Package 'CNLTreg' reference manual

Title:	Complex-Valued Wavelet Lifting for Signal Denoising
Description:	Implementations of recent complex-valued wavelet shrinkage procedures for smoothing irregularly sampled signals, see Hamilton et al (2018) <doi:10.1080/00401706.2017.1281846>.
Authors:	Matt Nunes [aut, cre], Marina Knight [aut], Jean Hamilton [ctb], Piotr Fryzlewicz [ctb]
Maintainer:	Matt Nunes <[email protected]>
License:	GPL-2
Version:	0.1-2
Built:	2024-12-19 06:25:52 UTC
Source:	CRAN

Complex-Valued Wavelet Lifting for Signal Denoising

Description

Implementations of recent complex-valued wavelet shrinkage procedures for smoothing irregularly sampled signals, see Hamilton et al (2018) <doi:10.1080/00401706.2017.1281846>.

Details

The DESCRIPTION file:

Package:	CNLTreg
Type:	Package
Title:	Complex-Valued Wavelet Lifting for Signal Denoising
Version:	0.1-2
Date:	2018-07-18
Author:	Matt Nunes [aut, cre], Marina Knight [aut], Jean Hamilton [ctb], Piotr Fryzlewicz [ctb]
Authors@R:	c(person("Matt","Nunes", role=c("aut","cre"),email="[email protected]"),person("Marina", "Knight", role="aut"), person("Jean", "Hamilton", role="ctb"), person("Piotr", "Fryzlewicz", role="ctb"))
Maintainer:	Matt Nunes <[email protected]>
Description:	Implementations of recent complex-valued wavelet shrinkage procedures for smoothing irregularly sampled signals, see Hamilton et al (2018) <doi:10.1080/00401706.2017.1281846>.
License:	GPL-2
Depends:	adlift, miscTools, nlt
Suggests:	MASS
NeedsCompilation:	no
Packaged:	2018-07-18 12:20:38 UTC; nunes
Repository:	CRAN
Date/Publication:	2018-07-18 12:40:08 UTC

Index of help topics:

CNLTreg-package         Complex-Valued Wavelet Lifting for Signal
                        Denoising
cnlt.reg                Performs 'nondecimated' complex-valued wavelet
                        lifting for signal denoising
denoisepermC            Denoises a signal using the complex-valued
                        lifting transform and multivariate soft
                        thresholding
denoisepermCh           Denoises a signal using the complex-valued
                        lifting transform and multivariate soft
                        thresholding and heteroscedastic variance
                        computation
fwtnppermC              Forward complex wavelet lifting transform
mthreshC                Function to perform 'multiwavelet style'
                        level-dependent soft thresholding for
                        complex-valued wavelet coefficients
orthpredfilters         Computes orthogonal filters

The main routines of the package are denoisepermC and cnlt.reg which perform complex-valued lifting-based denoising, using a single or a multiple (chosen) number of lifting trajectories, respectively.

Author(s)

Matt Nunes [aut, cre], Marina Knight [aut], Jean Hamilton [ctb], Piotr Fryzlewicz [ctb]

Maintainer: Matt Nunes <[email protected]>

References

Hamilton, J., Nunes, M. A, Knight, M. I. and Fryzlewicz, P. (2018) Complex-valued wavelet lifting and applications. Technometrics, 60 (1), 48-60, DOI 10.1080/00401706.2017.1281846.

For related literature on the lifting methodology adopted in the technique, see

Nunes, M. A., Knight, M. I and Nason, G. P. (2006) Adaptive lifting for nonparametric regression. Stat. Comput. 16 (2), 143–159.

Knight, M. I. and Nason, G. P. (2009) A 'nondecimated' wavelet transform. Stat. Comput. 19 (1), 1–16.

