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   -> Volume 6, Issue 2

Preprint: Preprints on shift-invariant decompositions
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Cohen Israel (

PostPosted: Thu Jan 30, 1997 11:37 am    
Subject: Preprint: Preprints on shift-invariant decompositions
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#4 Preprint: Preprints on shift-invariant decompositions

The following two preprints are available via www at:

Title: Orthonormal Shift-Invariant Adaptive Local Trigonometric

Authors: Israel Cohen, Shalom Raz and David Malah

Abstract: In this paper, an extended library of smooth local
trigonometric bases is defined, and an appropriate fast ``best-basis"
search algorithm is introduced. When compared with the standard local
cosine decomposition (LCD), the proposed algorithm is advantageous in
three respects. First, it leads to a best-basis expansion that is
shift-invariant. Second, the resulting representation is characterized
by a lower information cost. Third, the polarity of the folding
operator is adapted to the parity properties of the segmented signal
at the end-points. The shift-invariance stems from an adaptive relative
shift of expansions in distinct resolution levels. We show that at any
resolution level $ell$ it suffices to examine and select one of two
relative shift options --- a zero shift or a $2^{-ell-1}$ shift.
A variable folding operator, whose polarity is locally adapted to the
parity properties of the signal, further enhances the representation.
The computational complexity is manageable and comparable to that of
the LCD.

Note: To appear in Signal Processing, 57(1)


Title: Orthonormal Shift-Invariant Wavelet Packet Decomposition and

Authors: Israel Cohen, Shalom Raz and David Malah

Abstract: In this work, a shifted wavelet packet (SWP) library,
containing all the time shifted wavelet packet bases, is defined. A
corresponding shift-invariant wavelet packet decomposition (SIWPD)
search algorithm for a ``best basis" is introduced. The search
algorithm is representable by a binary tree, in which a node
symbolizes an appropriate subspace of the original signal. We prove
that the resultant ``best basis" is orthonormal and the associated
expansion, characterized by the lowest information cost, is
shift-invariant. The shift-invariance stems from an additional degree
of freedom, generated at the decomposition stage and incorporated into
the search algorithm. The added dimension is a relative shift between
a given parent-node and its respective children-nodes. We prove that
for any subspace it suffices to consider one of two alternative
decompositions, made feasible by the SWP library. These decompositions
correspond to a zero shift and a $2^{-ell}$ relative shift where
$ell$ denotes the resolution level. The optimal relative shifts,
which minimize the information cost, are estimated using finite depth
subtrees. By adjusting their depth, the quadratic computational
complexity associated with SIWPD may be controlled at the expense of
the attained information cost down to $O(N log_2 N)$.

Note: To appear in Signal Processing, 57(3)


Israel Cohen
Department of Electrical Engineering
Technion - Israel Institute of Technology
Technion City, Haifa 32000, Israel
Voice: 972 4 897 5033 Fax: 972 4 879 5315
All times are GMT + 1 Hour
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