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Computing the Scaling Function: The Cascade Algorithm

Module by: Phil Schniter. E-mail the author

Summary: This module shows how to compute the scaling function. It also has a section with a proof for an assumption made for the computation.

Given coefficients hn h n that satisfy the regularity conditions, we can iteratively calculate samples of φt φ t on a fine grid of points t t using the cascade algorithm. Once we have obtained φt φ t , the wavelet scaling equation can be used to construct ψt ψ t .

In this discussion we assume that Hz H z is causal with impulse response length NN. Recall, from our discussion of the regularity conditions, that this implies φt φ t will have compact support on the interval 0 N1 0 N 1 . The cascade algorithm is described below.

  1. Consider the scaling function at integer times t=m0N1 t m 0 N 1 : φm=2 n =0N1hnφ2mn φ m 2 n 0 N 1 h n φ 2 m n Knowing that φt=0 φ t 0 for t 0 N1 t 0 N 1 , the previous equation can be written using an NNxNN matrix. In the case where N=4 N 4 , we have
    ( φ0 φ1 φ2 φ3 )=2( h0000 h2h1h00 0h3h2h1 000h3 )( φ0 φ1 φ2 φ3 ) φ 0 φ 1 φ 2 φ 3 2 h 0 0 0 0 h 2 h 1 h 0 0 0 h 3 h 2 h 1 0 0 0 h 3 φ 0 φ 1 φ 2 φ 3
    (1)
    where H=( h0000 h2h1h00 0h3h2h1 000h3 ) where H h 0 0 0 0 h 2 h 1 h 0 0 0 h 3 h 2 h 1 0 0 0 h 3 The matrix HH is structured as a row-decimated convolution matrix. From the matrix equation above (Equation 1), we see that φ0φ1φ2φ3T φ 0 φ 1 φ 2 φ 3 must be (some scaled version of) the eigenvector of HH corresponding to eigenvalue 2-1 2 -1 . In general, the nonzero values of φn nZ φ n n , i.e., φ0φ1φN1T φ 0 φ 1 φ N 1 , can be calculated by appropriately scaling the eigenvector of the NNxNN row-decimated convolution matrix HH corresponding to the eigenvalue 2-1 2 -1 . It can be shown that this eigenvector must be scaled so that n =0N1φn=1 n 0 N 1 φ n 1 .
  2. Given φn nZ φ n n , we can use the scaling equation to determine φn2 nZ φ n 2 n :
    φm2=2n=0N1hnφmn φ m 2 2 n 0 N 1 h n φ m n
    (2)
    This produces the 2N1 2 N 1 non-zero samples φ0φ1/2φ1φ3/2φN1 φ 0 φ 12 φ 1 φ 32 φ N 1 .
  3. Given φn2 nZ φ n 2 n , the scaling equation can be used to find φn4 nZ φ n 4 n :
    φm4=2 n =0N1hnφm2n=2pp evenhp2φmp2=2pp h 2 p φ 1 2 mp φ m 4 2 n 0 N 1 h n φ m 2 n 2 p p even h p 2 φ m p 2 2 p p h 2 p φ 1 2 m p
    (3)
    where h 2 p h 2 p denotes the impulse response of Hz2 H z 2 , i.e., a 2-upsampled version of hn h n , and where φ 1 2 m=φm2 φ 1 2 m φ m 2 . Note that φn4 nZ φ n 4 n is the result of convolving h 2 n h 2 n with φ 1 2 n φ 1 2 n .
  4. Given φn4 nZ φ n 4 n , another convolution yields φn8 nZ φ n 8 n :
    φm8=2 n =0N1hnφm4n=2pp h 4 p φ 1 4 mp φ m 8 2 n 0 N 1 h n φ m 4 n 2 p p h 4 p φ 1 4 m p
    (4)
    where h 4 n h 4 n is a 4-upsampled version of hn h n and where φ 1 4 m=φm4 φ 1 4 m φ m 4 .
  5. At the th th stage, φn2 φ n 2 is calculated by convolving the result of the 1 th 1 th stage with a 21 2 1 -upsampled version of hn h n :
    φ 1 2 m=2pp h 2 1 p φ 1 2 1 mp φ 1 2 m 2 p p h 2 1 p φ 1 2 1 m p
    (5)
For 10 10 , this gives a very good approximation of φt φ t . At this point, you could verify the key properties of φt φ t , such as orthonormality and the satisfaction of the scaling equation.

In Figure 1 we show steps 1 through 5 of the cascade algorithm, as well as step 10, using Daubechies' db2 coefficients (for which N=4 N 4 ).

Figure 1
Figure 1 (daub_cascade.png)

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