.TH QCCWAVWAVELETREDUNDANTDWT1D 3 "QCCPACK" "".SH NAMEQccWAVWaveletRedundantDWT1D, QccWAVWaveletInverseRedundantDWT1D,QccWAVWaveletRedundantDWT1DSubsample \- redundant discrete wavelet transform and inverse transform for a 1D signal.SH SYNOPSIS.B #include "libQccPack.h".sp.BI "int QccWAVWaveletRedundantDWT1D(const QccVector " input_signal ", QccMatrix " output_signals ", int " signal_length ", int " num_scales ", const QccWAVWavelet *" wavelet );.br.BI "int QccWAVWaveletInverseRedundantDWT1D(const QccMatrix " input_signals ", QccVector " output_signal ", int " signal_length ", int " num_scales ", const QccWAVWavelet *" wavelet );.br.BI "int QccWAVWaveletRedundantDWT1DSubsample(const QccMatrix " input_signals ", QccVector " output_signal ", int " signal_length ", int " num_scales ", int " subsample_pattern ", const QccWAVWavelet *" wavelet );.SH DESCRIPTION.B QccWAVWaveletRedundantDWT1D()performs a redundantdiscrete wavelet transform (RDWT) of a one-dimensional signal..I num_scalesgives the number of scales, or levels, of the decomposition..BR QccWAVWaveletRedundantDWT1D()implements a dyadic decomposition of.IR input_signal with oversampling;that is, the lowpass subband is recursively decomposed into lowpass andhighpass bands for each level of decomposition.Unlike the usual critically sampled DWT(as implemented by.BR QccWAVWaveletDWT1D (3)),each subband has the same length as the original.IR input_signal ..LPThe RDWT produces .I num_scaleshighpass subbands and one baseband subband,for a total of.I num_scales + 1subbands.These subbands are returned in.I output_signalswhich is amatrix with .I "num_scales + 1"rows and .I signal_lengthcolumns.The first row of.I output_signalscontains the baseband signal;subsequent rows contain highpass bands of decreasing scale (increasingspatial resolution).Sufficient storage space for.IR output_signals must be allocated prior to calling.BR QccWAVWaveletRedundantDWT1D() ..LP.I waveletcan be either a filter-bank or lifting-schemeimplementation..LP.BR QccWAVWaveletInverseRedundantDWT1D()performs the inverse RDWT of.IR input_signals .Sufficient space for.I output_signalmust be allocated prior to calling.BR QccWAVWaveletInverseRedundantDWT1D() ..LPThe RDWT produces an oversampled DWT transform; that is,an overcomplete expansion of the original signal.However,the coefficients of the usual critically sampled DWTare amongst the RDWT coefficients.More accurately, there exist.I 2 ^ num_scalessets of RDWT coefficients that are identicalto a critically sampled, dyadic DWT.Each one of these.I 2 ^ num_scalesDWTs differs from the others in the subsampling phase choice(even or odd) at each level of the transform..BR QccWAVWaveletRedundantDWT1DSubsample()subsamples the coefficients output from.BR QccWAVWaveletRedundantDWT1D()to obtain the coefficients for a critically sampled DWT..IR subsample_pattern ,a value from 0 to.RI ( "2 ^ num_scales" ) " - 1" ,indicates which of the.I 2 ^ num_scalesDWT-coefficient sets to choose..I num_scalesgives the number of levels of decomposition that exist in.IR input_signals ,which is assumed to be a matrix of.I num_scales + 1rows and.I signal_lengthcolumns produced by.BR QccWAVWaveletRedundantDWT1D() .Sufficient space for.I output_signalmust be allocated prior to calling.BR QccWAVWaveletRedundantDWT1DSubsample() .Calling.BR QccWAVWaveletRedundantDWT1DSubsample()with .I subsample_pattern= 0will produce the same coefficients as would have beenproduced by.BR QccWAVWaveletDWT1D()..LP.BR QccWAVWaveletInverseRedundantDWT1D() is the "proper" procedure for inverting the RDWT.That is,.BR QccWAVWaveletInverseRedundantDWT1D() performs inverse filtering as wellas weighted "averaging" of the transform redundancies to properly reconstruct the original signal from theRDWT coefficients.A "quick and dirty" reconstruction is possible, however,by subsampling and then inverting using thecritically sampled inverse DWT, i.e., by calling.BR QccWAVWaveletRedundantDWT1DSubsample()and then.BR QccWAVWaveletInverseDWT1D() .Both these approaches to inverting the RDWT will producethe same results on unaltered RDWT coefficients; however,should the RDWT coefficients be processed in some fashion, say,through quantization, then.BR QccWAVWaveletInverseRedundantDWT1D()should be used to invert the transform.Additionally, this subsampling approach to inverting the RDWTworks only for .I subsample_pattern= 0, which is the only subsampling pattern whose phase choicesmatch those assumed by.BR QccWAVWaveletInverseDWT1D() ..SH NOTESThe notion of an RDWT apparently dates back to the work of Holschneider.IR "et al".and Dutilleux who devised a redundant transform implemented via the so-called.IR "algorithme a trous" .This filter-bank algorithm is similar to the usual Mallat algorithmfor the critically sampled DWT in that it implementsthe wavelet transform with filter banks. The key difference betweenthe two approaches is thatthe subsampling at the end of every scale of the transform as used inthe Mallat algorithm is not performed in the.IR "algorithm a trous" ,and the filters, which are the same at every scale in the Mallatalgorithm, change for every scale.Specifically, the.IR "algorithm a trous"calls for the insertion of "holes" ("trous" in French)between each filter tap; that is, the filters used in eachscale of the.IR "algorithm a trous"decompositionare the filters of the previous scale upsampled by a factor of 2..LPShensa points out that the Mallat and .I "a trous"implementations are very closely related; in fact, it is possible to obtain the coefficients for the Mallatalgorithm by subsampling, or decimating, the coefficientsresulting from the.IR "algorithm a trous" ,except possibly at the signal boundaries when symmetric extension is used.This observation leadsto an "alternative" implementation of the.IR "algorithm a trous" ,and it is this implementation that is used here in.BR QccWAVWaveletRedundantDWT1D() .This alternative implementation of the.IR "algorithm a trous" eliminates the above-mentioned symmetric-boundary inconsistencies;in addition, the alternative implementationpermits the direct use of lifting instead of filter banks for improvedcomputation efficiency..SH "RETURN VALUES"These routinesreturn 0 on success and 1 on error..SH "SEE ALSO".BR QccWAVWaveletDWT1D (3),.BR QccWAVWaveletInverseDWT1D (3),.BR QccPackWAV (3),.BR QccPack (3).LPM. Holschneider, R. Kronland-Martinet, J. Morlet, andP. Tchamitchian,"A Real-Time Algorithm for Signal Analysis with the Helpof the Wavelet Transform," in.IR "Wavelets: Time-Frequency Methods and Phase Space",Berlin: Springer-Verlag, pp. 286-297, 1989..LPP. Dutilleux,"An Implementation of the Algorithm A Trous to Compute the Wavelet Transform,"in.IR "Wavelets: Time-Frequency Methods and Phase Space",Berlin: Springer-Verlag, pp. 298-304, 1989..LPM. J. Shensa,"The Discrete Wavelet Transform: Wedding the A Trous and Mallat Algorithms,".IR "IEEE Trans. Signal Processing" ,vol. 40, no. 10, pp. 2464-2482, Oct. 1998..SH AUTHORCopyright (C) 1997-2005  James E. 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