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Wavelet and Scaling Function Moments

Module by: Ivan Selesnick. E-mail the author

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Define mk=∫tkφtdt m k t t k φ t μk=∑nnkhn μ k n n k h n Then m1=∫tφtdt m 1 t t φ t μ1=∑nnhn μ 1 n n h n It turns out we can relate m1 m 1 to the moments of hn h n by using the dilation equation.

m1=∫tφtdt=∫t2∑nhnφ2t−ndt=2∑nhn∫tφ2t−ndt=2∑nhn12∫α2+n2φαdα=24∑nhn∫αφαdα+24∑nhn∫nφαdα=24∑nhnm1+24∑nnhnm0=24m1∑nhn+24m0∑nnhn=24m1μ0+24m0μ1 m 1 t t φ t t t 2 n h n φ 2 t n 2 n h n t t φ 2 t n 2 n h n 1 2 α α 2 n 2 φ α 2 4 n h n α α φ α 2 4 n h n α n φ α 2 4 n h n m 1 2 4 n n h n m 0 2 4 m 1 n h n 2 4 m 0 n n h n 2 4 m 1 μ 0 2 4 m 0 μ 1 (1)
So we have m1=24m1μ0+24m0μ1 m 1 2 4 m 1 μ 0 2 4 m 0 μ 1 Solving for m1 m 1 gives m1=2m0μ14−2μ0 m 1 2 m 0 μ 1 4 2 μ 0 A valid scaling filter hn h n must have μ0=2 μ 0 2 . Using this, we get m1=m0μ12 m 1 m 0 μ 1 2 With the normalization m0=∫φtdt=1 m 0 t φ t 1 , we get
m1=μ12 m 1 μ 1 2 (2)

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