Adaptive Weights Smoothing

Description

We provide a collection of R-functions implementing adaptive smoothing procedures in 1D, 2D and 3D. This includes the Propagation-Separation Approach to adaptive smoothing, the Intersecting Confidence Intervals (ICI), variational approaches and a non-local means filter. The package is described in detail in Polzehl J, Papafitsoros K, Tabelow K (2020). Patch-Wise Adaptive Weights Smoothing in R. Journal of Statistical Software, 95(6), 1-27. <doi:10.18637/jss.v095.i06>, Usage of the package in MR imaging is illustrated in Polzehl and Tabelow (2023), Magnetic Resonance Brain Imaging, 2nd Ed. Appendix A, Springer, Use R! Series. <doi:10.1007/978-3-031-38949-8>.

Details

The DESCRIPTION file:

Index of help topics:

ICIcombined             Adaptive smoothing by Intersection of
                        Confidence Intervals (ICI) using multiple
                        windows
ICIsmooth               Adaptive smoothing by Intersection of
                        Confidence Intervals (ICI)
ICIsmooth-class         Class '"ICIsmooth"'
TV_denoising            TV/TGV denoising of image data
aws                     AWS for local constant models on a grid
aws-class               Class '"aws"'
aws-package             Adaptive Weights Smoothing
aws.gaussian            Adaptive weights smoothing for Gaussian data
                        with variance depending on the mean.
aws.irreg               local constant AWS for irregular (1D/2D) design
aws.segment             Segmentation by adaptive weights for Gaussian
                        models.
awsLocalSigma           3D variance estimation
awsdata                 Extract information from an object of class aws
awssegment-class        Class '"awssegment"'
awstestprop             Propagation condition for adaptive weights
                        smoothing
awsweights              Generate weight scheme that would be used in an
                        additional aws step
binning                 Binning in 1D, 2D or 3D
extract-methods         Methods for Function 'extract' in Package 'aws'
gethani                 Auxiliary functions (for internal use)
kernsm                  Kernel smoothing on a 1D, 2D or 3D grid
kernsm-class            Class '"kernsm"'
lpaws                   Local polynomial smoothing by AWS
nlmeans                 NLMeans filter in 1D/2D/3D
paws                    Adaptive weigths smoothing using patches
plot-methods            Methods for Function 'plot' from package
                        'graphics' in Package 'aws'
print-methods           Methods for Function 'print' from package
                        'base' in Package 'aws'
qmeasures               Quality assessment for image reconstructions.
risk-methods            Compute risks characterizing the quality of
                        smoothing results
show-methods            Methods for Function 'show' in Package 'aws'
smooth3D                Auxiliary 3D smoothing routines
smse3ms                 Adaptive smoothing in orientation space SE(3)
summary-methods         Methods for Function 'summary' from package
                        'base' in Package 'aws'
vaws                    vector valued version of function 'aws' The
                        function implements the propagation separation
                        approach to nonparametric smoothing (formerly
                        introduced as Adaptive weights smoothing) for
                        varying coefficient likelihood models with
                        vector valued response on a 1D, 2D or 3D grid.
vpaws                   vector valued version of function 'paws' with
                        homogeneous covariance structure

Author(s)

Joerg Polzehl [aut, cre], Felix Anker [ctb]

Maintainer: Joerg Polzehl <joerg.polzehl@wias-berlin.de>

References

J. Polzehl, K. Tabelow (2019). Magnetic Resonance Brain Imaging: Modeling and Data Analysis Using R. Springer, Use R! series. Appendix A. Doi:10.1007/978-3-030-29184-6.

J. Polzehl, K. Papafitsoros, K. Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06.

J. Polzehl and V. Spokoiny (2006) Propagation-Separation Approach for Local Likelihood Estimation, Prob. Theory and Rel. Fields 135(3), 335-362. DOI:10.1007/s00440-005-0464-1.

J. Polzehl, V. Spokoiny, Adaptive Weights Smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol. 62 , (2000) , pp. 335–354. DOI:10.1111/1467-9868.00235.

V. Katkovnik, K. Egiazarian and J. Astola (2006) Local Approximation Techniques in Signal and Image Processing, SPIE Press Monograph Vol. PM 157

A. Buades, B. Coll and J. M. Morel (2006). A review of image denoising algorithms, with a new one. Simulation, 4, 490-530. DOI:10.1137/040616024.

Rudin, L.I., Osher, S. and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Phys. D, 60, 259-268. DOI: 10.1016/0167-2789(92)90242-F.

Bredies, K., Kunisch, K. and Pock, T. (2010). Total Generalized Variation. SIAM J. Imaging Sci., 3, 492-526. DOI:10.1137/090769521.

Adaptive smoothing by Intersection of Confidence Intervals (ICI) using multiple windows

Description

The function performs adaptive smoothing by Intersection of Confidence Intervals (ICI) using multiple windows as described in Katkovnik et al (2006)

Usage

ICIcombined(y, hmax, hinc = 1.45, thresh = NULL, kern = "Gaussian", m = 0,
            sigma = NULL, nsector = 1, symmetric = FALSE, presmooth = FALSE,
            combine = "weighted", unit = c("SD","FWHM"))

Arguments

Details

This mainly follows Chapter 6.2 in Katkovnik et al (2006).

Value

An object of class ICIsmooth

Author(s)

Joerg Polzehl polzehl@wias-berlin.de

References

J. Polzehl, K. Papafitsoros, K. Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06.

V. Katkovnik, K. Egiazarian and J. Astola, Local Approximation Techniques in Signal And Image Processing, SPIE Society of Photo-Optical Instrumentation Engin., 2006, PM157

See Also

ICIsmooth, ICIsmooth-class, kernsm

Adaptive smoothing by Intersection of Confidence Intervals (ICI)

Description

The function performs adaptive smoothing by Intersection of Confidence Intervals (ICI) as described in Katkovnik et al (2006)

Usage

ICIsmooth(y, hmax, hinc = 1.45, thresh = NULL, kern = "Gaussian", m = 0,
          sigma = NULL, nsector = 1, sector = 1, symmetric = FALSE,
          presmooth = FALSE, unit = c("SD","FWHM"))

Arguments

Details

This mainly follows Chapter 6.1 in Katkovnik et al (2006).

Value

An object of class ICIsmooth

Author(s)

Joerg Polzehl polzehl@wias-berlin.de

References

J. Polzehl, K. Papafitsoros, K. Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06.

V. Katkovnik, K. Egiazarian and J. Astola, Local Approximation Techniques in Signal And Image Processing, SPIE Society of Photo-Optical Instrumentation Engin., 2006, PM157

See Also

ICIcombined, ICIsmooth-class, kernsm

Class "ICIsmooth"

Description

The "ICIsmooth" class is used for objects obtained by functions ICIsmooth and ICIcombined.

Objects from the Class

Objects can be created by calls of the form new("ICIsmooth", ...) or by functions ICIsmooth and ICIcombined.