Performs 'nondecimated' complex-valued wavelet lifting for signal denoising

Description

The transform-threshold-invert procedure for signal denoising is dependent on the trajectory (lifting order) used in the forward lifting transform. This procedure uses trajectory bootstrapping and averaging of estimates to gain denoising performance

Usage

cnlt.reg(x, f, P, returnall = FALSE, nkeep = 2, ...)
cnlt.reg(x, f, P, returnall = FALSE, nkeep = 2, ...)

Arguments

`x`	Vector of any length (not necessarily equally spaced) that gives the grid on which the signal is observed.
`f`	Vector of the same length as `x` that gives the signal values corresponding to the `x`-locations.
`P`	Number of trajectories to be used by the nondecimated lifting algorithm.
`returnall`	Indicates whether the function returns useful variables or just the denoised datapoints.
`nkeep`	Number of scaling points we want at the end of the transform. The usual choice is `nkeep`=2.
`...`	Any other arguments to be passed to `denoisepermC`, see the function documentation for more details.

Details

Essentially, this function applies the complex wavelet lifting denoising procedure denoisepermC P times, each with a different random lifting trajectory. This results in P estimates of the (unknown) true signal. The average of these estimators is the proposed estimator.

Value

If returnall=FALSE, the estimate of the function after denoising. If returnall=TRUE, a list with components:

`vec`	A matrix of dimension `P x (n - nkeep)`, each row corresponding to a different lifting trajectory.
`aveghat`	Estimated signal after removing the noise.

Warning

Using a large number of trajectories for long datasets could take a long time!

Author(s)

Matt Nunes

References

Hamilton, J., Nunes, M. A., Knight, M. I. and Fryzlewicz, P. (2018) Complex-valued wavelet lifting and applicati$ Technometrics, bold60 (1), 48-60, DOI 10.1080/00401706.2017.1281846.

For the real-valued equivalent procedure, see also
Knight, M. I. and Nason, G. P. (2009) A 'nondecimated' wavelet transform. Stat. Comput. 19 (1), 1–16.

Examples

 
    library(adlift)

    # construct an (irregular) observation grid
     x<-runif(256) 
     
     #construct the true, normally unknown, signal
     g<-make.signal2("blocks",x=x) 
     
     #generate noise with mean 0 and signal-to-noise ratio 5
     noise<-rnorm(256,mean=0,sd=sqrt(var(g))/5)
     
     #generate a noisy version of g 
     f<-g+noise 
     
     #decide on a number of random trajectories to be used (e.g. J=5 below), and apply
     # the nondecimated lifting transform to the noisy signal (x,f): 
## Not run: 
     est<-cnlt.reg(x,f,P=50,LocalPred=AdaptPred,neighbours=1,returnall=FALSE) 

## End(Not run)
library(adlift)

    # construct an (irregular) observation grid
     x<-runif(256) 
     
     #construct the true, normally unknown, signal
     g<-make.signal2("blocks",x=x) 
     
     #generate noise with mean 0 and signal-to-noise ratio 5
     noise<-rnorm(256,mean=0,sd=sqrt(var(g))/5)
     
     #generate a noisy version of g 
     f<-g+noise 
     
     #decide on a number of random trajectories to be used (e.g. J=5 below), and apply
     # the nondecimated lifting transform to the noisy signal (x,f): 
## Not run: 
     est<-cnlt.reg(x,f,P=50,LocalPred=AdaptPred,neighbours=1,returnall=FALSE) 

## End(Not run)

Denoises a signal using the complex-valued lifting transform and multivariate soft thresholding

Description

Denoises an input signal contaminated by noise. First the signal is decomposed using the complex-valued lifting scheme (see fwtnppermC) using an order of point removal. The resulting complex-valued wavelet coefficients are then thresholded using a soft thresholding rule on the details' magnitude. The transform is inverted and an estimate of the noisy signal is obtained.

Usage

denoisepermC(x, f, returnall = FALSE, sdtype = "adlift", verbose = FALSE, ...)
denoisepermC(x, f, returnall = FALSE, sdtype = "adlift", verbose = FALSE, ...)

