Mathematical theorems of adaptive wavelet transform
Harold H Szu, Brian A. Telfer · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1994
The computational efficiency of the adaptive wavelet transform (AWT) is due both to the compact support closely matching with signal characteristics, and to a larger redundancy factor of the superposition-mother (s(x), or in short super-mother, created adaptively by a linear superposition of other admissible mother wavelets. We prove that the super-mother always forms a complete basis, but usually associated with a higher redundancy number than its constituent C.O.N. bases. Then, in terms of Daubechies frame redundancy, we prove that the robustness of super-mother in suffering less noise contamination (since noise is everywhere, and a redundant sampling by band-passings can suppress the noise and enhance the signal). Since the continuous function of super- mother has been created with least-mean-squares (LMS) off-line using neural nets and is formulated in discrete approximation in terms of high-pass and low-pass filter bank coefficients, then such a digital subband coding via QMF saves the in-situ computational time of AWT. Moreover, the power of such an adaptive wavelet transform is due to the potential of massive parallel real-time implementation by means of artificial neural networks, where each node is a daughter wavelet similar to a radial basis function using dyadic affine scaling.