Slots

.Data:

Object of class "list", usually empty.

y:

Object of class "array" containing the original (response) data

dy:

Object of class "numeric" dimension attribute of y

x:

Object of class "numeric" if provided the design points

hmax:

Object of class "numeric" maximum bandwidth

hinc:

Object of class "numeric" initial bandwidth

thresh:

Object of class "numeric" threshold used for bandwidth selection

kern:

Object of class "character" kernel, can be any of c("Gaussian","Uniform","Triangle","Epanechnicov","Biweight","Triweight"). Defaults to kern="Gaussian".

m:

Object of class "integer" vector of length length(dy) determining the order of derivatives specified for the coordinate directios.

nsector:

Object of class "integer" number of sectors to use.

sector:

Object of class "integer" sector used.

symmetric:

Object of class "logical" sectors are symmetric with respect to the origin.

yhat:

Object of class "array" smoothed response variable

vhat:

Object of class "array" estimated variance of smoothed response variable

hbest:

Object of class "array" selected bandwidth(s))

sigma:

Object of class "numeric" estimated standard deviation of errors in y

call:

Object of class "call" that created the object.

Methods

extract

signature(x = "ICIsmooth"): ...

risk

signature(y = "ICIsmooth"): ...

plot

Method for Function ‘plot’ in Package ‘aws’.

show

Method for Function ‘show’ in Package ‘aws’.

print

Method for Function ‘print’ in Package ‘aws’.

summary

Method for Function ‘summary’ in Package ‘aws’.

Author(s)

Joerg Polzehl polzehl@wias-berlin.de

References

V. Katkovnik, K. Egiazarian and J. Astola, Local Approximation Techniques in Signal And Image Processing, SPIE Society of Photo-Optical Instrumentation Engin., 2006, PM157

See Also

ICIsmooth, ICIcombined, kernsm, aws

Examples

showClass("ICIsmooth")

TV/TGV denoising of image data

Description

Total variation and total generalized variation are classical energy minimizing methods for image denoising.

Usage

TV_denoising(datanoisy, alpha, iter = 1000, tolmean = 1e-06,
             tolsup = 1e-04, scale = 1, verbose=FALSE)
TGV_denoising(datanoisy, alpha, beta, iter = 1000, tolmean = 1e-06,
              tolsup = 1e-04, scale = 1, verbose=FALSE)
TV_denoising_colour(datanoisy, alpha, iter = 1000, tolmean = 1e-06,
                    tolsup = 1e-04, scale = 1, verbose=FALSE)
TGV_denoising_colour(datanoisy, alpha, beta, iter = 1000, tolmean = 1e-06,
                     tolsup = 1e-04, scale = 1, verbose=FALSE)

Arguments

Details

Reimplementation of original matlab code by Kostas Papafitsoros (WIAS).

Value

TV/TGV reconstructed image data (2D array)

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de, https://www.wias-berlin.de/people/polzehl/

References

J. Polzehl, K. Papafitsoros, K. Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06.

Rudin, L.I., Osher, S. and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Phys. D, 60, 259-268. DOI: 10.1016/0167-2789(92)90242-F.

Bredies, K., Kunisch, K. and Pock, T. (2010). Total Generalized Variation. SIAM J. Imaging Sci., 3, 492-526. DOI:10.1137/090769521.

Auxiliary functions (for internal use)

Description

Function gethani determines a bandwidth that leads to, for the specified kernel, a variance reduction for a non-adaptive kernel estimate by a factor of value. getvofh calculates the sum of location weights for a given bandwidth vector and kernel. sofmchi precomputes the variance of a non-central chi distribution with 2*L degrees of freedom as a function of the noncentrality parameter for an interval c(0,to). Functions residualVariance and residualSpatialCorr are used in package fmri to calculate variances and spatial correlations from residual objects.

Usage

gethani(x, y, lkern, value, wght, eps = 0.01)
getvofh(bw, lkern, wght)
sofmchi(L, to = 50, delta = 0.01)
residualVariance(residuals, mask, resscale = 1, compact = FALSE)
residualSpatialCorr(residuals, mask, lags = c(5, 5, 3), compact = FALSE)

Arguments

Details

These are auxiliary functions not to be used by the user. They are only exported to be available for internal use in packages fmri, dti, qMRI and adimpro.

Value

gethani returns a vector of bandwidths, getvofh returns the variance reduction that would be obtained with a kernel estimate employing the specified kernel and bandwidth, sofmchi returns a list with, e.g., components ncp and s2 containing vectors of noncentralityparameter values and corresponding variances, respectively, for the specified noncentral Chi distribution, residualVariance returns a vector (compact==TRUE) or array(compact==FALSE) of voxelwise residual variances, residualSpatialCorr returns an array of dimension lags containing spatial correlations.

Note

These functions are for internal use only. They are only exported to be available in other packages.

Author(s)

Joerg Polzehl polzehl@wias-berlin.de

AWS for local constant models on a grid

Description

The function implements the propagation separation approach to nonparametric smoothing (formerly introduced as Adaptive weights smoothing) for varying coefficient likelihood models on a 1D, 2D or 3D grid. For "Gaussian" models, i.e. regression with additive "Gaussian" errors, a homoskedastic or heteroskedastic model is used depending on the content of sigma2

Usage

aws(y,hmax=NULL, mask=NULL, aws=TRUE, memory=FALSE, family="Gaussian",
                lkern="Triangle", aggkern="Uniform",
                sigma2=NULL, shape=NULL, scorr=0, spmin=0.25,
		            ladjust=1,wghts=NULL,u=NULL,graph=FALSE,demo=FALSE,
                testprop=FALSE,maxni=FALSE)

Arguments

Details

The function implements the propagation separation approach to nonparametric smoothing (formerly introduced as Adaptive weights smoothing) for varying coefficient likelihood models on a 1D, 2D or 3D grid. For "Gaussian" models, i.e. regression with additive "Gaussian" errors, a homoskedastic or heteroskedastic model is used depending on the content of sigma2. aws==FALSE provides the stagewise aggregation procedure from Belomestny and Spokoiny (2004). memory==FALSE provides Adaptive weights smoothing without control by stagewise aggregation.

The essential parameter in the procedure is a critical value lambda. This parameter has an interpretation as a significance level of a test for equivalence of two local parameter estimates. Optimal values mainly depend on the choosen family. Values set internally are choosen to fulfil a propagation condition, i.e. in case of a constant (global) parameter value and large hmax the procedure provides, with a high probability, the global (parametric) estimate. More formally we require the parameter lambda to be specified such that \bf{E} |\hat{\theta}^k - \theta| \le (1+\alpha) \bf{E} |\tilde{\theta}^k - \theta| where \hat{\theta}^k is the aws-estimate in step k and \tilde{\theta}^k is corresponding nonadaptive estimate using the same bandwidth (lambda=Inf). The value of lambda can be adjusted by specifying the factor ladjust. Values ladjust>1 lead to an less effective adaptation while ladjust<<1 may lead to random segmentation of, with respect to a constant model, homogeneous regions.

