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Unser-Blu Scaling Function / Spline Factorization Theorem

Module by: Rebecca Willett. E-mail the author

Summary: In this module, the reader will be introduced to the scaling function spline factorization theorm by M. Unser and T. Blu. This factorization allows many key wavelet properties to be derived in a relatively straightforward manner.

Factorization Theorem

The central theorem of the paper Wavelet Theory Demystified by M. Unser and T. Blu is that every scaling function can be factorized into two components: a B-spline component and a distribution. The following theorem is a precise statement of this concept:

Theorem 1: B-spline Factorization of scaling functions

φφ is a valid scaling function of order γγ if and only iff its Fourier transform φ ^ φ ^ can be factorized as φ ^ ⁢ω= β ^ + γ - 1 ⁢ω⁢ φ ^ 0 ⁢ω φ ^ ω β ^ + γ - 1 ω φ ^ 0 ω for φ 0 ∈ S ′ φ 0 S ′ and ∫ φ 0 ⁢xdx=1 x φ 0 x 1 . In time domain, this means φ⁢x= β ^ + γ - 1 * φ 0 ⁢x φ x β ^ + γ - 1 φ 0 x .

The parameter γγ is the order of approximation of the scaling function φφ. For the purposes of this module, the condition that φ 0 ∈ S ′ φ 0 S ′ essentially means that φ0φ0 is Lebesgue integrable and has a Fourier transform.

This factorization is significant because it makes several desirable properties of wavelet analysis immediately transparent. For example, this factorization makes it easy to prove that φφ can reproduce polynomials of order up to γγ and has γγ vanishing moments. This can also be used to derive that wavelets form an unconditional basis for Besov spaces.

Reproduction of Polynomials

Recall from spline theory that fractional B-splines reproduce polynomials of order less than or equal to the ceiling of the spline order. This means that any polynomial of a certain order can be expressed as a linear combination of B-splines; that is, the B-splines form a basis for polynomials. Specifically, for some pp which is a nonnegative integer no greater than α+1α1, we have

∑kk∈Zkp⁢ β + α ⁢x−k=xp+ a p , 1 ⁢xp−1+…+ a p , p - 1 ⁢x+ a p , p k k k p β + α x k x p a p , 1 x p 1 … a p , p - 1 x a p , p
(1)
The collection of all these polynomials for all pp no greater than α+1α1 forms a basis for polynomials of order no greater than α+1α1.

This combined with the Unser-Blu Scaling Function/Spline Factorization Theorem leads to a straighforward proof of the fact that scaling functions reproduce polynomials up to a degree proportional to their smoothness.

Proposition 1

Let φ 0 φ 0 be any distribution such that ∫ φ 0 ⁢xdx=1 x φ 0 x 1 adn ∫xi⁢ φ 0 ⁢xdx<∞ x x i φ 0 x , for all i∈1…n i 1 … n , where n=⌈α⌉ n α . Then φ⁢x= β + α * φ 0 ⁢x φ x β + α φ 0 x reproduces polynomials of degree lesser or equal to n=⌈α⌉ n α .

Proof

  1. First note that
    ∑kk∈Zkp⁢φ⁢x−k=∑kk∈Zkp⁢ β + γ - 1 * φ 0 ⁢(x−k)= φ 0 *∑k=0p a p , k ⁢xp−k k k k p φ x k k k k p β + γ - 1 φ 0 x k φ 0 k 0 p a p , k x p k
    (2)
    because B-splines have been shown to reproduce polynomials. This can be further simplified by studying the convolution of φ 0 φ 0 with a polynomial.
  2. Now we have
    φ 0 ⁢x*xp=∫−∞∞x−up⁢ φ 0 ⁢ud u=∑k=0pc⁢pk⁢xp−k⁢−1k⁢∫−∞∞uk⁢ φ 0 ⁢ud u=xp+ b 1 ⁢xp−1+…+ b p φ 0 x x p u x up φ 0 u k 0 p c p k x p k 1 k u uk φ 0 u x p b 1 x p 1 … b p
    (3)
    where bk=(c⁢pk⁢−1k⁢∫−∞∞uk⁢ φ 0 ⁢ud u) bk c p k 1 k u uk φ 0 u

Vanishing Moments

Recall from wavelet theory that the number of vanishing moments that a wavelet has dictates whether the inner product of that wavelet with a polynomial of degree less than the number of vanishing moments has value. If a wavelet has three vanishing moments, then the inner product of that wavelet with a quadratic portion of a signal will be zero. It is this feature that makes wavelets form such sparse representations of piecewise smooth signals. The theory of vanishing moments also allows the characterizations of function singularities based on the decay of wavelet coefficients across wavelet scales.

Using the spline factorization theorem discussed in this module, it is easy to see how wavelet regularity and order of approximation is directly related to the number of vanishing wavelet moments.

Proposition 2

If the scaling function φφ reproduces the polynomials of degree nn, then the analysis wavelet ψψ has n+1n1 vanishing moments.

Proof

Since the scaling function reproduces polynomials of degree nn, and the analysis wavelet is orthogonal to the scaling functions, then no polynomial of degree up to nn can be contained in the space spanned by the analysis wavelet functions, and hence the analysis wavelet has ψψ has n+1n1 vanishing moments.

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