Arguments

`x`	Vector of any length (not necessarily equally spaced) that gives the grid on which the signal is observed.
`f`	Vector of the same length as `x` that gives the signal values corresponding to the `x`-locations.
`returnall`	Indicates whether the function returns useful variables or just the denoised datapoints.
`sdtype`	Options are either `"adlift"` or `"complex"`, indicating whether the noise variance is estimated with the average of the mean absolute deviations of both real and imaginary components of the finest wavelet coefficients, or just the real component, as in `denoise`.
`verbose`	Indicates whether useful messages should be printed to the console during the procedure.
`...`	Any other arguments to be passed to `fwtnppermC`, see documentation for this function for more details.

Details

After the complex lifting transform is applied, the wavelet coeficients are divided into artificial levels. The details from the lifting scheme have different variances, and will therefore be normalized to have the same variance as the noise, by using the lifting matrix. Those normalized details falling into the finest artificial level will be used for estimating the standard deviation of the noise that contaminated the signal. The variable sdtype is used for this estimate, see Appendix B of Hamilton et al. (2018) for more details. Using this estimate, the normalized details can then be thresholded and un-normalized. The transform is then inverted to give an estimate of the signal.

Value

If returnall=FALSE, the estimate of the function after denoising. If returnall=TRUE, a list with components:

`fhat`	Estimated signal after removing the noise.
`w`	This is the matrix associated to the modified lifting transform.
`indsd`	Vector giving the standard deviations of the detail and scaling coefficients.
`al`	List giving the split of points between the artificial levels.
`sd`	Estimated standard deviation of the noise.

Author(s)

Matt Nunes, Marina Knight

References

Hamilton, J., Nunes, M. A., Knight, M. I. and Fryzlewicz, P. (2018) Complex-valued wavelet lifting and applications. Technometrics, 60 (1), 48-60, DOI 10.1080/00401706.2017.1281846.

Examples

    library(adlift)

    # construct an (irregular) observation grid
     x<-runif(256)

     #construct the true, normally unknown, signal
     g<-make.signal2("blocks",x=x)

     #generate noise with mean 0 and signal-to-noise ratio 5
     noise<-rnorm(256,mean=0,sd=sqrt(var(g))/5)

     #generate a noisy version of g
     f<-g+noise

     # perform the complex-valued denoising procedure to the noisy signal (x,f):
     est<-denoisepermC(x,f,LocalPred=AdaptPred,neigh=1,returnall=FALSE)

library(adlift)

    # construct an (irregular) observation grid
     x<-runif(256)

     #construct the true, normally unknown, signal
     g<-make.signal2("blocks",x=x)

     #generate noise with mean 0 and signal-to-noise ratio 5
     noise<-rnorm(256,mean=0,sd=sqrt(var(g))/5)

     #generate a noisy version of g
     f<-g+noise

     # perform the complex-valued denoising procedure to the noisy signal (x,f):
     est<-denoisepermC(x,f,LocalPred=AdaptPred,neigh=1,returnall=FALSE)

Denoises a signal using the complex-valued lifting transform and multivariate soft thresholding and heteroscedastic variance computation

Description

Denoises an input signal contaminated by noise. First the signal is decomposed using the complex-valued lifting scheme (see fwtnppermC) using an order of point removal. A sliding window approach is then used on these wavelet coefficients to estimate a local noise variance. The resulting complex-valued wavelet coefficients are then thresholded using a soft thresholding rule on the details' magnitude. The transform is inverted and an estimate of the noisy signal is obtained.

Usage

denoisepermCh(x, f, returnall = FALSE, verbose = FALSE, ...)
denoisepermCh(x, f, returnall = FALSE, verbose = FALSE, ...)