The numerical complexity of the procedure is mainly determined by hmax. The number of iterations is approximately Const*d*log(hmax)/log(1.25) with d being the dimension of y and the constant depending on the kernel lkern. Comlexity in each iteration step is Const*hakt*n with hakt being the actual bandwith in the iteration step and n the number of design points. hmax determines the maximal possible variance reduction.

Value

returns anobject of class aws with slots

Note

use setCores='number of threads' to enable parallel execution.

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de, https://www.wias-berlin.de/people/polzehl/

References

J. Polzehl, K. Tabelow (2019). Magnetic Resonance Brain Imaging: Modeling and Data Analysis Using R. Springer, Use R! series. Appendix A. Doi:10.1007/978-3-030-29184-6.

J. Polzehl, K. Papafitsoros, K. Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06.

J. Polzehl, V. Spokoiny, Adaptive Weights Smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol. 62 , (2000) , pp. 335–354. DOI:10.1111/1467-9868.00235.

J. Polzehl, V. Spokoiny, Propagation-separation approach for local likelihood estimation, Probab. Theory Related Fields 135 (3), (2006) , pp. 335–362. DOI:10.1007/s00440-005-0464-1.

See Also

See also paws, lpaws, vaws,link{awsdata}, aws.irreg, aws.gaussian

Examples

require(aws)
# 1D local constant smoothing
## Not run: demo(aws_ex1)
## Not run: demo(aws_ex2)
# 2D local constant smoothing
## Not run: demo(aws_ex3)

Class "aws"

Description

The "aws" class is used for objects obtained by functions aws, lpaws, aws.irreg and aws.gaussian.

Objects from the Class

Objects are created by calls to functions aws, lpaws, aws.irreg and aws.gaussian.

Slots

.Data:

Object of class "list", usually empty.

y:

Object of class "array" containing the original (response) data

dy:

Object of class "numeric" dimension attribute of y

nvec:

Object of class "integer" leading dimension of y in vector valued data.

x:

Object of class "numeric" if provided the design points

ni:

Object of class "numeric" sum of weights used in final estimate

mask:

Object of class "logical" mask of design points where computations are performed

theta:

Object of class "array" containes the smoothed object and in case of function lpaws its derivatives up to the specified degree. Dimension is dim(theta)=c(dy,p)

hseq:

Sequence of bandwidths employed.

mae:

Object of class "numeric" Mean absolute error with respect to array in argument u if provided.

psnr:

Object of class "numeric" Peak Signal to Noise Ratio (PSNR) with respect to array in argument u if provided.

var:

Object of class "numeric" pointwise variance of theta[...,1]

xmin:

Object of class "numeric" min of x in case of irregular design

xmax:

Object of class "numeric" max of x in case of irregular design

wghts:

Object of class "numeric" weights used in location penalty for different coordinate directions, corresponds to ratios of distances in coordinate directions 2 and 3 to and distance in coordinate direction 1.

degree:

Object of class "integer" degree of local polynomials used in function lpaws

hmax:

Object of class "numeric" maximal bandwidth

sigma2:

Object of class "numeric" estimated error variance

scorr:

Object of class "numeric" estimated spatial correlation

family:

Object of class "character" distribution of y, can be any of c("Gaussian","Bernoulli","Poisson","Exponential", "Volatility","Variance")

shape:

Object of class "numeric" possible shape parameter of distribution of y

lkern:

Object of class "integer" location kernel, can be any of c("Triangle","Quadratic","Cubic","Plateau","Gaussian"), defauts to "Triangle"

lambda:

Object of class "numeric" scale parameter used in adaptation

ladjust:

Object of class "numeric" factor to adjust scale parameter with respect to its predetermined default.

aws:

Object of class "logical" Adaptation by Propagation-Separation

memory:

Object of class "logical" Adaptation by Stagewise Aggregation

homogen:

Object of class "logical" detect regions of homogeneity (used to speed up the calculations)

earlystop:

Object of class "logical" further speedup in function lpaws estimates are fixed if sum of weigths does not increase with iterations.

varmodel:

Object of class "character" variance model used in function aws.gaussian

vcoef:

Object of class "numeric" estimates variance parameters in function aws.gaussian

call:

Object of class "call" that created the object.

Methods

extract

signature(x = "aws"): ...

risk

signature(y = "aws"): ...

plot

Method for Function ‘plot’ in Package ‘aws’.

show

Method for Function ‘show’ in Package ‘aws’.

print

Method for Function ‘print’ in Package ‘aws’.

summary

Method for Function ‘summary’ in Package ‘aws’.

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de

References

Joerg Polzehl, Vladimir Spokoiny, Adaptive Weights Smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol. 62 , (2000) , pp. 335–354

Joerg Polzehl, Vladimir Spokoiny, Propagation-separation approach for local likelihood estimation, Probab. Theory Related Fields 135 (3), (2006) , pp. 335–362.

See Also

aws, lpaws, aws.irreg, aws.gaussian

Examples

showClass("aws")

Adaptive weights smoothing for Gaussian data with variance depending on the mean.

Description

The function implements an semiparametric adaptive weights smoothing algorithm designed for regression with additive heteroskedastic Gaussian noise. The noise variance is assumed to depend on the value of the regression function. This dependence is modeled by a global parametric (polynomial) model.

Usage

aws.gaussian(y, hmax = NULL, hpre = NULL, aws = TRUE, memory = FALSE,
             varmodel = "Constant", lkern = "Triangle",
             aggkern = "Uniform", scorr = 0, mask=NULL, ladjust = 1,
             wghts = NULL, u = NULL, varprop = 0.1, graph = FALSE, demo = FALSE)

Arguments

Details

The function implements the propagation separation approach to nonparametric smoothing (formerly introduced as Adaptive weights smoothing) for varying coefficient likelihood models on a 1D, 2D or 3D grid. In contrast to function aws observations are assumed to follow a Gaussian distribution with variance depending on the mean according to a specified global variance model. aws==FALSE provides the stagewise aggregation procedure from Belomestny and Spokoiny (2004). memory==FALSE provides Adaptive weights smoothing without control by stagewise aggregation.

The essential parameter in the procedure is a critical value lambda. This parameter has an interpretation as a significance level of a test for equivalence of two local parameter estimates. Values set internally are choosen to fulfil a propagation condition, i.e. in case of a constant (global) parameter value and large hmax the procedure provides, with a high probability, the global (parametric) estimate. More formally we require the parameter lambda to be specified such that \bf{E} |\hat{\theta}^k - \theta| \le (1+\alpha) \bf{E} |\tilde{\theta}^k - \theta| where \hat{\theta}^k is the aws-estimate in step k and \tilde{\theta}^k is corresponding nonadaptive estimate using the same bandwidth (lambda=Inf). The value of lambda can be adjusted by specifying the factor ladjust. Values ladjust>1 lead to an less effective adaptation while ladjust<<1 may lead to random segmentation of, with respect to a constant model, homogeneous regions.

