Wim Sweldens' Wavelet Cascade Applet

How to Operate the Applet

This Applet graphs scaling functions and wavelets with two parameters even and odd. If you are interested, check the mathematics behind the Applet. The applet shows the graph of the scaling function or wavelet, which is defined on [0,3], on top and the parameter space below. The circle and lines in the parameter space have a specific meaning which you can find in the mathematical background. The red dot indicates the current value of the parameters.

You probably already figured out that:

You can use the slide bar to set the number of iterations of the algorithm. Use it to find a compromise between quality of the graph and speed (6 works best for me).
You can move the red dot in parameter space. In fact, you can even move it outside the paremeter space.
The Mirror button flips the function.
The History radio button allows you to trace the execution of the program. This originally was a bug in the program, but I now added it as an option as one can make pretty pictures with it. How about this as a multi wavelet ;-) Clicking History again clears the screan. By switching history on and changing the level, you can check the convergence speed of the algorithm. The smoother the limit function, the faster the convergence.
The buttons below give you well known functions.
The buttons on the left allow you to switch between wavelet and scaling function.
You only get smooth graphs inside the shaded region. The closer to the upper right corner (1,1), the more fractal the function becomes.
Using the buttons on the bottom left, you can switch between piecewise linear and piecewise constant drawing. Linear is best when drawing smooth functions, while constant is better while outside the region of continuity. With constant drawing you can see how the function almost falls into two pieces around the point (1,1). (Try it with the symmetric constraint on.)
You can contrain the parameter space using the buttons on the right. See if you can converge to the Daubechies by alternating the orthogonality and order 2 constraints.

Some notes on the implementation:

The Applet uses double buffering to avoid flicker both for the graph and the parameter space.
The Applet is fully resizable, so if you would like a larger canvas, simply adjust the HEIGHT and WIDTH of the APPLET tag.
The Applet does not rely on garbadge collection and will never increase memory after initialization.
In order to increase speed while dragging the mouse, only every other mouse position is read.

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

[ Wavelet Cascade Applet | Mathematical background | My home page ]