Arguments

`x`	Vector of any length (not necessarily equally spaced) that gives the grid on which the signal is observed.
`f`	Vector of the same length as `x` that gives the signal values corresponding to the `x`-locations.
`returnall`	Indicates whether the function returns useful variables or just the denoised datapoints.
`verbose`	Indicates whether useful messages should be printed to the console during the procedure.
`...`	Any other arguments to be passed to `fwtnpperm` and `fwtnppermC`.

Details

After the complex lifting transform is applied, the wavelet coeficients are divided into artificial levels. The details from the lifting scheme have different variances, and will therefore be normalized to have the same variance as the noise, by using the lifting matrix. A sliding window is used to compute a local 'heteroscedastic' noise variance by taking the MAD of those normalized details falling into the window, see Nunes et al. (2006) for more details. Given the noise estimates for each observation, the normalized details can then be thresholded and un-normalized. The transform is then inverted to give an estimate of the signal.

Value

If returnall=FALSE, the estimate of the function after denoising. If returnall=TRUE, a list with components:

`fhat`	Estimated signal after removing the noise.
`w`	This is the matrix associated to the modified lifting transform.
`al`	List giving the split of points between the artificial levels.
`sd`	Estimated heteroscedastic standard deviation of the noise.

Author(s)

Matt Nunes, Marina Knight

References

Hamilton, J., Nunes, M. A, Knight, M. I. and Fryzlewicz, P. (2018) Complex-valued wavelet lifting and applications. Technometrics, 60 (1), 48-60, DOI 10.1080/00401706.2017.1281846.

Nunes, M. A., Knight, M. I and Nason, G. P. (2006) Adaptive lifting for nonparametric regression. Stat. Comput. 16 (2), 143–159.

Knight, M. I. and Nason, G. P. (2009) A 'nondecimated' wavelet transform. Stat. Comput. 19 (1), 1–16.

Examples


library(MASS)	# where the motorcyle data lives

mcycleu<-mcycle[which(duplicated(mcycle$times)=='FALSE'),]
time<-mcycleu[,1]
accel<-mcycleu[,2]

set.seed(200)
est1<-denoisepermCh(time,accel)

library(MASS)	# where the motorcyle data lives

mcycleu<-mcycle[which(duplicated(mcycle$times)=='FALSE'),]
time<-mcycleu[,1]
accel<-mcycleu[,2]

set.seed(200)
est1<-denoisepermCh(time,accel)

Forward complex wavelet lifting transform

Description

Performs the complex-valued lifting transform on a signal with grid x and corresponding function values f.

Usage

fwtnppermC(x, f, LocalPred = LinearPred, neighbours = 1, 
intercept = TRUE, closest = FALSE, nkeep = 2, 
mod = sample(1:length(x), (length(x) - nkeep), FALSE))
fwtnppermC(x, f, LocalPred = LinearPred, neighbours = 1, 
intercept = TRUE, closest = FALSE, nkeep = 2, 
mod = sample(1:length(x), (length(x) - nkeep), FALSE))

Arguments

`x`	A vector of grid values. Can be of any length, not necessarily equally spaced.
`f`	A vector of function values corresponding to x. Must be of the same length as x.
`LocalPred`	The type of regression to be performed in the prediction lifting step. Possible options are `LinearPred`, `QuadPred`, `CubicPred`, `AdaptPred` and `AdaptNeigh`.
`neighbours`	The number of neighbours over which the regression is performed at each step. If closest is `FALSE`, then this in fact denotes the number of neighbours on each side of the removed point.
`intercept`	Indicates whether or not the regression prediction includes an intercept.
`closest`	Refers to the configuration of the chosen neighbours. If closest is `FALSE`, the neighbours will be chosen symmetrically around the removed point. Otherwise, the closest neighbours (in distance) will be chosen.
`nkeep`	The number of scaling coefficients to be kept in the final representation of the initial signal. This must be at least two.
`mod`	Vector of length (length(`x`)-`nkeep`). This gives the trajectory for the lifting algorithm to follow, i.e. it gives the order of point removal.