The numerical complexity of the procedure is mainly determined by hmax. The number of iterations is approximately Const*d*log(hmax)/log(1.25) with d being the dimension of y and the constant depending on the kernel lkern. Comlexity in each iteration step is Const*hakt*n with hakt being the actual bandwith in the iteration step and n the number of design points. hmax determines the maximal possible variance reduction.

Value

returns anobject of class aws with slots

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de, https://www.wias-berlin.de/people/polzehl/

References

Joerg Polzehl, Vladimir Spokoiny, Adaptive Weights Smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol. 62 , (2000) , pp. 335–354

Joerg Polzehl, Vladimir Spokoiny, Propagation-separation approach for local likelihood estimation, Probab. Theory Related Fields 135 (3), (2006) , pp. 335–362.

Joerg Polzehl, Vladimir Spokoiny, in V. Chen, C.; Haerdle, W. and Unwin, A. (ed.) Handbook of Data Visualization Structural adaptive smoothing by propagation-separation methods Springer-Verlag, 2008, 471-492

See Also

See also aws, link{awsdata}, aws.irreg

Examples

require(aws)

local constant AWS for irregular (1D/2D) design

Description

The function implements the propagation separation approach to nonparametric smoothing (formerly introduced as Adaptive weights smoothing) for varying coefficient Gaussian models on a 1D or 2D irregulat design. The function allows for a paramertic (polynomial) mean-variance dependence.

Usage

aws.irreg(y, x, hmax = NULL, aws=TRUE, memory=FALSE, varmodel = "Constant",
          lkern = "Triangle", aggkern = "Uniform", sigma2 = NULL, nbins = 100,
          hpre = NULL, henv = NULL, ladjust =1, varprop = 0.1, graph = FALSE)

Arguments

Details

Data are first binned (1D/2D), then aws is performed on all datapoints within distance <= henv of nonempty bins.

Value

returns anobject of class aws with slots

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de

References

J. Polzehl, V. Spokoiny, in V. Chen, C.; Haerdle, W. and Unwin, A. (ed.) Handbook of Data Visualization Structural adaptive smoothing by propagation-separation methods. Springer-Verlag, 2008, 471-492. DOI:10.1007/978-3-540-33037-0_19.

See Also

See also lpaws, link{awsdata}, lpaws

Examples

require(aws)
# 1D local constant smoothing
## Not run: demo(irreg_ex1)
# 2D local constant smoothing
## Not run: demo(irreg_ex2)

Segmentation by adaptive weights for Gaussian models.

Description

The function implements a modification of the adaptive weights smoothing algorithm for segmentation into three classes. The

Usage

aws.segment(y, level, delta = 0, hmax = NULL, hpre = NULL, mask =NULL,
            varmodel = "Constant", lkern = "Triangle", scorr = 0, ladjust = 1,
            wghts = NULL, u = NULL, varprop = 0.1, ext = 0, graph = FALSE,
            demo = FALSE, fov=NULL)

Arguments

Details

The image is segmented into three parts by performing multiscale tests of the hypotheses H1 value >= level - delta and H2 value <= level + delta. Pixel where the first hypotesis is rejected are classified as -1 (segment 1) while rejection of H2 results in classification 1 (segment 3). Pixel where neither H1 or H2 are rejected ar assigned to a value 0 (segment 2). Critical values for the tests are adjusted for smoothness at the different scales inspected in the iteration process using results from multiscale testing, see e.g. Duembgen and Spokoiny (2001). Critical values also depend on the size of the region of interest specified in parameter fov.

Within segment 2 structural adaptive smoothing is performed while if a pair of pixel belongs to segment 1 or segment 3 the corresponding weight will be nonadaptive.

Value

returns anobject of class aws with slots

Note

This function is still experimental and may be changes considerably in future.

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de, https://www.wias-berlin.de/people/polzehl/

References

J. Polzehl, H.U. Voss, K. Tabelow (2010). Structural adaptive segmentation for statistical parametric mapping, NeuroImage, 52, pp. 515–523. DOI:10.1016/j.neuroimage.2010.04.241

Duembgen, L. and Spokoiny, V. (2001). Multiscale testing of qualitative hypoteses. Ann. Stat. 29, 124–152.

Polzehl, J. and Spokoiny, V. (2006). Propagation-Separation Approach for Local Likelihood Estimation. Probability Theory and Related Fields. 3 (135) 335 - 362. DOI:10.1007/s00440-005-0464-1

See Also

aws, aws.gaussian

Examples

require(aws)

3D variance estimation

Description

Functions for 3D variance estimation. awsLocalSigma implements the local adaptive variance estimation procedure introduced in Tabelow, Voss and Polzehl (2015). awslinsd uses a parametric model for varianc/mesn dependence. Functions AFLocalSigma and estGlobalSigma implement various proposals for local and global variance estimates from Aja-Fernandez (2009, 2013) and a global variant of the approach from Tabelow, Voss and Polzehl (2015).

Usage

awsLocalSigma(y, steps, mask, ncoils, vext = c(1, 1), lambda = 5,
    minni = 2, hsig = 5, sigma = NULL, family = c("NCchi", "Gauss"),
    verbose = FALSE, trace = FALSE, u = NULL)
awslinsd(y, hmax = NULL, hpre = NULL, h0 = NULL, mask = NULL,
    ladjust = 1, wghts = NULL, varprop = 0.1, A0, A1)
AFLocalSigma(y, ncoils, level = NULL, mask = NULL, h = 2, hadj = 1,
    vext = c(1, 1))
estGlobalSigma(y, mask = NULL, ncoils = 1, steps = 16, vext = c(1, 1),
    lambda = 20, hinit = 2, hadj = 1, q = 0.25, level = NULL,
    sequence = FALSE, method = c("awsVar", "awsMAD", "AFmodevn",
                "AFmodem1chi", "AFbkm2chi", "AFbkm1chi"))
estimateSigmaCompl(magnitude, phase, mask, kstar = 20, kmin = 8, hsig = 5,
        lambda = 12, verbose = TRUE)

Arguments

Value

all functions return lists with variance estimates in component sigma

Author(s)

J\"org Polzehl polzehl@wias-berlin.de

References

K. Tabelow, H.U. Voss, J. Polzehl, Local estimation of the noise level in MRI using structural adaptation, Medical Image Analysis, 20 (2015), 76–86. DOI:10.1016/j.media.2014.10.008.

S. Aja-Fernandez, V. Brion, A. Tristan-Vega, Effective noise estimation and filtering from correlated multiple-coil MR data. Magn Reson Imaging, 31 (2013), 272-285. DOI:10.1016/j.mri.2012.07.006

S. Aja-Fernandez, A. Tristan-Vega, C. Alberola-Lopez, Noise estimation in single- and multiple-coil magnetic resonance data based on statistical models. Magn Reson Imaging, 27 (2009), 1397-1409. DOI:10.1016/j.mri.2009.05.025.

