Daubechies wavelet

Scaling and wavelet function of continuous D4 wavelet

Daubechies wavelets are a family of orthogonal wavelets named after Belgian physicist and mathematician Ingrid Daubechies. They are used in discrete wavelet transform.

Definition[edit]

Scale function coefficients (low pass filter in orthogonal filter banks) must satisfy following conditions ( $N$ is length of filter).

Normalization: $\sum _{n=0}^{N-1}h_{0}[n]={\sqrt {2}}$ or $\sum _{n=0}^{N-1}h_{0}[n]=2$ (then coefficients must be divided by factor of ${\sqrt {2}}$ )

which implies

\sum _{n=0}^{N-1}(h_{0}[n])^{2}=1

or

\sum _{n=0}^{N-1}(h_{0}[n])^{2}=2

(then coefficients must be divided by factor of

{\sqrt {2}}

)

Orthogonality: $\sum _{n=0}^{N-1}h_{0}[n]h_{0}[n-2k]=0$ for $k\not =0$

Vanishing moments: $\sum _{n=0}^{N-1}(-1)^{n}h_{0}[n]n^{m}=0$ for $0\leq m<N/2$

There is more than one solution (except case of $N=2$ ). 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 ( $N$ ) or number of vanishing moments ( $N/2$ ). First case of notation (number of coefficients) is more recent and more frequented (e.g. D8 is wavelet with 8 coefficients).

Example[edit]

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):

h0	h1	h2	h3
-0.129409522551260	0.224143868042014	0.836516303737808	0.482962913144534
0.482962913144534	0.836516303737808	0.224143868042014	-0.129409522551260

Related pages[edit]

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