qccwavwaveletdwt1d.3

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.TH QCCWAVWAVELETDWT1D 3 "QCCPACK" ""
.SH NAME
QccWAVWaveletDWT1D, QccWAVWaveletInverseDWT1D \- 
discrete wavelet transform and inverse transform for a 1D signal
.SH SYNOPSIS
.B #include "libQccPack.h"
.sp
.BI "int QccWAVWaveletDWT1D(QccVector " signal ", int " signal_length ", int " signal_origin ", int " subsample_pattern ", int " num_scales ", const QccWAVWavelet *" wavelet );
.br
.BI "int QccWAVWaveletInverseDWT1D(QccVector " signal ", int " signal_length ", int " signal_origin ", int " subsample_pattern ", int " num_scales ", const QccWAVWavelet *" wavelet );
.br
.BI "int QccWAVWaveletDWTSubbandLength(int " original_length ", int " level ", int " highband ", int " signal_origin ", int " subsample_pattern );
.SH DESCRIPTION
.B QccWAVWaveletDWT1D()
performs a discrete wavelet transform (DWT) of a one-dimensional signal.
.I num_scales
gives the number of scales, or levels, of the decomposition.
.BR QccWAVWaveletDWT1D()
implements a dyadic decomposition of
.IR signal ;
that is, the lowpass subband is recursively decomposed into lowpass and
highpass bands for each level of decomposition.
The transform is critically sampled; that is,
each subband produced in each decomposition level
has roughly half as many samples as the lowpass band of the preceding level.
The subbands output from the DWT are returned in
.IR signal ,
overwriting the original input signal.
The output subbands are nested in
.I signal
starting with the lowpass subband of the lowest (coarsest) level of
decomposition (i.e., the baseband) with subsequent highpass subbands
of increasing resolution following.
.LP
Essentially,
.BR QccWAVWaveletDWT1D()
calls
.BR QccWAVWaveletAnalysis1D (3)
for each level of decomposition.
.BR QccWAVWaveletAnalysis1D (3)
in turn calls either
.BR QccWAVFilterBankAnalysis (3)
or
.BR QccWAVLiftingAnalysis (3)
depending on whether
.I wavelet
specifies a filter-bank or
lifting implementation.
In either case,
.I signal_origin
indicates the sample index at which 
.I signal
starts.
.LP
.B QccWAVWaveletInverseDWT1D()
performs the inverse DWT of
.IR signal
which is assumed to have been produced
by
.BR QccWAVWaveletDWT1D() .
.I num_scales
gives the number of levels of decomposition that exist in
.IR signal .
Essentially,
.B QccWAVWaveletInverseDWT1D()
calls
.BR QccWAVWaveletSynthesis1D (3)
for each level of synthesis;
.BR QccWAVWaveletSynthesis1D (3)
in turn calls either
.BR QccWAVFilterBankSynthesis (3)
or
.BR QccWAVLiftingSynthesis (3)
depending on whether
.I wavelet
specifies a filter-bank or
lifting implementation.
In either case,
.I signal_origin
indicates the sample index at which
.I signal
starts.
.LP
.I subsample_pattern
indicates the even- or odd-phase subsampling to be used at each level
of decomposition. In most applications, even subsampling at all
levels is desired, in which case
.I subsample_pattern
should be set to zero.
In more general settings, when some mixture of even- and odd-phase subsampling
is desired, 
.I subsample_pattern
can be an integer between 0 and
.RI "2^" num_levels " - 1."
In this integer, the 
.IR j th
bit (where
.I j
= 1 is the least-significant bit) indicates whether the
.IR j th
level of decomposition employs
even or odd subsampling (0 = even, 1 = odd).
For example, if
.I subsample_pattern
is 5, then the first and third decompositions use odd-phase
subsampling, while all others use even subsampling.
.LP
.BR QccWAVWaveletDWTSubbandLength()
gives the length of a subband produced by
.BR QccWAVWaveletDWT1D() .
.I original_length
gives the original length of the undecomposed original signal.
.I level
and
.I highpass
indicates for which subband the length is to be returned.
.I level
gives the level of decomposition of the subband of interest,
with 0 denoting the highest resolution (i.e., the level of the original
signal) and
.I num_scales
being the lowest resolution scale (where 
.I num_scales
is the number of scales used by
.BR QccWAVWaveletDWT1D()
in the decomposition).
.I highpass
indicates whether the highpass
.RI ( highpass
= 1)
or lowpass
.RI ( highpass
= 0)
subband at the given
.I level
is of interest.
.I signal_origin
indicates the sample index at which the original signal started.
.SH "RETURN VALUES"
.BR QccWAVWaveletDWTSubbandLength()
returns the subband length;
the other routines
return 0 on success and 1 on error.
.SH "SEE ALSO"
.BR QccWAVWaveletAnalysis1D (3),
.BR QccWAVWaveletSynthesis1D (3),
.BR QccWAVFilterBankAnalysis (3),
.BR QccWAVFilterBankSynthesis (3),
.BR QccWAVLiftingAnalysis (3),
.BR QccWAVLiftingSynthesis (3),
.BR QccWAVWaveletRedundantDWT1D (3),
.BR QccWAVWavelet (3),
.BR QccPackWAV (3),
.BR QccPack (3)
.LP
M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies,
"Image Coding Using Wavelet Transform,"
.IR "IEEE Transactions on Image Processing" ,
vol. 1, pp. 205-220, April 1992.
.LP
I. Daubechies and W. Sweldens,
"Factoring Wavelet Transforms Into Lifting Steps,"
.IR "J. Fourier Anal. Appl." ,
vol. 4, no. 3, pp. 245-267, 1998.
.SH AUTHOR
Copyright (C) 1997-2007  James E. Fowler
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