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

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

  1. Consider the scaling function at integer times t=m∈0…N−1 t m 0 … N 1 : φ⁢m=2⁢∑ n =0N−1h⁢n⁢φ⁢2⁢m−n φ m 2 n 0 N 1 h n φ 2 m n Knowing that φ⁢t=0 φ t 0 for t∉ 0 N−1 t 0 N 1 , the previous equation can be written using an NNxNN matrix. In the case where N=4 N 4 , we have
    ( φ⁢0 φ⁢1 φ⁢2 φ⁢3 )=2⁢( h⁢0000 h⁢2h⁢1h⁢00 0h⁢3h⁢2h⁢1 000h⁢3 )⁢( φ⁢0 φ⁢1 φ⁢2 φ⁢3 ) 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=( h⁢0000 h⁢2h⁢1h⁢00 0h⁢3h⁢2h⁢1 000h⁢3 ) 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 φ⁢0φ⁢1φ⁢2φ⁢3T 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 φ⁢n n∈Z φ n n , i.e., φ⁢0φ⁢1…φ⁢N−1T 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 =0N−1φ⁢n=1 n 0 N 1 φ n 1 .
  2. Given φ⁢n n∈Z φ n n , we can use the scaling equation to determine φ⁢n2 n∈Z φ n 2 n :
    φ⁢m2=2⁢∑n=0N−1h⁢n⁢φ⁢m−n φ m 2 2 n 0 N 1 h n φ m n
    (2)
    This produces the 2⁢N−1 2 N 1 non-zero samples φ⁢0φ⁢1/2φ⁢1φ⁢3/2…φ⁢N−1 φ 0 φ 12 φ 1 φ 32 … φ N 1 .
  3. Given φ⁢n2 n∈Z φ n 2 n , the scaling equation can be used to find φ⁢n4 n∈Z φ n 4 n :
    φ⁢m4=2⁢∑ n =0N−1h⁢n⁢φ⁢m2−n=2⁢∑pp evenh⁢p2⁢φ⁢m−p2=2⁢∑pp h ↑ 2 ⁢p⁢ φ 1 2 ⁢m−p 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 H⁢z2 H z 2 , i.e., a 2-upsampled version of h⁢n h n , and where φ 1 2 ⁢m=φ⁢m2 φ 1 2 m φ m 2 . Note that φ⁢n4 n∈Z 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 φ⁢n4 n∈Z φ n 4 n , another convolution yields φ⁢n8 n∈Z φ n 8 n :
    φ⁢m8=2⁢∑ n =0N−1h⁢n⁢φ⁢m4−n=2⁢∑pp h ↑ 4 ⁢p⁢ φ 1 4 ⁢m−p 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 h⁢n h n and where φ 1 4 ⁢m=φ⁢m4 φ 1 4 m φ m 4 .
  5. At the ℓ th ℓ th stage, φ⁢n2ℓ φ n 2 ℓ is calculated by convolving the result of the ℓ − 1 th ℓ − 1 th stage with a 2ℓ−1 2 ℓ 1 -upsampled version of h⁢n h n :
    φ 1 2 ℓ ⁢m=2⁢∑pp h ↑ 2 ℓ − 1 ⁢p⁢ φ 1 2 ℓ − 1 ⁢m−p φ 1 2 ℓ m 2 p p h ↑ 2 ℓ − 1 p φ 1 2 ℓ − 1 m p
    (5)
For ℓ≃10 ℓ 10 , this gives a very good approximation of φ⁢t φ t . At this point, you could verify the key properties of φ⁢t 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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