Package rwavelet. September 12, 2018

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1 Type Package Title Wavelet Analysis Version Date Author F. Navarro and C. Chesneau Package rwavelet September 12, 2018 Maintainer Navarro Fabien Perform wavelet analysis (orthogonal and translation invariant transforms) with applications to data compression or denoising. Most of the code is a port of 'MATLAB' Wavelab toolbo written by D. Donoho, A. Maleki and M. Shahram (< URL BugReports License LGPL (>= 2) Encoding UTF-8 LazyData true RoygenNote Imports dplyr, signal Suggests knitr, rmarkdown, imager VignetteBuilder knitr NeedsCompilation no Repository CRAN Date/Publication :10:03 UTC R topics documented: aconv DownDyadHi DownDyadLo dyad

2 2 aconv dyadlength FWT2_PO FWT_PO FWT_TI GWN HardThresh iconvv IWT2_PO IWT_PO IWT_TI lshift MAD MakeONFilter MakeSignal MirrorFilt packet PlotSpikes PlotWaveCoeff quadlength RaphNMR rshift ShapeAsRow SNR SoftThresh UpDyadHi UpDyadLo UpSampleN Inde 24 aconv Convolution Tool for Two-Scale Transform. Filtering by periodic convolution of with the time-reverse of f. aconv(f, ) f filter. 1-d signal.

3 DownDyadHi 3 y filtered result. iconvv, UpDyadHi, UpDyadLo, DownDyadHi, DownDyadLo. qmf <- MakeONFilter('Haar') <- MakeSignal('HeaviSine',2^3) aconv(qmf,) DownDyadHi Hi-Pass Downsampling operator (periodized) Hi-Pass Downsampling operator (periodized) DownDyadHi(, qmf) qmf 1-d signal at fine scale. filter. y 1-d signal at coarse scale. DownDyadLo, UpDyadHi, UpDyadLo, FWT_PO, iconvv. qmf <- MakeONFilter('Haar') <- MakeSignal('HeaviSine',2^3) DownDyadHi(, qmf)

4 4 dyad DownDyadLo Lo-Pass Downsampling operator (periodized) Lo-Pass Downsampling operator (periodized) DownDyadLo(, qmf) qmf 1-d signal at fine scale. filter. d 1-d signal at coarse scale. DownDyadHi, UpDyadHi, UpDyadLo, FWT_PO, aconv. qmf <- MakeONFilter('Haar') <- MakeSignal('HeaviSine',2^3) DownDyadLo(,qmf) dyad Inde entire j-th dyad of 1-d wavelet form Inde entire j-th dyad of 1-d wavelet form dyad(j) j integer.

5 dyadlength 5 i list of all indices of wavelet coeffts at j-th level. dyad(0) dyadlength Find length and dyadic length of array Find length and dyadic length of array dyadlength() array of length n = 2^J (hopefully). n length(). J least power of two greater than n. quadlength, dyad <- MakeSignal('Ramp', 8) dyadlength()

6 6 FWT_PO FWT2_PO 2-d tensor wavelet transform (periodized, orthogonal). A two-dimensional Wavelet Transform is computed for the array. qmf filter may be obtained from MakeONFilter. To reconstruct, use IWT2_PO. FWT2_PO(, L, qmf) L qmf 2-d image (n by n array, n dyadic). coarse level. quadrature mirror filter. wc 2-d wavelet transform. IWT2_PO, MakeONFilter. ## Not run: wc <- FWT2_PO(,L,qmf) ## End(Not run) FWT_PO Forward Wavelet Transform (periodized, orthogonal). 1. qmf filter may be obtained from MakeONFilter. 2. usually, length(qmf) < 2^(L+1). 3. To reconstruct use IWT_PO. FWT_PO(, L, qmf)

7 FWT_TI 7 1-d signal; length() = 2^J. L Coarsest Level of V_0; L «J. qmf quadrature mirror filter (orthonormal). wc 1-d wavelet transform of. IWT_PO, MakeONFilter. <- MakeSignal('Ramp', 8) L <- 0 qmf <- MakeONFilter('Haar') wc <- FWT_PO(, L, qmf) FWT_TI Translation Invariant Forward Wavelet Transform. 1. qmf filter may be obtained from MakeONFilter. 2. usually, length(qmf) < 2^(L+1). 3. To reconstruct use IWT_TI. FWT_TI(, L, qmf) L qmf array of dyadic length n=2^j. degree of coarsest scale. orthonormal quadrature mirror filter. TIWT stationary wavelet transform table. IWT_TI, MakeONFilter.

