.TH QCCWAVLIFTINGANALYSIS 3 "QCCPACK" "".SH NAMEQccWAVLiftingAnalysis, QccWAVLiftingSynthesis \- perform lifting analysis/synthesis of a signal.SH SYNOPSIS.B #include "libQccPack.h".sp.BI "int QccWAVLiftingAnalysis(const QccVector " input_signal ", QccVector " output_signal ", int " signal_length ", int " phase ", const QccWAVLiftingScheme *" lifting_scheme ", int " boundary );.br.BI "int QccWAVLiftingSynthesis(QccVector " input_signal ", const QccVector " output_signal ", int " signal_length ", int " phase ", const QccWAVLiftingScheme *" lifting_scheme ", int " boundary );.SH DESCRIPTION.B QccWAVLiftingAnalysis()essentially performs one level of a wavelet decomposition.Specifically,lifting analysis is employed to produce a lowpass subbandand a highpass subband of.IR input_signal .The lowpass subband is returned as thefirst half of.IR output_signal ;the highpass subband is returned as the last half of.IR output_signal ..I signal_lengthgives the length of.I input_signaland may be even or odd..I output_signalis assumed to be allocated to sufficient size priorto calling.BR QccWAVLiftingAnalysis() ..LPIf.I phaseis.BR QCCWAVWAVELET_PHASE_EVEN ,then.I input_signalis assumed to start with a even-indexed sample.Otherwise, if.I phaseis.BR QCCWAVWAVELET_PHASE_ODD ,it indicates that.I input_signalstarts with an odd-indexed sample..I phaseis passed to.BR QccWAVWaveletLWT (3),the lazy wavelet transform,the first step of lifting analysis, to indicate thesignal origin..LPIn the case that.I signal_lengthis even, both the odd and even subbands of produced by lifting analysisare the same length.  On the other hand, if.I signal_lengthis odd, one of the two subbands will be one sample longerthan the other. Which subband will be longer will depend onthe value of.IR phase .Specifically, if.IR phaseis.BR QCCWAVWAVELET_PHASE_EVEN ,then the lowpass subband is one sample longer than the highpasssubband. If.IR phaseis.BR QCCWAVWAVELET_PHASE_ODD ,the the highpass subband is one sample longer than the lowpasssubband..LP.I lifting_schemegives the particular lifting scheme to employ.Lifting implementations ofwavelet analysis and synthesis are "hard-coded" into the QccPack libraryfor purposes of execution speed and ease of implementation(this is opposed to the situation for filter-based analysis and synthesiswhich employ generic filter routines and read coefficients from files).As a consequence, only a limited number of waveletsare current supported, and this list cannot be extended by the user(without modifying the QccPack source code, or course).The currently supported lifting schemes and their corresponding.B LFTfiles (see.BR QccWAVLiftingScheme (3))are.RSLWT.lft - Lazy Wavelet transform.brCohenDaubechiesFeauveau.5-3.lft - length 9/5 biorthogonal wavelet.brCohenDaubechiesFeauveau.9-7.lft - length 9/7 biorthogonal wavelet.brDaubechies.4.lft - length 4 orthogonal wavelet.RE.LP.B QccWAVLiftingSynthesis()performs one level of wavelet synthesis.  The first half of.I input_signalis assumed to contain the lowpass subband while the second half containsthe highpass subband.Lifting synthesis is performedto produce.IR output_signal ,including a call to.BR QccWAVWaveletInverseLWT (3)to perform an inverse lazy wavelet transform.As with.BR QccWAVLiftingAnalysis() ,.I signal_lengthmay be even or odd, andsufficient space for.I output_signalmust be allocated prior to calling.BR QccWAVLiftingSynthesis() ..I phaseindicates whether.I output_signalis to start with an even- or odd-indexed sample..LP.IR boundaryindicates how lifting should be handled at the ends of the signal andcan be one of the following:.B QCCWAVWAVELET_BOUNDARY_SYMMETRIC_EXTENSION(symmetric extension, valid only for biorthogonal wavelets),.B QCCWAVWAVELET_BOUNDARY_PERIODIC_EXTENSION(periodic extension, valid for orthogonal and biorthogonal wavelets), or.B QCCWAVWAVELET_BOUNDARY_BOUNDARY_WAVELET(boundary wavelet, valid for orthogonal and biorthogonal wavelets).Check the comments at the start of each.B LFTfile for permitted values for.I boundaryfor particular lifting schemes.Note that, if periodic extension is used,.IR signal_lengthmust be even (this is due to mathematical constraints)..LPThe case in which.IR signal_lengthis equal to 1 is degenerate. A wavelet transform is technically not welldefined for this situation because it is not clear how to subsamplethe signal. However, in practice, this degenerate case can be handled inseveral ways with no problems. In QccPack, the length-one signal ismerely multiplied by sqrt(2) and placed in the lowpass subband (in whichcase the highpass subband has zero length) for .IR phaseequal to.BR QCCWAVWAVELET_PHASE_EVEN ,or divided by sqrt(2) and placed in the highpass subband(in which case the lowpass subband has zero length) for.IR phaseequal to.BR QCCWAVWAVELET_PHASE_ODD ..LPNote:In general, you will probably want to use.BR QccWAVWaveletDWT1D (3)and.BR QccWAVWaveletInverseDWT1D (3)instead of these routinesfor implementing a discrete wavelet transform and its inverse since.BR QccWAVWaveletDWT1D (3)and.BR QccWAVWaveletInverseDWT1D (3)allow any number of scales, or levels, of decomposition to beperformed, and can be used with either filter-bank or lifting implementationsof wavelets..SH "RETURN VALUES"These routinesreturn 0 on success and 1 on error..SH "SEE ALSO".BR QccWAVLiftingScheme (3),.BR QccWAVWavelet (3),.BR QccWAVWaveletDWT1D (3),.BR QccWAVWaveletInverseDWT1D (3),.BR QccPackWAV (3),.BR QccPack (3).LPI. Daubechies and W. Sweldens,"Factoring Wavelet Transforms Into Lifting Steps,".IR "J. Fourier Anal. Appl." ,vol. 4, no. 3, pp. 245-267, 1998..SH AUTHORCopyright (C) 1997-2005  James E. Fowler.\"  The programs herein are free software; you can redistribute them an.or.\"  modify them under the terms of the GNU General Public License.\"  as published by the Free Software Foundation; either version 2.\"  of the License, or (at your option) any later version..\"  .\"  These programs are distributed in the hope that they will be useful,.\"  but WITHOUT ANY WARRANTY; without even the implied warranty of.\"  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the.\"  GNU General Public License for more details..\"  .\"  You should have received a copy of the GNU General Public License.\"  along with these programs; if not, write to the Free Software.\"  Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.