Mathematics of adaptive wavelet transforms: relating continuous with   discrete transforms

Harold H. Szu; Brian A. Telfer

doi:10.1117/12.173205

1 July 1994 Mathematics of adaptive wavelet transforms: relating continuous with discrete transforms

Harold H. Szu, Brian A. Telfer

Author Affiliations +

Optical Engineering, Vol. 33, Issue 7, (July 1994). https://doi.org/10.1117/12.173205

Abstract

We prove several theorems and construct explicitly the bridge between the continuous and discrete adaptive wavelet transform (AWT). The computational efficiency of the AWT is a result of its compact support closely matching linearly the signal's time-frequency characteristics, and is also a result of a larger redundancy factor of the superposition-mother s(x) (super-mother), created adaptively by a linear superposition of other admissible mother wavelets. The super-mother always forms a complete basis, but is usually associated with a higher redundancy number than its constituent complete orthonormal (CON) bases. The robustness 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), we wish to accomplish fast AWT on line. Thus, we formulate AWT in discrete high-pass (H) and low-pass (L) filter bank coefficients via the quadrature mirror filter (QMF), a digital subband lossless coding. A linear combination of two special cases of the complete biorthogonal normalized (Cbi-ON) QMF [L(z),H(z),L⁺(z),H⁺(z)], called α-bank and β-bank, becomes a hybrid aα + bβ-bank (for any real positive constants a and b) that is still admissible, meaning Cbi-ON and lossless. Finally, the power of AWT is the implementation by means of wavelet chips and neurochips, in which each node is a daughter wavelet similar to a radial basis function using dyadic affine scaling.

Citation Download Citation

Harold H. Szu and Brian A. Telfer "Mathematics of adaptive wavelet transforms: relating continuous with discrete transforms," Optical Engineering 33(7), (1 July 1994). https://doi.org/10.1117/12.173205

Published: 1 July 1994

ACCESS THE FULL ARTICLE

INSTITUTIONAL
Select your institution to access the SPIE Digital Library.

SELECT YOUR INSTITUTION

PERSONAL
Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.

PERSONAL SIGN IN

No SPIE Account? Create one

PURCHASE THIS CONTENT

SUBSCRIBE TO DIGITAL LIBRARY

50 downloads per 1-year subscription

Members: $195

Non-members: $335 ADD TO CART

25 downloads per 1 - year subscription

Members: $145

Non-members: $250 ADD TO CART

PURCHASE SINGLE ARTICLE

Includes PDF, HTML & Video, when available