15 March 1994 Mathematical theorems of adaptive wavelet transform
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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.
© (1994) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
Harold H. Szu, Brian A. Telfer, "Mathematical theorems of adaptive wavelet transform", Proc. SPIE 2242, Wavelet Applications, (15 March 1994); doi: 10.1117/12.170061; https://doi.org/10.1117/12.170061

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