8 8 HardThresh <- MakeSignal('Ramp', 8) L <- 0 qmf <- MakeONFilter('Haar') TIWT <- FWT_TI(, L, qmf) GWN Generation of Gaussian White Noise. Generation of Gaussian White Noise. GWN(n, sigma) n sigma sample size. standard deviation. epsilon resulting noise. GWN(10,0.1) HardThresh Apply Hard Threshold. Apply Hard Threshold. HardThresh(y, t) y t Noisy Data. Threshold.

9 iconvv 9 filtered result (y 1_ y >t). ## Not run: HardThresh(y,t) ## End(Not run) iconvv Convolution Tool for Two-Scale Transform. Filtering by periodic convolution of with f. iconvv(f, ) f filter. 1-d signal. y filtered result. aconv, UpDyadHi, UpDyadLo, DownDyadHi, DownDyadLo. qmf <- MakeONFilter('Haar') <- MakeSignal('HeaviSine',2^3) iconvv(qmf,)

10 10 IWT_PO IWT2_PO Inverse 2-d MRA wavelet transform (periodized, orthogonal). If wc is the result of a forward 2d wavelet transform, with wc <- FWT2_PO(,L,qmf). then <- IWT2_PO(wc,L,qmf) reconstructs eactly qmf is a nice qmf, e.g. one made by MakeONFilter. IWT2_PO(wc, L, qmf) wc L qmf 2-d wavelet transform (n by n array, n dyadic). coarse level. quadrature mirror filter. 2-d signal reconstructed from wc. FWT2_PO, MakeONFilter. ## Not run: <- IWT2_PO(wc,L,qmf) ## End(Not run) IWT_PO Inverse Wavelet Transform (periodized, orthogonal). Suppose wc <- FWT_PO(,L,qmf) where qmf is an orthonormal quad. mirror filter, e.g. one made bymakeonfilter. Then can be reconstructed by <- IWT_PO(wc,L,qmf). IWT_PO(wc, L, qmf)

11 IWT_TI 11 wc 1-d wavelet transform: length(wc) = 2^J. L Coarsest scale (2^(-L) = scale of V_0); L «J. qmf quadrature mirror filter (orthonormal). 1-d signal reconstructed from wc. FWT_PO, MakeONFilter. <- MakeSignal('Ramp', 8) L <- 0 qmf <- MakeONFilter('Haar') wc <- FWT_PO(, L, qmf) r <- IWT_PO(wc,L,qmf) IWT_TI Translation Invariant Iorward Wavelet Transform. Translation Invariant Iorward Wavelet Transform. IWT_TI(pkt, qmf) pkt qmf translation-invariant wavelet transform table (TIWT). orthonormal quadrature mirror filter. 1-d signal reconstructed from translation-invariant transform TIWT. FWT_TI, MakeONFilter.

12 12 MAD <- MakeSignal('Ramp', 8) L <- 0 qmf <- MakeONFilter('Haar') TIWT <- FWT_TI(, L, qmf) r <- IWT_TI(TIWT,qmf) lshift Circular left shift of 1-d signal Circular left shift of 1-d signal lshift(a) a 1-d signal. l 1-d signal l(i) = (i+1) ecept l(n) = (1). <- MakeSignal('HeaviSine',2^3) lshift() MAD Median Absolute Deviation Compute the median absolute deviation. MAD() 1-d signal.

