Custom Design of Wavelets in ( ) RL

The existence in L2(R) of biorthogonal wavelet bases associated with perfect reconstruction 2 channel filter banks comprising finite, real filters is determined by the eigenvalues of two related Lawton matrices. Each Lawton matrix is required to have a simple eigenvalue at one and all remaining eigenvalues must have modulus less than one. This being the case then the wavelet bases are said to be stable. If the filter banks are changed in any way then the eigenvalues of the associated Lawton matrices must be re-calculated to determine the stability of the new wavelet bases. This is a numerically intensive task. In this paper we present a new stability criterion that is far simpler to test. Starting with stable biorthogonal wavelet bases we perturb the associated filter banks, preserving perfect reconstruction and ensuring that the new Lawton matrices continue to have an eigenvalue at one. We show that stability of the new biorthogonal wavelet bases first breaks down, not just when a second eigenvalue attains a modulus of one, but rather when a second eigenvalue actually equals one. This leads to the numerically simpler stability criterion of counting eigenvalues at one in the Lawton matrices. The new criterion is used in conjunction with the lifting scheme to construct a perfect reconstruction filter bank with associated biorthogonal wavelet bases. This provides a general-purpose algorithm for the custom design of filter banks that are guaranteed to be stable.