Details

Given $n$ points on a line, x, each with a corresponding envf value this function computes the complex-valued lifting transform of the (x,f) data. This is similar in spirit to the real-valued lifting transform in fwtnpperm, except that the algorithm constructs *two* orthogonally linked prediction filters, as in Section 2.2 of Hamilton et al. (2018). A summary of the procedure is as follows:

Step One. Compute "integrals" associated to each point, representing the intervals that each gridpoint x_i spans.

Then for each point index in the lifting trajectory mod,

Step Two(a). The neighbours of the removed point are identified using the specified neighbour configuration. The value of f at the removed point is predicted using the specified regression over the neighbours, unless an adaptive procedure is chosen. In this case, the algorithm chooses the regression which produces the minimal detail coefficient (in magnitude) from a range of regression types (see AdaptPred or AdaptNeigh for more information). In either case, the regression specifies a local filter of the function values over the neighbourhood, L.

Step Two(b). A second filter, M, is then constructed orthogonal to L, such that it has unit norm, see Hamilton et al. (2018) for more details.

The differences between the removed point's f value and the predictions using the two filters are computed, which constitute the real and imaginary parts of the complex-valued wavelet coefficient. This coefficient is then stored

Step Three. The integrals and the scaling function values (neighbouring coeffv values) are updated according to the filter L.

The algorithm continues until all points in mod are removed.

Value

`coeff`	matrix of detail and scaling coefficients in the wavelet decomposition of the signal; first column: real component, second column: imaginary component.
`lengthsremove`	vector of interval lengths corresponding to the points removed during the transform (in removelist).
`pointsin`	indices into X of the scaling coefficients in the wavelet decomposition. These are the indices of the X values which remain after all points in removelist have been predicted and removed. This has length nkeep.
`removelist`	a vector of indices into X of the lifted coefficients during the transform (in the order of removal).
`gamlist`	a list of all the prediction weights used at each step of the transform; each list entry is a matrix of two rows, corresponding to the filters `L` and `M`.
`alphalist`	a list of the update coefficients used in the update step of the decomposition.
`W`	The complex-valued lifting matrix associated to the transform.
`reo`	An index into the observations indicating a reordering to give `1:n`. This is reported for convenience for other functions, and is not intended for use by the user.
`coeffv`	vector of complex-valued detail and scaling coefficients in the wavelet decomposition of the signal; contains the same information as coeff.
`Ialpha`	Vector of "irregularity degree" measures corresponding to each lifting step of the transform. Note that this is returned for convenience in other functions, and is not intended for use by the user.

Author(s)

Matt Nunes, Marina Knight

References

Hamilton, J., Knight, Nunes, M. A. and Fryzlewicz (2018) Complex-valued wavelet lifting and applications. Technometrics, 69 (1), 48-60, DOI 10.1080/00401706.2017.1281846.

For related literature on the lifting methodology adopted in the technique, see

Nunes, M. A., Knight, M. I and Nason, G. P. (2006) Adaptive lifting for nonparametric regression. Stat. Comput. 16 (2), 143–159.

Knight, M. I. and Nason, G. P. (2009) A 'nondecimated' wavelet transform. Stat. Comput. 19 (1), 1–16.

Examples

    library(adlift)

    # construct an (irregular) observation grid
     x<-runif(256)

     #construct a signal
     f<-make.signal2("blocks",x=x)

     fwd<-fwtnppermC(x,f,LocalPred=AdaptPred,neigh=1,closest=FALSE)

     # have a look at the complex-valued coefficients and the removal trajectory:

     fwd$coeffv

     fwd$removelist	
library(adlift)

    # construct an (irregular) observation grid
     x<-runif(256)

     #construct a signal
     f<-make.signal2("blocks",x=x)

     fwd<-fwtnppermC(x,f,LocalPred=AdaptPred,neigh=1,closest=FALSE)

