0.6 Regularity, moments, and wavelet system design (Page 11/13)

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μ (k) = \frac{1}{2 \sqrt{2}} \sum_{ℓ = 1}^{k - 1} (\begin{matrix} k \\ ℓ \end{matrix}) {(- 1)}^{ℓ} μ (ℓ) μ (k - ℓ)

which allows calculating the even-order discrete scaling function moments in terms of the lower odd-order discrete scaling function moments for $k = 2, 4, \dots, 2 K - 2$ . For example:

\begin{matrix} μ (2) & = & \frac{1}{\sqrt{2}} μ^{2} (1) \\ μ (4) & = & \frac{- 1}{2 \sqrt{2}} [8 μ (1) μ (3) - 3 μ^{4} (1)] \\ \dots & \dots \end{matrix}

which can be seen from values in [link] .

Johnson [link] noted from Beylkin [link] and Unser [link] that by using the moments of the autocorrelation function of the scaling function, a relationship of the continuous scaling functionmoments can be derived in the form

\sum_{ℓ = 0}^{k} (\begin{matrix} k \\ ℓ \end{matrix}) {(- 1)}^{k - ℓ} m (ℓ) m (k - ℓ) = 0

where $0 < k < 2 K$ if $K - 1$ wavelet moments are zero. Solving for $m (k)$ in terms of lower order moments gives for $k$ even

m (k) = \frac{- 1}{2} \sum_{ℓ = 1}^{k - 1} (\begin{matrix} k \\ ℓ \end{matrix}) {(- 1)}^{ℓ} m (ℓ) m (k - ℓ)

which allows calculating the even-order scaling function moments in terms of the lower odd-order scaling function moments for $k = 2, 4, \dots, 2 K - 2$ . For example [link] :

\begin{matrix} m (2) & = & m^{2} (1) \\ m (4) & = & 4 m (3) m (1) - 3 m^{4} (1) \\ m (6) & = & 6 m (5) m (1) + 10 m^{2} (3) + 60 m (3) m^{3} (1) + 45 m^{6} (1) \\ m (8) & = & 8 m (7) m (1) + 56 m (5) m (3) - 168 m (5) m^{3} (1) \\ + 2520 m (3) m^{5} (1) - 840 m (3) m^{2} (1) - 1575 m^{8} (1) \\ \dots & \dots \end{matrix}

Coiflet Scaling Function and Wavelet Coefficients plus their Discrete Moments

	Length- $N = 6$ ,	Degree $L = 2$
$n$	$h (n)$	$h_{1} (n)$	$μ (k)$	$μ_{1} (k)$	$k$
-2	-0.07273261951285	0.01565572813546	1.414213	0	0
-1	0.33789766245781	-0.07273261951285	0	0	1
0	0.85257202021226	-0.38486484686420	0	-1.163722	2
1	0.38486484686420	0.85257202021226	-0.375737	-3.866903	3
2	-0.07273261951285	-0.33789766245781	-2.872795	-10.267374	4
3	-0.01565572813546	-0.07273261951285
	Length- $N = 8$ ,	Degree $L = 3$
$n$	$h (n)$	$h_{1} (n)$	$μ (k)$	$μ_{1} (k)$	$k$
-4	0.04687500000000	0.01565572813546	1.414213	0	0
-3	-0.02116013576461	-0.07273261951285	0	0	1
-2	-0.14062500000000	-0.38486484686420	0	0	2
-1	0.43848040729385	1.38486484686420	-2.994111	0.187868	3
0	1.38486484686420	-0.43848040729385	0	11.976447	4
1	0.38486484686420	-0.14062500000000	-45.851020	-43.972332	5
2	-0.07273261951285	0.02116013576461	63.639610	271.348747	6
3	-0.01565572813546	0.04687500000000
	Length- $N = 12$ ,	Degree $L = 4$
$n$	$h (n)$	$h_{1} (n)$	$μ (k)$	$μ_{1} (k)$	$k$
-4	0.016387336463	0.000720549446	1.414213	0	0
-3	-0.041464936781	0.001823208870	0	0	1
-2	-0.067372554722	-0.005611434819	0	0	2
-1	0.386110066823	-0.023680171946	0	0	3
0	0.812723635449	0.059434418646	0	11.18525	4
1	0.417005184423	0.076488599078	-5.911352	175.86964	5
2	-0.076488599078	-0.417005184423	0	1795.33634	6
3	-0.059434418646	-0.812723635449	-586.341304	15230.54650	7
4	0.023680171946	-0.386110066823	3096.310009	117752.68833	8
5	0.005611434819	0.067372554722
6	-0.001823208870	0.041464936781
7	-0.000720549446	-0.016387336463

if the wavelet moments are zero up to $k = K - 1$ . Notice that setting $m (1) = m (3) = 0$ causes $m (2) = m (4) = m (6) = m (8) = 0$ if sufficient wavelet moments are zero. This explains the extra zero moments in [link] . It also shows that the traditional specification of zero scaling function moments is redundant. In [link] $m (8)$ would be zero if more wavelet moments were zero.

Discrete and Continuous Moments for the Coiflet Systems

	$N = 6$ ,	$L = 2$
$k$	$μ (k)$	$μ_{1} (k)$	$m (k)$	$m_{1} (k)$
0	1.4142135623	0	1.0000000000	0
1	0	0	0	0
2	0	-1.1637219122	0	-0.2057189138
3	-0.3757374752	-3.8669032118	-0.0379552166	-0.3417891854
4	-2.8727952940	-10.2673737288	-0.1354248688	-0.4537580992
5	-3.7573747525	-28.0624304008	-0.0857053279	-0.6103378310
	$N = 8$ ,	$L = 3$
$k$	$μ (k)$	$μ_{1} (k)$	$m (k)$	$m_{1} (k)$
0	1.4142135623	0	1.0000000000	0
1	0	0	0	0
2	0	0	0	0
3	-2.9941117777	0.1878687376	-0.3024509630	0.0166054072
4	0	11.9764471108	0	0.5292891854
5	-45.8510203537	-43.9723329775	-1.0458570134	-0.9716604635

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Source: OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6

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