The cascade algorithm is an interesting numerical method for calculating the basic scaling function or wavelets uses an iterative algorithm, which computes wavelet coefficients at one scale from those at another. Because it applies the same operation over and over to the output of the previous application, it is known as the cascade algorithm. Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... A wavelet is a kind of mathematical function used to divide a given function into different frequency components and study each component with a resolution that matches its scale. ...

Successive approximation

The iterative algorithm generate successive approximations to ψ(t) or φ(t) from {h} and {g} filter coefficients. If the algorithm converges to a fixed point, then that fixed point is the basic scaling function or wavelet.

The iterations are defined by

For the k^th iteration, where an initial φ⁽⁰⁾(t) must be given.

The frequency domain estimates of the basic scaling function is given by

and the limit can be viewed as an infinite product in the form

If such a limit exists, the spectrum of the scaling function is

The limit does not depends on the initial shape assume for φ⁽⁰⁾(t). This algorithm converges reliably to φ(t), even if it is discontinuous.

From this scaling function, the wavelet can be generated from

Plots of the function at each iteration is shown in Figure 1.

Successive approximation can also be derived in the frequency domain.

Image:Figure cascade.png

References

• C.S. Burrus, R.A. Gopinath, H. Guo, Introduction to Wavelets and Wavelet Transforms: A Primer, Prentice-Hall, 1988, ISBN 0124896009.
• http://cnx.org/content/m10486/latest/
• http://cm.bell-labs.com/cm/ms/who/wim/cascade/index.html