13 MakeONFilter 13 <- c(1, 1, 2, 2, 4, 6, 9) MAD() MakeONFilter Generate Orthonormal QMF Filter for Wavelet Transform. The Haar filter (which could be considered a Daubechies-2) was the first wavelet, though not called as such, and is discontinuous. MakeONFilter(Type, Par) Type Par string, Haar, Beylkin, Coiflet, Daubechies Symmlet, Vaidyanathan, Battle. integer, it is a parameter related to the support and vanishing moments of the wavelets, eplained below for each wavelet. Details The Beylkin filter places roots for the frequency response function close to the Nyquist frequency on the real ais. The Coiflet filters are designed to give both the mother and father wavelets 2*Par vanishing moments; here Par may be one of 1,2,3,4 or 5. The Daubechies filters are minimal phase filters that generate wavelets which have a minimal support for a given number of vanishing moments. They are indeed by their length, Par, which may be one of 4,6,8,10,12,14,16,18 or 20. The number of vanishing moments is par/2. Symmlets are also wavelets within a minimum size support for a given number of vanishing moments, but they are as symmetrical as possible, as opposed to the Daubechies filters which are highly asymmetrical. They are indeed by Par, which specifies the number of vanishing moments and is equal to half the size of the support. It ranges from 4 to 10. The Vaidyanathan filter gives an eact reconstruction, but does not satisfy any moment condition. The filter has been optimized for speech coding. The Battle-Lemarie filter generate spline orthogonal wavelet basis. The parameter Par gives the degree of the spline. The number of vanishing moments is Par+1. qmf quadrature mirror filter.

14 14 MakeSignal FWT_PO, IWT_PO, FWT2_PO, IWT2_PO. Type <- 'Coiflet' Par <- 1 qmf <- MakeONFilter(Type, Par) MakeSignal Make artificial signal. Make artificial signal. MakeSignal(name, n) name n string, HeaviSine, Bumps, Blocks, Doppler, Ramp, Cusp, Sing, Hi- Sine, LoSine, LinChirp, TwoChirp, QuadChirp, MishMash, Werner- Sorrows (Heisenberg), Leopold (Kronecker), Riemann, HypChirps, LinChirps, Chirps, Gabor, sineoneover, Cusp2, SmoothCusp, Piece-Regular (Piece- Wise Smooth), Piece-Polynomial (Piece-Wise 3rd degree polynomial), desired signal length sig 1-d signal. FWT_PO, IWT_PO, FWT2_PO, IWT2_PO. name <- 'Cusp' n <- 2^5 sig <- MakeSignal(name,n)

15 MirrorFilt 15 MirrorFilt Apply (-1)^t modulation h(t) = (-1)^(t-1) * (t), 1 <= t <= length() MirrorFilt() 1-d signal. h 1-d signal with DC frequency content shifted. to Nyquist frequency DownDyadHi. <- MakeSignal('HeaviSine',2^3) h <- MirrorFilt() packet Packet table indeing. Packet table indeing. packet(d, b, n) d depth of splitting in packet decomposition. b block inde among 2^d possibilities at depth d. n length of signal

16 16 PlotSpikes p linear indices of all coeff s in that block. ## Not run: packet(d, b, n) ## End(Not run) PlotSpikes Plot 1-d signal as baseline with series of spikes. Plot 1-d signal as baseline with series of spikes. PlotSpikes(base, t,, L, J) base number, baseline level. t ordinate values. 1-d signal, specifies spike deflections from baseline. L level of coarsest scale. J least power of two greater than n. A plot of spikes on a baseline. PlotWaveCoeff. ## Not run: PlotSpikes(base,t,, L, J) ## End(Not run)

17 PlotWaveCoeff 17 PlotWaveCoeff Spike-plot display of wavelet coefficients. Spike-plot display of wavelet coefficients. PlotWaveCoeff(wc, L, scal) wc L scal 1-d wavelet transform. level of coarsest scale. scale factor (0 ==> autoscale). A display of wavelet coefficients (coarsest level NOT included) by level and position. FWT_PO, IWT_PO, PlotSpikes. <- MakeSignal('Ramp', 128) qmf <- MakeONFilter('Daubechies', 10) L <- 3 scal <- 1 wc <- FWT_PO(, L, qmf) PlotWaveCoeff(wc,L,scal) quadlength Find length and dyadic length of square matri. h(t) = (-1)^(t-1) * (t), 1 <= t <= length() quadlength()

