Skip to content Skip to navigation

Connexions

You are here: Home » Content » Computing the Scaling Function: The Cascade Algorithm

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

Computing the Scaling Function: The Cascade Algorithm

Module by: Phil Schniter

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.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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 0N−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 : φm=2∑n=0N−1hnφ2m−n φ m 2 n 0 N 1 h n φ 2 m n Knowing that φt=0 φ t 0 for t∉0N−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
    φ0φ1φ2φ3=2h0000h2h1h000h3h2h1000h3φ0φ1φ2φ3 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=h0000h2h1h000h3h2h1000h3 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|n∈ℤ} φ n n , i.e., φ0φ1…φN−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−1φn=1 n 0 N 1 φ n 1 .
  2. Given {φn|n∈ℤ} φ n n , we can use the scaling equation to determine {φn2|n∈ℤ} φ n 2 n :
    φm2=2∑n=0N−1hnφm−n φ m 2 2 n 0 N 1 h n φ m n (2)
    This produces the 2N−1 2 N 1 non-zero samples φ0φ1/2φ1φ3/2…φN−1 φ 0 φ 12 φ 1 φ 32 … φ N 1 .
  3. Given {φn2|n∈ℤ} φ n 2 n , the scaling equation can be used to find {φn4|n∈ℤ} φ n 4 n :
    φm4=2∑n=0N−1hnφm2−n=2∑p evenhp2φm−p2=2∑p 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 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|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 {φn4|n∈ℤ} φ n 4 n , another convolution yields {φn8|n∈ℤ} φ n 8 n :
    φm8=2∑n=0N−1hnφm4−n=2∑p 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 hn h n and where φ 1 4 m=φm4 φ 1 4 m φ m 4 .
  5. At the ℓ th ℓ th stage, φn2ℓ 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 hn h n :
    φ 1 2 ℓ m=2∑p 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 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)

Comments, questions, feedback, criticisms?

Send feedback