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Scaling Function Order of Approximation

Module by: Rebecca Willett. E-mail the author

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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.

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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.

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