18 18 RaphNMR 2-d image; size(n,n), n = 2^J (hopefully). n length(). J least power of two greater than n. quadlength(matri(1:16,ncol=4)) RaphNMR Nuclear magnetic resonance (NMR) signal. A dataset containing a NMR signal data(raphnmr) Format A numeric vector of length 1024 Source MRS Unit, VA Medical Center, San Francisco. Adrain Maudsley, Ph.D., Professor of Radiology. This NMR signal was obtained from Chris Raphael, then a postdoctoral fellow in the Department of Statistics at Stanford University who was working on Hidden Markov Models for restoring NMR Spectra.

19 rshift 19 rshift Circular right shift of 1-d signal Circular right shift of 1-d signal rshift(a) a 1-d signal. r 1-d signal r(i) = (i-1) ecept r(1) = (n). <- MakeSignal('HeaviSine',2^3) rshift() ShapeAsRow Make signal a row vector Make signal a row vector ShapeAsRow(sig) sig a row or column vector. row a row vector. sig <- matri(1:4) row <- ShapeAsRow(sig)

20 20 SoftThresh SNR Signal/Noise ratio Signal/Noise ratio SNR(, y) y Original reference signal. Restored or noisy signal Signal/Noise ratio. n <- 2^4 <- MakeSignal('HeaviSine', n) y <- + rnorm(n, mean=0, sd=1) SNR(, y) SoftThresh Apply Soft Threshold. Apply Soft Threshold. SoftThresh(y, t) y t Noisy Data. Threshold. filtered result (y 1_ y >t).

21 UpDyadHi 21 ## Not run: SoftThresh(y,t) ## End(Not run) UpDyadHi Hi-Pass Upsampling operator; periodized Hi-Pass Upsampling operator; periodized UpDyadHi(, qmf) qmf 1-d signal at coarser scale. filter. u 1-d signal at finer scale. DownDyadLo, DownDyadHi, UpDyadLo, IWT_PO, aconv. qmf <- MakeONFilter('Haar') <- MakeSignal('HeaviSine',2^3) UpDyadHi(,qmf)

22 22 UpSampleN UpDyadLo Lo-Pass Upsampling operator; periodized Lo-Pass Upsampling operator; periodized UpDyadLo(, qmf) qmf 1-d signal at coarser scale. filter. y 1-d signal at finer scale. DownDyadLo, DownDyadHi, UpDyadHi, IWT_PO, iconvv. qmf <- MakeONFilter('Haar') <- MakeSignal('HeaviSine',2^3) UpDyadLo(,qmf) UpSampleN Upsampling operator Upsampling operator UpSampleN(, s) 1-d signal, of length n. s upsampling scale, default = 2.

23 UpSampleN 23 y 1-d signal, of length s*n with zeros interpolating alternate samples y(s*i-1) = (i), i=1,...,n

24 Inde Topic datasets RaphNMR, 18 aconv, 2, 4, 9, 21 UpDyadHi, 3, 4, 9, 21, 22 UpDyadLo, 3, 4, 9, 21, 22 UpSampleN, 22 DownDyadHi, 3, 3, 4, 9, 15, 21, 22 DownDyadLo, 3, 4, 9, 21, 22 dyad, 4, 5 dyadlength, 5 FWT2_PO, 6, 10, 14 FWT_PO, 3, 4, 6, 11, 14, 17 FWT_TI, 7, 11 GWN, 8 HardThresh, 8 iconvv, 3, 9, 22 IWT2_PO, 6, 10, 14 IWT_PO, 6, 7, 10, 14, 17, 21, 22 IWT_TI, 7, 11 lshift, 12 MAD, 12 MakeONFilter, 6, 7, 10, 11, 13 MakeSignal, 14 MirrorFilt, 15 packet, 15 PlotSpikes, 16, 17 PlotWaveCoeff, 16, 17 quadlength, 5, 17 RaphNMR, 18 rshift, 19 ShapeAsRow, 19 SNR, 20 SoftThresh, 20 24