K-Regularity of Scaling Filter 2.0 2003/05/09 2003/05/09 Feng Qiao fqiao@rice.edu Rachael Milam rmilam@rice.edu Charlet Reedstrom charlet@rice.edu Rachael Milam rmilam@rice.edu Feng Qiao fqiao@rice.edu regularity scaling filter K-Regularity A unitary 2-band scaling filter is said to be K-regular if it has a polynomial factor of the form P K z , with P z 1 z 2 . H 0 z 1 z 2 K Q z Here a unitary filter is defined as a FIR filter with coefficients h n from the basic recursive equation φ t n n h n 2 φ 2 t n satisfying the admissibility conditions and orthogonality conditions, i.e., n n h n 2 and k k h k h k 2 m δ m Any unitary filter is at least 1-regular according to the above definition. In what range can K take its value? The admissibility conditions and orthogonality conditions take N 2 1 degrees of freedom out of the total N degrees of freedom in designing the filter h n , where N is the length of the filter. Since K is at least 1, K-regularity is equivalent to a set of K 1 linear constraints on the scaling filter, this takes another K 1 degrees of freedom. So we have 1 K N 2 . K-regularity describes the flatness of H ω , the Fourier transform of h n , at ω 0 and 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. shows the frequency response of Daubechies scaling filters with increasing regularities.

Frequency response of Daubechies scaling filters

It can be proved that putting K-regularity on scaling filter is equivalent to the following statements: All moments of the wavelet filters are zero, μ 1 k 0 , for k 0 1 … K 1 . All moments of the wavelets are zero, m 1 k 0 , for k 0 1 … K 1 . The partial moments of the scaling filter are equal for k 0 1 … K 1 . The frequency response of the scaling filter has a zero of order K at ω . The k th derivative of the magnitude-squared frequency response of the scaling filter is zero at ω 0 , for k 1 2 … 2 K 1 . All polynomial sequences up to degree K 1 can be expressed as a linear combination of shifted scaling filters. All polynomials of degree up to 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 K-regularity. Daubechies filters utilize all the remaining N 2 1 degrees of freedom to get maximum number of vanishing wavelet moments, i.e., 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 2 3 … 14 .