Extract information from an object of class aws

Description

Extract data and estimates from an object of class aws

Usage

awsdata(awsobj, what)

Arguments

Details

The returned object is formatted as an array if appropriate. The returned object may be NULL if the information is not available.

Value

an vector or array containing the specified information.

Author(s)

Joerg Polzehl polzehl@wias-berlin.de

References

Joerg Polzehl, Vladimir Spokoiny, Adaptive Weights Smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol. 62 , (2000) , pp. 335–354

Joerg Polzehl, Vladimir Spokoiny, Propagation-separation approach for local likelihood estimation, Probab. Theory Related Fields 135 (3), (2006) , pp. 335–362.

Joerg Polzehl, Vladimir Spokoiny, in V. Chen, C.; Haerdle, W. and Unwin, A. (ed.) Handbook of Data Visualization Structural adaptive smoothing by propagation-separation methods Springer-Verlag, 2008, 471-492

See Also

link{awsdata},aws, aws.irreg

Examples

require(aws)
# 1D local constant smoothing
## Not run: demo(aws_ex1)
## Not run: demo(aws_ex2)
# 2D local constant smoothing
## Not run: demo(aws_ex3)
# 1D local polynomial smoothing
## Not run: demo(lpaws_ex1)
# 2D local polynomial smoothing
## Not run: demo(lpaws_ex2)
# 1D irregular design
## Not run: demo(irreg_ex1)
# 2D irregular design 
## Not run: demo(irreg_ex2)

Class "awssegment"

Description

The "aws" class is used for objects obtained by functions aws.segment

Objects from the Class

Objects are created by calls to functions aws.segment

Slots

.Data:

Object of class "list", usually empty.

y:

Object of class "array" containing the original (response) data

dy:

Object of class "numeric" dimension attribute of y

x:

Object of class "numeric" if provided the design points

ni:

Object of class "numeric" sum of weights used in final estimate

mask:

Object of class "logical" mask of design points where computations are performed

segment:

Object of class "array" segmentation results (3 segments coded by c(-1, 0, 1))

level:

Object of class "numeric" center of segment 0

delta:

Object of class "numeric" half width of segment 0

theta:

Object of class "array" ~~

theta:

Object of class "array" containes the smoothed object and in case of function lpaws its derivatives up to the specified degree. Dimension is dim(theta)=c(dy,p)

mae:

Object of class "numeric" Mean absolute error with respect to array in argument u if provided.

var:

Object of class "numeric" pointwise variance of theta[...,1]

xmin:

Object of class "numeric" not used

xmax:

Object of class "numeric" not used

wghts:

Object of class "numeric" weights used in location penalty for different coordinate directions

degree:

not used

hmax:

Object of class "numeric" maximal bandwidth

sigma2:

Object of class "numeric" estimated error variance

scorr:

Object of class "numeric" estimated spatial correlation

family:

Object of class "character" distribution of y, can be any of c("Gaussian","Bernoulli","Poisson","Exponential", "Volatility","Variance")

shape:

Object of class "numeric" possible shape parameter of distribution of y

lkern:

Object of class "integer" location kernel, can be any of c("Triangle","Quadratic","Cubic","Plateau","Gaussian"), defauts to "Triangle"

lambda:

Object of class "numeric" scale parameter used in adaptation

ladjust:

Object of class "numeric" factor to adjust scale parameter with respect to its predetermined default.

aws:

Object of class "logical" Adaptation by Propagation-Separation

memory:

Object of class "logical" Adaptation by Stagewise Aggregation

homogen:

Object of class "logical" detect regions of homogeneity (used to speed up the calculations) currently FALSE

earlystop:

Object of class "logical" currently FALSE

varmodel:

Object of class "character" variance model used currently "Gaussian"

vcoef:

Object of class "numeric" contains NULL

call:

Object of class "call" that created the object.

Methods

extract

signature(x = "awssegment"): ...

plot

signature(x = "awssegment"): ...

print

signature(x = "awssegment"): ...

risk

signature(y = "awssegment"): ...

show

signature(object = "awssegment"): ...

summary

signature(object = "awssegment"): ...

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de

See Also

aws.segment

Examples

showClass("awssegment")

Propagation condition for adaptive weights smoothing

Description

The function enables testing of the propagation condition in order to select appropriate values for the parameter lambda in function aws.

Usage

awstestprop(dy, hmax, theta = 1, family = "Gaussian", lkern = "Triangle",
            aws = TRUE, memory = FALSE, shape = 2, homogeneous=TRUE, varadapt=FALSE,
            ladjust = 1, spmin=0.25, seed = 1, minlevel=1e-6, maxz=25, diffz=.5,
            maxni=FALSE, verbose=FALSE)
pawstestprop(dy, hmax, theta = 1, family = "Gaussian", lkern = "Triangle",
             aws = TRUE, patchsize=1, shape = 2,
             ladjust = 1, spmin = 0.25, seed = 1, minlevel = 1e-6,
             maxz = 25, diffz = .5, maxni = FALSE, verbose = FALSE)

Arguments

Details

Estimates exceedence probabilities

Results for intermediate steps are provided as contour plots. For a good choice of lambda (ladjust) the contours up to probabilities of 1e-5 should be vertical.

Value

A list with components

Author(s)

Joerg Polzehl polzehl@wias-berlin.de

References

S. Becker, P. Mathe, Electron. J. Statist. (2013), 2702-2736, doi:10.1214/13-EJS860

See Also

aws

Generate weight scheme that would be used in an additional aws step

Description

Utility function to create a weighting scheme for an additional aws step. Inteded to be used for illustrations only.

Usage

awsweights(awsobj, spmin = 0.25, inx = NULL)

Arguments

Value

an array of either dimension awsobj@dy x number of points or awsobj@dy x awsobj@dy

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de, https://www.wias-berlin.de/people/polzehl/

References

Joerg Polzehl, Vladimir Spokoiny, Adaptive Weights Smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol. 62 , (2000) , pp. 335–354

Joerg Polzehl, Vladimir Spokoiny, Propagation-separation approach for local likelihood estimation, Probab. Theory Related Fields 135 (3), (2006) , pp. 335–362.

Joerg Polzehl, Kostas Papafitsoros, Karsten Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06.

See Also

See also aws

Binning in 1D, 2D or 3D

Description

The function performs a binning in 1D, 2D or 3D.

Usage

binning(x, y, nbins, xrange = NULL)

Arguments

Value

A list with components

Note

This function has been adapted from the code of function binning in package sm.

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de

See Also

See Also as aws.irreg

Methods for Function extract in Package aws

Description

The method extract and/or compute specified statistics from object of class "aws", "awssegment", ICIsmooth and "kernsm".

