Daubechies wavelet

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Named after Ingrid Daubechies, the Daubechies wavelets are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support. With each wavelet type of this class, there is a scaling function (also called father wavelet) which generates an orthogonal multiresolution analysis.

Contents 1 Properties 2 Construction 3 Implementation 3.1 Transform, D4 3.2 Inverse transform, D4 4 See also 5 References 6 External links

[edit] Properties

In general the Daubechies wavelets are chosen to have the highest number A of vanishing moments, (this does not imply the best smoothness) for given support width N=2A, and among the 2^A-1 possible solutions the one is chosen whose scaling filter has extremal phase. The wavelet transform is also easy to put into practice using the fast wavelet transform. Daubechies wavelets are widely used in solving a broad range of problems, e.g. self-similarity properties of a signal or fractal problem, signal discontinuities, etc.

scaling and wavelet functions
amplitudes of the frequency spectrum

Daubechies orthogonal wavelets D2-D20 (even index numbers only) are commonly used. The index number refers to the number N of coefficients. Each wavelet has a number of zero moments or vanishing moments equal to half the number of coefficients. For example D2 (the Haar wavelet) has one vanishing moments, D4 has two moments, etc. A vanishing moment refers to the wavelets ability to represent polynomial behaviour or information in a signal. For example, D2, with one moment, easily encodes polynomials of one coefficient, ie. constant signal components. D4 encodes polynomials of two coefficients, ie constant and linear signal components, D6 encodes 3-polynomials, ie constant, linear and quadratic signal components.

[edit] Construction

Both the scaling sequence (Low-Pass Filter) and the wavelet sequence (Band-Pass Filter) as used above must be normalized to have sum equal 2 and sum of squares equal 2. In some applications they must be renormalised to have sum , so that both sequences and all shifts of them by an even number of coefficients are orthonormal to each other. Using the above representation for a scaling sequence with approximation order A,

, with N=2A, p having real coefficients, p(1)=1 and degree(p)=A-1,

one can write the orthogonality condition as

, or equally as (*),

with the Laurent-polynomial generating all symmetric sequences and X( − Z) = 2 − X(Z). Further, P(X) stands for the symmetric Laurent-polynomial P(X(Z)) = p(Z)p(Z ^{− 1}). Since X(e^iw) = 1 − cos(w) and p(e^iw)p(e ^{− iw}) = | p(e^iw) | ², P takes nonnegative values on the segment [0,2].

Equation (*) has one minimal solution for each A, which can be obtained by division in the ring of truncated power series in X,

.

Obviously, this has positive values on (0,2)

The homogeneous equation for (*) is antisymmetric about X=1 and has thus the general solution X^A(X − 1)R((X − 1)²), with R some polynomial with real coefficients. That the sum

P(X) = P_A(X) + X^A(X − 1)R((X − 1)²)

shall be nonnegative on the interval [0,2] translates into a set of linear restrictions on the coefficients of R. The values of P on the interval [0,2] are bounded by some quantity 4^{A − r}, maximizing r results in a linear program with infinitely many inequality conditions.

To solve P(X(Z)) = p(Z)p(Z ^{− 1}) for p one uses a technique called spectral factorization resp. Fejer-Riesz-algorithm. The polynomial P(X) splits into linear factors , N=A+1+2deg(R). Each linear factor represents a Laurent-polynomial that can be factored into two linear factors. One can assign either one of the two linear factors to p(Z), thus one obtains 2^N possible solutions. For extremal phase one chooses the one that has all complex roots of p(Z) inside or on the unit circle and is thus real.

Below are the coefficients for the scaling functions for D2-20. The wavelet coefficients are derived by reversing the order of the scaling function coefficients and then reversing the sign of every second one, (ie. D4 wavelet = {-0.1830127, -0.3169873, 1.1830127, -0.6830127}) Mathematically, this looks like b_k = ( − 1)^ka_{N − 1 − k} where k is the coefficient index, b is a coefficient of the wavelet sequence and a a coefficient of the scaling sequence. N is the wavelet index, ie 2 for D2.

Parts of the construction are also used to derive the biorthogonal Cohen-Daubechies-Feauveau wavelets (CDFs).

[edit] Implementation

Here are two, very concise, examples of implementing a Daubechies wavelet transform in Matlab (in this case, Daubechies 4):

[edit] Transform, D4

s1 = S(1:2:N-1) + sqrt(3)*S(2:2:N);
d1 = S(2:2:N) - sqrt(3)/4*s1 - (sqrt(3)-2)/4*[s1(N/2) s1(1:N/2-1)];
s2 = s1 - [d1(2:N/2) d1(1)];
s = (sqrt(3)-1)/sqrt(2) * s2;
d = (sqrt(3)+1)/sqrt(2) * d1;

[edit] Inverse transform, D4

d1 = d / ((sqrt(3)+1)/sqrt(2));
s2 = s / ((sqrt(3)-1)/sqrt(2));
s1 = s2 + cpv(d1,1);
S(2:2:N) = d1 + sqrt(3)/4*s1 + (sqrt(3)-2)/4*cpv(s1,-1);
S(1:2:N-1) = s1 - sqrt(3)*S(2:2:N);

cpv() refers to the cyclic permutation of a vector, a non-standard Matlab function. (Jensen & la Cour-Harbo, 2001)

[edit] See also

• Fast wavelet transform

[edit] References

• Jensen, la Cour-Harbo (2001). Ripples in Mathematics. Berlin: Springer, 157-160. ISBN 3-540-41662-5.

[edit] External links

• Ingrid Daubechies: Ten Lectures on Wavelets, SIAM 1992,
• Carlos Cabrelli, Ursula Molter: Generalized Self-similarity", Journal of Mathematical Analysis and Applications, 230: 251 - 260, 1999.

• Hardware implementation of wavelets