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K-Regularity of Scaling Filter

Module by: Feng Qiao, Rachael Milam

Definition 1: K-Regularity
A unitary 2-band scaling filter is said to be KK-regular if it has a polynomial factor of the form P K z P K z , with Pz=1+z-12 P z 1 z 2 .
H 0 z=1+z-12KQz H 0 z 1 z 2 K Q z (1)
Here a unitary filter is defined as a FIR filter with coefficients hn h n from the basic recursive equation φt=∑nhn2φ2t-n φ t n n h n 2 φ 2 t n satisfying the admissibility conditions and orthogonality conditions, i.e., ∑nhn=2 n n h n 2 and ∑khkhk+2m=δm k k h k h k 2 m δ m

note:

Any unitary filter is at least 1-regular according to the above definition.

question:

In what range can KK take its value?

recall:

The admissibility conditions and orthogonality conditions take N2+1 N 2 1 degrees of freedom out of the total NN degrees of freedom in designing the filter hn h n , where NN is the length of the filter.
Since KK is at least 1, KK-regularity is equivalent to a set of K-1 K 1 linear constraints on the scaling filter, this takes another K-1 K 1 degrees of freedom. So we have 1≤K≤N2 1 K N 2 .

KK-regularity describes the flatness of Hω H ω , the Fourier transform of hn h n , at ω=0 ω 0 and ω=π ω . The flatness at these two locations indicates how well the scaling filter performs as a low-pass filter, and is related to the smoothness of corresponding scaling and wavelet functions. Figure 1 shows the frequency response of Daubechies scaling filters with increasing regularities.

Figure 1: Frequency response of Daubechies scaling filters
Figure 1 (h.png)

It can be proved that putting KK-regularity on scaling filter is equivalent to the following statements:

  1. All moments of the wavelet filters are zero, μ 1 k=0 μ 1 k 0 , for k∈01…K-1 k 0 1 … K 1 .
  2. All moments of the wavelets are zero, m 1 k=0 m 1 k 0 , for k∈01…K-1 k 0 1 … K 1 .
  3. The partial moments of the scaling filter are equal for k∈01…K-1 k 0 1 … K 1 .
  4. The frequency response of the scaling filter has a zero of order KK at ω=π ω .
  5. The k th k th derivative of the magnitude-squared frequency response of the scaling filter is zero at ω=0 ω 0 , for k∈12…2K-1 k 1 2 … 2 K 1 .
  6. All polynomial sequences up to degree K-1 K 1 can be expressed as a linear combination of shifted scaling filters.
  7. All polynomials of degree up to K-1 K 1 can be expressed as a linear combination of shifted scaling functions at any scale.

Statement 1-3 relate the regularity of scaling filter to vanishing wavelet moments and the moments of scaling filter. Statement 4 and 5 describes the flatness of the frequency response of the scaling filter. The last two statements show the power of polynomial representation as a consequence of KK-regularity.

Daubechies filters utilize all the remaining N2-1 N 2 1 degrees of freedom to get maximum number of vanishing wavelet moments, i.e., K=N2 K N 2 for Daubechies filters.

Here is a LabVIEW VI showing the Daubechies scaling and wavelet functions, along with the frequency response of scaling and wavelet filters for k∈23…14 k 2 3 … 14 .

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