Usage

  ## S4 method for signature 'aws'
extract(x, what="y")
  ## S4 method for signature 'awssegment'
extract(x, what="y")
  ## S4 method for signature 'ICIsmooth'
extract(x, what="y")
  ## S4 method for signature 'kernsm'
extract(x, what="y")

Arguments

Methods

signature(x = "ANY")

Returns a message that method extract is not defined.

signature(x = "aws")

Returns a list with components containing the requested statistics. Component names correspond to tolower(what)

signature(x = "awssegment")

Returns a list with components containing the requested statistics. Component names correspond to tolower(what)

signature(x = "ICIsmooth")

Returns a list with components containing the requested statistics. Component names correspond to tolower(what).

signature(x = "kernsm")

Returns a list with components containing the requested statistics. Component names correspond to tolower(what).

Kernel smoothing on a 1D, 2D or 3D grid

Description

Performs Kernel smoothing on a 1D, 2D or 3D grid by fft

Usage

kernsm(y, h = 1, kern = "Gaussian", m = 0, nsector = 1, sector = 1,
       symmetric = FALSE, unit = c("SD","FWHM"))

Arguments

Details

In case of any(m>0) derivative kernels are generated and applied for the corresponding coordinate directions. If nsector>1 the support of the kernel is restricted to a circular sector determined by sector.

Value

An object of class kernsm

Author(s)

Joerg Polzehl polzehl@wias-berlin.de

References

J. Polzehl, K. Papafitsoros, K. Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06 .

V. Katkovnik, K. Egiazarian and J. Astola, Local Approximation Techniques in Signal And Image Processing, SPIE Society of Photo-Optical Instrumentation Engin., 2006, PM157

See Also

kernsm-class, ICIsmooth,ICIcombined

Class "kernsm"

Description

This class refers to objects created by function kernsm. These objects contain

Objects from the Class

Objects can be created by calls of the form new("kernsm", ...). they are usually created by a call to function{kernsm}.

Slots

.Data:

Object of class "list", usually empty.

y:

Object of class "array" containing the response in nonparametric regression. The design is assumed to be a 1D, 2D or 3D grid, with dimensionality determined by dim(y).

dy:

Object of class "numeric" containing dim(y).

x:

Object of class "numeric" currently not used.

h:

Object of class "numeric" containing the bandwidth employed.

kern:

Object of class "character" determining the kernel that was used, can be one of c("Gaussian","Uniform","Triangle","Epanechnikov","Biweight","Triweight")

m:

Object of class "integer" with length length(dy) determining the order of derivatives in the corresponding coordinate directions. If m[i6>0] a dirivative kernel derived from kern has been used for the corresponding coordinate direction.

nsector:

Object of class "integer". If nsector>1 positive weights are restricted to a segment of a circle (1D or 2D only). The segment is given by sector.

sector:

Object of class "integer" containing the number of the segment used in case of nsector>1

symmetric:

Object of class "logical" determines if the sector is mirrored at the origin.

yhat:

Object of class "array" with same size and dimension as y providing the convolution of y with the chosen kernel.

vred:

Object of class "array" Variance reduction achieved by convolution assuming independence.

call:

Object of class "function", call that created the object.

Methods

extract

signature(x = "aws"): ...

risk

signature(y = "aws"): ...

plot

Method for Function ‘plot’ in Package ‘aws’.

show

Method for Function ‘show’ in Package ‘aws’.

print

Method for Function ‘print’ in Package ‘aws’.

summary

Method for Function ‘summary’ in Package ‘aws’.

Author(s)

J\"org Polzehl polzehl@wias-berlin.de

See Also

kernsm, ICIsmooth, ICIcombined, ICIsmooth

Examples

showClass("kernsm")

Local polynomial smoothing by AWS

Description

The function allows for structural adaptive smoothing using a local polynomial (degree <=2) structural assumption. Response variables are assumed to be observed on a 1 or 2 dimensional regular grid.

Usage

lpaws(y, degree = 1, hmax = NULL, aws = TRUE, memory = FALSE, lkern = "Triangle",
      homogen = TRUE, earlystop = TRUE, aggkern = "Uniform", sigma2 = NULL,
      hw = NULL, ladjust = 1, u = NULL, graph = FALSE, demo = FALSE)

Arguments

Value

returns anobject of class aws with slots

Note

If you specify graph=TRUE for 2D problems avoid using the default X11() on systems build with cairo, use X11(type="Xlib") instead (faster by a factor of 30).

Author(s)

Joerg Polzehl polzehl@wias-berlin.de

References

J. Polzehl, K. Papafitsoros, K. Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06 .

J. Polzehl, V. Spokoiny, in V. Chen, C.; Haerdle, W. and Unwin, A. (ed.) Handbook of Data Visualization Structural adaptive smoothing by propagation-separation methods. Springer-Verlag, 2008, 471-492. DOI:10.1007/978-3-540-33037-0_19.

See Also

link{awsdata},aws, aws.irreg

Examples

library(aws)
# 1D local polynomial smoothing
## Not run: demo(lpaws_ex1)
# 2D local polynomial smoothing
## Not run: demo(lpaws_ex2)

NLMeans filter in 1D/2D/3D

Description

Implements the Non-Local-Means Filter of Buades et al 2005

Usage

nlmeans(x, lambda, sigma, patchhw = 1, searchhw = 7, pd = NULL)

Arguments

Details

The implementation follows the description of the Non-Local-Means Filter of Buades et al 2005 on http://www.numerical-tours.com/matlab/denoisingadv_6_nl_means/#biblio that incorporates dimension reduction for patch comparisons by PCA.

Value

A list of class "nlmeans" with components

Note

use setCores='number of threads' to enable parallel execution.

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de, https://www.wias-berlin.de/people/polzehl/

References

J. Polzehl, K. Papafitsoros, K. Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06 .

A. Buades, B. Coll and J. M. Morel (2006). A review of image denoising algorithms, with a new one. Simulation, 4, 490-530. DOI:10.1137/040616024.

http://www.numerical-tours.com/matlab/denoisingadv_6_nl_means/#biblio

Adaptive weigths smoothing using patches

Description

The function implements a version the propagation separation approach that uses patches instead of individuel voxels for comparisons in parameter space. Functionality is analog to function aws. Using patches allows for an improved handling of locally smooth functions and in 2D and 3D for improved smoothness of discontinuities at the expense of increased computing time.

Usage

paws(y, hmax = NULL, mask=NULL, onestep = FALSE, aws = TRUE, family = "Gaussian",
     lkern = "Triangle", aggkern = "Uniform", sigma2 = NULL, shape = NULL,
     scorr = 0, spmin = 0.25, ladjust = 1, wghts = NULL, u = NULL,
     graph = FALSE, demo = FALSE, patchsize = 1)

Arguments

Details

see aws. The procedure is supposed to produce superior results if the assumption of a local constant image is violated or if smooothness of discontinuities is desired.

Value

returns an object of class aws with slots

Note

use setCores='number of threads' to enable parallel execution.

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de, https://www.wias-berlin.de/people/polzehl/

References

J. Polzehl, K. Tabelow (2019). Magnetic Resonance Brain Imaging: Modeling and Data Analysis Using R. Springer, Use R! series. Appendix A. Doi:10.1007/978-3-030-29184-6.

