MATHEMATICS OF ADAPTIVE WAVELET TRANSFORMS - RELATING CONTINUOUS WITHDISCRETE-TRANSFORMS

We prove several theorems and construct explicitly the bridge between the continuous and discrete adaptive wavelet transform (AWT). The comp utational efficiency of the AWT is a result of its compact support clo sely matching linearly the signal's time-frequency characteristics, an d is also a result of a larger redundancy factor of the superposition- mother s(x) (super-mother), created adaptively by a linear superpositi on of other admissible mother wavelets. The super-mother always forms a complete basis, but is usually associated with a higher redundancy n umber than its constituent complete orthonormal (CON) bases. The robus tness of super-mother suffers less noise contamination (since noise is everywhere, and a redundant sampling by bandpassings can suppress the noise and enhance the signal). Since the continuous super-mother has been created off-line by AWT (using least-mean-squares neural nets), w e wish to accomplish fast AWT on line. Thus, we formulate AWT in discr ete high-pass (H) and low-pass (L) filter bank coefficients via the qu adrature mirror filter (QMF), a digital subband lossless coding. A lin ear combination of two special cases of the complete biorthogonal norm alized (Cbi-ON) QMF [L(z),H(z),L(dagger)(z),H(dagger)(z)], called alph a-bank and beta-bank, becomes a hybrid aalpha + bbeta-bank (for any re al positive constants a and b) that is still admissible, meaning Cbi-O N and lossless. Finally, the power of AWT is the implementation by mea ns of wavelet chips and neurochips, in which each node is a daughter w avelet similar to a radial basis function using dyadic affine scaling.