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   -> Volume 5, Issue 2

Note: Wavelets with binary coefficients
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Gil Strang (

PostPosted: Wed Dec 04, 2002 9:49 am    
Subject: Note: Wavelets with binary coefficients
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Note: Wavelets with binary coefficients

This note is to call attention to wavelet filters in which the
coefficients are BINARY (integers divided by powers of 2). They can be
implemented by adds and shifts, without roundoff error. Symmetry with
biorthogonality is available, and the filters are mostly reversible.
The Haar filter and all spline filters are included, when the transfer
function is a power of (1 + z^-1)/2.

Nguyen is combining the very simple Haar filter with long biorthogonal
filters for lossless compression (often no roundoff in medical
applications). Sweldens discovered and the writer rediscovered pairs
like h_0= [1 0 -8 16 46 16 -8 0 1]/64 f_0 = [-1 0 9 16 9 0 -1]/32.
Factors of f_0 give Daubechies filters but here f_0 itself is used in
synthesis. It is binary and halfband (interpolating), with four zeros
at z = -1. The "FBI" 9/7 has four zeros at -1 also in the 9-tap analysis
filter but its coefficients are nonbinary. In compression the new pair
may be slightly better at edges. A first comparison is in our textbook
Wavelets and Filter Banks (Gilbert Strang and Truong Nguyen, Wellesley-
Cambridge Press 1996: email for Table of Contents
or see ).

Correspondence about applications of binary filters is invited.

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