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.
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:
The parameter
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
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
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.
Let
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.
If the scaling function
Since the scaling function reproduces polynomials of degree