J. Polzehl, K. Papafitsoros, K. Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06 .

See Also

See also aws, lpaws, vpaws,link{awsdata}

Examples

## Not run: 
setCores(2)
y <- array(rnorm(64^3),c(64,64,64))
yhat <- paws(y,hmax=6)

## End(Not run)

Methods for Function ‘plot’ from package 'graphics' in Package ‘aws’

Description

Visualization of objects of class "aws", "awsswgment", "kernsm" and "ICIsmooth"

Methods

signature(x = "ANY")

Generic function: see plot.

signature(x = "aws")

Visualization of objects of class "aws"

signature(x = "awssegment")

Visualization of objects of class "awssegment"

signature(x = "ICIsmooth")

Visualization of objects of class "ICIsmooth"

signature(x = "kernsm")

Visualization of objects of class "kernsm"

Author(s)

J\"org Polzehl polzehl@wias-berlin.de

See Also

aws, awssegment, ICIsmooth kernsm

Methods for Function ‘print’ from package 'base' in Package ‘aws’

Description

The function provides information on data dimensions, creation of the object and existing slot-names for objects of class "aws", "awssegment", "ICIsmooth" and "kernsm"

Methods

signature(x = "ANY")

Generic function: see print.

signature(x = "aws")

Provide information on data dimensions, creation of the object and existing slot-names for objects of class "aws"

signature(x = "awssegment")

Provide information on data dimensions, creation of the object and existing slot-names for objects of class "awssegment"

signature(x = "ICIsmooth")

Provide information on data dimensions, creation of the object and existing slot-names for objects of class "ICIsmooth"

signature(x = "kernsm")

Provide information on data dimensions, creation of the object and existing slot-names for objects of class "kernsm"

Author(s)

J\"org Polzehl polzehl@wias-berlin.de

See Also

aws, awssegment, ICIsmooth kernsm

Quality assessment for image reconstructions.

Description

Computes selected criteria for quality assessments of

Usage

qmeasures(img, ref,
  which = c("PSNR", "MAE", "MSE", "RMSE", "SSIM", "MAGE", "RMSGE"),
  mask = FALSE)

Arguments

Details

Calculates specified quality indices.

Value

A vector with names as specified in which.

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de, https://www.wias-berlin.de/people/polzehl/

Compute risks characterizing the quality of smoothing results

Description

Methods function risk in package aws. For an given array u the following statistics are computed : Root Mean Squared Error RMSE <- sqrt(mean((y-u)^2)), Signal to Noise Ratio SNR <- 10*log(mean(u^2)/MSE,10), Peak Signal to Noise Ratio PSNR <- 10*log(max(u^2)/MSE,10), Mean Absolute Error MAE <- mean(abs(y-u)), Maximal Absolute Error MaxAE <- max(abs(y-u)), Universal Image Quality Index (UIQI) (Wang and Bovik (2002)).

Usage

  ## S4 method for signature 'array'
risk(y, u=0)
  ## S4 method for signature 'aws'
risk(y, u=0)
  ## S4 method for signature 'awssegment'
risk(y, u=0)
  ## S4 method for signature 'ICIsmooth'
risk(y, u=0)
  ## S4 method for signature 'kernsm'
risk(y, u=0)
  ## S4 method for signature 'numeric'
risk(y, u=0)

Arguments

Methods

signature(y = "ANY")

The method extract and/or compute specified statistics from object of class

signature(y = "array")

Returns a list with components RMSE, SNR, PSNR, MAE, MaxAE, UIQI

signature(y = "aws")

Returns a list with components RMSE, SNR, PSNR, MAE, MaxAE, UIQI

signature(y = "awssegment")

Returns a list with components RMSE, SNR, PSNR, MAE, MaxAE, UIQI

signature(y = "ICIsmooth")

Returns a list with components RMSE, SNR, PSNR, MAE, MaxAE, UIQI

signature(y = "kernsm")

Returns a list with components RMSE, SNR, PSNR, MAE, MaxAE, UIQI

signature(y = "numeric")

Returns a list with components RMSE, SNR, PSNR, MAE, MaxAE, UIQI

Author(s)

Joerg Polzehl polzehl@wias-berlin.de

References

V. Katkovnik, K. Egiazarian and J. Astola, Local Approximation Techniques in Signal And Image Processing, SPIE Society of Photo-Optical Instrumentation Engin., 2006, PM157

Z. Wang and A. C. Bovik, A universal image quality index, IEEE Signal Processing Letters, vol. 9, N3, pp. 81-84, 2002.

Methods for Function ‘show’ in Package ‘aws’

Description

The function provides information on data dimensions, data source and existing slot-names for objects of class "aws", "awssegment", "ICIsmooth" and "kernsm" in package aws

Methods

signature(object = "ANY")

Generic function.

signature(object = "aws")

Provide information on data dimensions, data source and existing slot-names for objects of class "dti" and classes that extent "aws".

signature(object = "awssegment")

Provide information on data dimensions, data source and existing slot-names for objects of class "dti" and classes that extent "awssegment".

signature(object = "ICIsmooth")

Provide information on data dimensions, data source and existing slot-names for objects of class "dti" and classes that extent "ICIsmooth".

signature(object = "kernsm")

Provide information on data dimensions, data source and existing slot-names for objects of class "dti" and classes that extent "kernsm".

Author(s)

Karsten Tabelow tabelow@wias-berlin.de
J\"org Polzehl polzehl@wias-berlin.de

See Also

aws, awssegment, ICIsmooth kernsm

Auxiliary 3D smoothing routines

Description

smooth3D and medianFilter3D are auxiliary functions for non-adaptive smoothing of 3D image data using kernel or median smoothing. Both function restrict to sub-areas determined by a mask. The functions are used in packages dti and qMRI.

Functions aws3Dmask and aws3Dmaskfull perform adaptive weights smoothing on statistical parametric maps in fMRI. Variability of results is determined from smoothed (using the same weighting schemes) residuals in order to correctly account for spatial correlation. These functions are intended to be used internally in package fmri. They have been moved here because they share significant parts of the openMP parallelized Fortran code underlying function aws.

Usage

smooth3D(y, h, mask, lkern = "Gaussian", weighted = FALSE, sigma2 = NULL,
         wghts = NULL)
medianFilter3D(y, h = 10, mask = NULL)
aws3Dmask(y, mask, lambda, hmax, res = NULL, sigma2 = NULL, lkern = "Gaussian",
   skern = "Plateau", weighted = TRUE, u = NULL, wghts = NULL,
   h0 = c(0, 0, 0), testprop = FALSE)
aws3Dmaskfull(y, mask, lambda, hmax, res = NULL, sigma2 = NULL, lkern = "Gaussian",
       skern = "Plateau", weighted = TRUE, u = NULL, wghts = NULL,
       testprop = FALSE)

Arguments

Value

Functions smooth3D and medianFilter3D return a 3D array. Functions awsmask* return a list with smoothed values of y in component theta and smoothed residuals in component res.

