Wavelet Cascade Applet: Mathematical Background

Cascading (or subdivision) is one of the standard methods to build wavelets and scaling functions. With this Applet you can graph all scaling functions and wavelets that satisfy refinement relations with four coefficients. The scaling function is defined on [0,3] and satisfies
phi(x) = c0 phi(2x) + c1 phi(2x-1) + c2 phi(2x-2) + c3 phi(2x-3).
\int_0^3 phi(x) dx = 1.
The associated (QMF) wavelet is given by
psi(x) = c3 phi(2x) - c2 phi(2x-1) + c1 phi(2x-2) - c0 phi(2x-3).
It is well known that a continuous solution can only exist in case the refinement coefficients satisfy:
c0 + c2 = c1 + c3 = 1.
This leaves us with two degrees of freedom. We choose them to be the first coefficient even and the last one odd. We then have
c0 = even, c1 = 1-odd, c2 = 1-even, c3 = odd.

Several properties of the scaling function immediately follow from the refinement coefficients:

Several well-known functions are part of this class.

Please feel free to contact me if you have questions or comments.

Note: the rendering of the mathematical equations on this page is done with Robert Miner's WebEq.

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Last modified: Thu Oct 9 11:26:54 EDT 1997