficients must be divided by factor of ) which implies or (then coefficients must be divided by factor of )
Orthogonality
for
Vanishing moments
for
There is more than one solution (except case of ). However, it is necessary to distinguish between low pass and high pass filter.
Wavelets are denoted like Dx, where x is either number of coefficients () or number of vanishing moments (). First case of notation (number of coefficients) is more recent and more frequented (e.g. D8 is wavelet with 8 coefficients).
Example
MATLAB code for enumeration of wavelet with 4 coefficients (denoted as D4). t = solve( 'h0*h0 + h1*h1 + h2*h2 + h3*h3 = 1', % normalization 'h2*h0 + h3*h1 = 0', % orthogonality '+(0^0)*h0 -(1^0)*h1 +(2^0)*h2 -(3^0)*h3 = 0', % zero '+(0^1)*h0 -(1^1)*h1 +(2^1)*h2 -(3^1)*h3 = 0' % and first vanishing moments );
Solutions (low pass filters only):