Skip to content Skip to navigation

Connexions

You are here: Home » Content » Scaling Function Order of Approximation

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.

    • External bookmarks
  • E-mail the author

Recently Viewed

This feature requires Javascript to be enabled.

Scaling Function Order of Approximation

Module by: Rebecca Willett

Summary: Wavelets can be characterized in terms of how well scaling functions can approximate functions in certain smoothness classes. In this module, this relationship is defined.

The order of approximation of a scaling function in a wavelet system determines how well the scaling fuction can appoximate smooth functions. (The smoothness is determined by the Sobolev space in which the function lies.) Having a scaling function with a high order of approximation implies that the wavelet decomposition of a smooth function will have very few wavelet coefficients with high energy. Conventional wisdom tells us that good appoximation leads to good denoising and good compression.

definition 1

A scaling function φφ has order of approximation γγ if for all f∈ W 2 γ f W 2 γ

∥f- P a f∥≤Caγ∥∂f∥ f P a f C a γ f (1)
where P a f P a f is the projection of ff onto the space spanned by all wavelets scaled to have support aa for 0≤a0a.

Example

This example highlights the tradeoff between order of approximation and wavelet filter length.

Wavelets and Their Orders of Approximation

  • Haar: γ=1 γ 1 . This is barely enough for the error to decay as a→∞ a .
  • Daub9/7 Biorthogonal Wavelets:γ=4 γ 4 .
  • Cubic Spline Wavelets:γ=4 γ 4 .

The desired order of approximation translates into filter design contraints when designing a wavelet system, as specified by the following theorem:

theorem 1

A valid scaling function φφ has a γth γ th order of approximation if and only if its refinement filter Hz H z can be factorized as Hz=1+z-12γQz H z 1 z 2 γ Q z

This theorem is particularly enlightening when considered with respect to the Unser-Blu Scaling Function/B-Spline Factorization Theorem.

Comments, questions, feedback, criticisms?

Send feedback