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
and
The associated (QMF) wavelet is given by
It is well known that a continuous solution can only exist
in case the refinement coefficients satisfy:
c_{0}
+
c_{2}
=
c_{1}
+
c_{3}
= 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
c_{0} = even,
c_{1} = 1odd,
c_{2} = 1even,
c_{3} = odd.
Several properties of the scaling function immediately follow
from the refinement coefficients:

In case even=odd, the scaling function is symmetric
and the wavelet is antisymmetric.
 By flipping even and odd, the function
flips around x=3/2.

In case even (1even) + odd (1odd) = 0 , the
scaling function and wavelet are orthogonal.
In the (even,odd) plane
this is a circle which goes trough the corners of the unit
square.

In case even + odd = 1/2, the order of the scaling
functions is two, i.e., the scaling function and its translates
can reproduce linears. In the orthogonal case, the wavelet
then has two vanishing moments.

In case even or odd is zero, then the support of
the function is [0,2] or [1,3] respectively. The function then
also is interpolating in the sense that it takes the value 1 at
x=1 or x=2 respectively and zero at the other
integers.

In case 0 < even < 1 and
0 < odd < 1, i.e., the red dot is in the unit
square, the cascade algorithm will only use convex combinations.
As a result the scaling function will only take on values
between 0 and 1, while the wavelet only takes values between 1
and 1.

A lot of research has been done on how the smoothness of the
scaling function depends on the refinements coefficients. In
fact the cascade algorithm is an instance of a 1D subdivision
algorithm. For a fixed length, the smoothest solution is always
the Bspline. Colella and Heil study the four coefficient case
in detail and are able to outline the area in the
(even,odd) plane where the scaling function is
continuous. This area is draw shaded in light gray. For more
information, check D. Colella and C. Heil, Characterizations of
scaling functions: Continuous Solutions, SIAM J. Matrix
Anal. Appl., vol. 15, pp. 496518, 1994. The idea for
this applet is partly inspired by their work. Special
thanks to Chris Heil for providing the polygon outlining the
area of continuity.
Several wellknown functions are part of this class.

If even = 1 and odd = 0, then the scaling function
is a box function which is one on the interval [0,1] and zero
elsewhere. Box functions on [1,2] and [2,3] have respectively
even = odd = 0 and even = 0 and odd = 1.
The box functions are orthogonal and lie on 3 of the corners
of the unit square. These are the only scaling functions that
are orthogonal and symmetric.
A box scaling function leads to a Haar wavelet.

If even = 1/2 and odd = 0 the refinement coeffients
are [1/2 1 1/2 0], and the scaling function is a hat function (linear
Bspline).

If even = odd = 1/4
the refinement coefficients are [1/4 3/4 3/4 1/4] and the
scaling function
is a quadratic Bspline. The quadratic Bspline has order 3.

If even =
and odd = 1/2  even, you get
the Daubechies D4 orthogonal scaling function and wavelet. This
and its flipped version are the only solutions which are orthogonal
and have order 2.

If even = odd = 1, the cascade algorithm diverges.
However, in a weak (distributional) sense, it converges
and the scaling function is 1/3 times the indicator function on [0,3].
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