     # have a look at the complex-valued coefficients and the removal trajectory:

     fwd$coeffv

     fwd$removelist

Function to perform 'multiwavelet style' level-dependent soft thresholding for complex-valued wavelet coefficients

Description

This function uses chi^2 statistics similar to Barber and Nason (2004) to threshold wavelet coefficients based on their magnitude

Usage

mthreshC(coeffv, Sigma, rl, po, ali, verbose = FALSE)
mthreshC(coeffv, Sigma, rl, po, ali, verbose = FALSE)

Arguments

`coeffv`	A matrix of complex-valued wavelet coefficients (columns are real and imaginary parts of the coefficients respectively).
`Sigma`	An array of dimension `2 x 2 x n` describing the covariance between real and imaginary parts of the wavelet coefficients. In particular, `Sigma[,,i]` represents the covariance between real and imaginary parts of the `i`th lifted wavelet coefficient (see rl argument).
`rl`	The removelist (trajectory of lifted points) corresponding to a forward lifting transform.
`po`	A vector of indices describing the unlifted scaling coefficients in a forward lifting transform.
`ali`	A list of indices of observations, each entry corresponding to an 'artificial level' (finest to coarsest), see `artlev` for more details.
`verbose`	Indicates whether helpful messages should be printed to the console during the procedure.

Details

The procedure in Downie and Silverman (1998) or Barber and Nason (2004) makes use of the magnitude of wavelet coefficients to threshold them. In particular, the covariance between the components of the wavelet coefficients (contained in Sigma is taken into account to compute a thresholding statistic, the distribution of which is chi-squared_2 distributed, see cthresh for more details. These statistics are then compared with level-dependent universal thresholds computed by counting the numnber of coefficients in specific artificial levels.

Value

A list with the following components:

`chi`	the vector of chi-squared statistics used in the thresholding procedure.
`coeffvt`	the matrix of thresholded coefficients, columns representing the real and imaginary components respectively.

Author(s)

Matt Nunes, Marina Knight

References

Hamilton, J., Knight, M. I., Nunes, M. A. and Fryzlewicz (2018) Complex-valued wavelet lifting and applications. Technometrics, 60 (1), 48-60,DOI 10.1080/00401706.2017.1281846. Barber, S. and Nason, G. P. (2004) Real nonparametric regression using complex wavelets. J. Roy. Stat. Soc. B 66 (4), 927–939. Downie, T. R. and Silverman, B. W. (1998) The discrete multiple wavelet tranform and thresholding methods. IEEE Trans. Sig. Proc. 46 2558–2561.

Examples


library(adlift)

set.seed(100)

# construct an (irregular) sampling structure:

x<-sort(runif(200))

g<-make.signal2("bumps",x=x)

# generate IID noise with a particular sd
noise<-rnorm(200,0,sd=0.5)

f<-g+noise

# perform forward complex lifting transform

out<-fwtnppermC(x,f,LocalPred=LinearPred,neigh=1)

# have a look at some of the coefficients

out$coeffv[1:10]

# extract lifting matrix and induced lifting variances
W <- out$W

Gpre<-tcrossprod(W,Conj(W))

indsd<-sqrt(diag(Gpre))

# now estimate noise sd using the first artificial level:

al<-artlev(out$lengthsremove,out$removelist)

fine<-(out$coeffv/indsd)[al[[1]]]

varest<-mad(Re(fine))^2

# now compute coefficient covariance structure, see
# Hamilton et al. (2018), Appendix B

C = varest * tcrossprod(W)
    G = varest * Gpre
    P = Conj(G) - t(Conj(C)) 
    Sigma <- array(0, dim = c(2, 2, length(out$coeffv)))
    Sigma[1, 1, ] <- diag(Re(G + C)/2)
    Sigma[2, 2, ] <- diag(Re(G - C)/2)
    Sigma[1, 2, ] <- -diag(Im(G - C)/2)
    Sigma[2, 1, ] <- diag(Im(G + C)/2)