Note

Functions awsmask* are used intenally in package fmri. They refer to the situation, typical for fMRI, where the data are spatially correlated and this correlation can be accessed using residuals with respect to a model.

Author(s)

Joerg Polzehl polzehl@wias-berlin.de, Karsten Tabelow tabelow@wias-berlin.de

Adaptive smoothing in orientation space SE(3)

Description

The functions perform adaptive weights smoothing for data in orientation space SE(3), e.g. diffusion weighted MR data, with spatial coordinates given by voxel location within a mask and spherical information given by gradient direction. Observations can belong to different shells characterized by b-value bv. The data provided should only refer to voxel within mask.

Usage

smse3ms(sb, s0, bv, grad, kstar, lambda, kappa0, mask, sigma,
    ns0 = 1, ws0 = 1, vext = NULL, ncoils = 1, verbose = FALSE, usemaxni = TRUE)
smse3(sb, s0, bv, grad, mask, sigma, kstar, lambda, kappa0,
    ns0 = 1, vext = NULL, vred = 4, ncoils = 1, model = 0, dist = 1,
    verbose = FALSE)

Arguments

Value

The functions return lists with main results in components th and th0 containing the smoothed data.

Note

These functions are intended to be used internally in package dti only.

Author(s)

J\"org Polzehl polzehl@wias-berlin.de

References

Joerg Polzehl, Karsten Tabelow (2019). Magnetic Resonance Brain Imaging: Modeling and Data Analysis Using R. Springer, Use R! series. Doi:10.1007/978-3-030-29184-6.

S. Becker, K. Tabelow, H.U. Voss, A. Anwander, R. Heidemann, J. Polzehl. Position-orientation adaptive smoothing of diffusion weighted magnetic resonance data (POAS). Medical Image Analysis, 2012, 16, 1142-1155. DOI:10.1016/j.media.2012.05.007.

S. Becker, K. Tabelow, S. Mohammadi, N. Weiskopf, J. Polzehl. Adaptive smoothing of multi-shell diffusion-weighted magnetic resonance data by msPOAS. Neuroimage, 2014, 95, 90-105. DOI:10.1016/j.neuroimage.2014.03.053.

Methods for Function ‘summary’ from package 'base' in Package ‘aws’

Description

The method provides summary information for objects of class "aws".

Arguments

Methods

signature(object = "ANY")

Generic function: see summary.

signature(object = "aws")

The function provides summary information for objects of class "aws"

signature(object = "awssegment")

The function provides summary information for objects of class "awssegment"

signature(object = "ICIsmooth")

The function provides summary information for objects of class "ICIsmooth"

signature(object = "kernsm")

The function provides summary information for objects of class "kernsm"

Author(s)

J\"org Polzehl polzehl@wias-berlin.de

See Also

aws, awssegment, ICIsmooth kernsm

vector valued version of function aws The function implements the propagation separation approach to nonparametric smoothing (formerly introduced as Adaptive weights smoothing) for varying coefficient likelihood models with vector valued response on a 1D, 2D or 3D grid.

Description

The function implements a version the propagation separation approach that uses vector valued instead of scalar responses.

Usage

vaws(y, kstar = 16, sigma2 = 1, mask = NULL, scorr = 0, spmin = 0.25,
     ladjust = 1, wghts = NULL, u = NULL, maxni = FALSE)
vawscov(y, kstar = 16, invcov = NULL, mask = NULL, scorr = 0, spmin = 0.25,
          ladjust = 1, wghts = NULL, u = NULL, maxni = FALSE)

Arguments

Details

see aws. Expets vector valued responses. Currently only implements the case of additive Gaussian errors.

Value

returns anobject of class aws with slots

Note

use setCores='number of threads' to enable parallel execution.

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de, https://www.wias-berlin.de/people/polzehl/

References

J. Polzehl, K. Tabelow (2019). Magnetic Resonance Brain Imaging: Modeling and Data Analysis Using R. Springer, Use R! series. Appendix A. Doi:10.1007/978-3-030-29184-6.

J. Polzehl, V. Spokoiny, Adaptive Weights Smoothing with applications to image restoration, J. R. Stat. Soc. Ser. B Stat. Methodol. 62 , (2000) , pp. 335–354. DOI:10.1111/1467-9868.00235.

J. Polzehl, V. Spokoiny, Propagation-separation approach for local likelihood estimation, Probab. Theory Related Fields 135 (3), (2006) , pp. 335–362. DOI:10.1007/s00440-005-0464-1.

See Also

See also aws, vpaws,link{awsdata}

Examples

## Not run: 
setCores(2)
y <- array(rnorm(4*64^3),c(4,64,64,64))
yhat <- vaws(y,kstar=20)

## End(Not run)

vector valued version of function paws with homogeneous covariance structure

Description

The function implements a vector-valued version the propagation separation approach that uses patches instead of individuel voxels for comparisons in parameter space. Functionality is analog to function vaws. Using patches allows for an improved handling of locally smooth functions and in 2D and 3D for improved smoothness of discontinuities at the expense of increased computing time.

Usage

vpaws(y, kstar = 16, sigma2 = 1, invcov = NULL, mask = NULL, scorr = 0, spmin = 0.25,
      ladjust = 1, wghts = NULL, u = NULL, patchsize = 1)
vpawscov(y, kstar = 16, invcov = NULL, mask = NULL, scorr = 0, spmin = 0.25, ladjust = 1, 
      wghts = NULL, maxni = FALSE, patchsize = 1)
vpawscov2(y, kstar = 16, invcov = NULL, mask = NULL, scorr = 0, spmin = 0.25,
      lambda = NULL, ladjust = 1, wghts = NULL, patchsize = 1,
      data = NULL, verbose = TRUE)

Arguments

Details

see vaws. Parameter y The procedure is supposed to produce superior results if the assumption of a local constant image is violated or if smooothness of discontinuities is desired.

Function vpawscov2 is intended for internal use in package qMRI only.

Value

function vpaws returns returns an object of class aws with slots

If y contained only information (condensed data) for positions within a mask, then the returned object only contains results for these positions.

Note

use setCores='number of threads' to enable parallel execution.

Author(s)

Joerg Polzehl, polzehl@wias-berlin.de, https://www.wias-berlin.de/people/polzehl/

References

J. Polzehl, K. Tabelow (2019). Magnetic Resonance Brain Imaging: Modeling and Data Analysis Using R. Springer, Use R! series. Appendix A. Doi:10.1007/978-3-030-29184-6.

J. Polzehl, K. Papafitsoros, K. Tabelow (2020). Patch-Wise Adaptive Weights Smoothing in R, Journal of Statistical Software, 95(6), 1-27. doi:10.18637/jss.v095.i06 .

See Also

See also vaws, lpaws, vawscov,link{awsdata}

Examples

## Not run: 
setCores(2)
y <- array(rnorm(4*64^3),c(4,64,64,64))
yhat <- vpaws(y,kstar=20)

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