# now threshold complex coefficients according to this structure:

coeff.thresh<-mthreshC(out$coeffv,Sigma,out$removelist,out$pointsin,al)

# have a look at some of these coefficients

coeff.thresh$coeffv[1:10]

library(adlift)

set.seed(100)

# construct an (irregular) sampling structure:

x<-sort(runif(200))

g<-make.signal2("bumps",x=x)

# generate IID noise with a particular sd
noise<-rnorm(200,0,sd=0.5)

f<-g+noise

# perform forward complex lifting transform

out<-fwtnppermC(x,f,LocalPred=LinearPred,neigh=1)

# have a look at some of the coefficients

out$coeffv[1:10]

# extract lifting matrix and induced lifting variances
W <- out$W

Gpre<-tcrossprod(W,Conj(W))

indsd<-sqrt(diag(Gpre))

# now estimate noise sd using the first artificial level:

al<-artlev(out$lengthsremove,out$removelist)

fine<-(out$coeffv/indsd)[al[[1]]]

varest<-mad(Re(fine))^2

# now compute coefficient covariance structure, see
# Hamilton et al. (2018), Appendix B

C = varest * tcrossprod(W)
    G = varest * Gpre
    P = Conj(G) - t(Conj(C)) 
    Sigma <- array(0, dim = c(2, 2, length(out$coeffv)))
    Sigma[1, 1, ] <- diag(Re(G + C)/2)
    Sigma[2, 2, ] <- diag(Re(G - C)/2)
    Sigma[1, 2, ] <- -diag(Im(G - C)/2)
    Sigma[2, 1, ] <- diag(Im(G + C)/2)

# now threshold complex coefficients according to this structure:

coeff.thresh<-mthreshC(out$coeffv,Sigma,out$removelist,out$pointsin,al)

# have a look at some of these coefficients

coeff.thresh$coeffv[1:10]

Computes orthogonal filters

Description

Given a filter L, finds a second filter M, orthogonal to L and with unit norm

Usage

orthpredfilters(filter = c(0.5, 1, 0.5))
orthpredfilters(filter = c(0.5, 1, 0.5))

Arguments

filter

An initial filter L

Details

See Hamilton et al. (2018), section 2.2.

Value

A matrix with two rows, the first row corresponding to L, the second corresponding to the orthogonal filter M.

Warning

At present only works with odd length filters

Author(s)

Marina Knight, Matt Nunes

References

Hamilton, J., Nunes, M. A., Knight, M. I. and Fryzlewicz, P. (2018) Complex-valued wavelet lifting and applications. Technometrics, 60 (1), 48-60, DOI 10.1080/00401706.2017.1281846.

Examples


# create a vector representing a filter for one neighbour either side of a removed point
# (equally weighted):

L = c(0.5, 1, 0.5)

# now work out a unit-norm filter orthogonal to L

out <- orthpredfilters(L)

# M should be the second row:

out[2,]

# create a vector representing a filter for one neighbour either side of a removed point
# (equally weighted):

L = c(0.5, 1, 0.5)

# now work out a unit-norm filter orthogonal to L

out <- orthpredfilters(L)

# M should be the second row:

out[2,]

Package 'CNLTreg'

Help Index

Complex-Valued Wavelet Lifting for Signal Denoising

Description

Details

Author(s)

References

See Also

Performs 'nondecimated' complex-valued wavelet lifting for signal denoising

Description

Usage

Arguments

Details

Value

Warning

Author(s)

References

See Also

Examples

Denoises a signal using the complex-valued lifting transform and multivariate soft thresholding

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Denoises a signal using the complex-valued lifting transform and multivariate soft thresholding and heteroscedastic variance computation

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Forward complex wavelet lifting transform

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Function to perform 'multiwavelet style' level-dependent soft thresholding for complex-valued wavelet coefficients

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Computes orthogonal filters

Description

Usage

Arguments

Details

Value

Warning

Author(s)

References

See Also

Examples