| Title: | 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>. |
| Authors: | Joerg Polzehl [aut, cre], Felix Anker [ctb] |
| Maintainer: | Joerg Polzehl <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 2.5-6 |
| Built: | 2024-11-18 06:29:51 UTC |
| Source: | CRAN |
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>.
The DESCRIPTION file:
| Package: | aws |
| Version: | 2.5-6 |
| Date: | 2024-09-29 |
| Title: | Adaptive Weights Smoothing |
| Authors@R: | c(person("Joerg","Polzehl",role=c("aut","cre"),email="[email protected]"),person("Felix","Anker",role=c("ctb"))) |
| Author: | Joerg Polzehl [aut, cre], Felix Anker [ctb] |
| Maintainer: | Joerg Polzehl <[email protected]> |
| Depends: | R (>= 3.4.0), awsMethods (>= 1.1-1) |
| Imports: | methods, gsl |
| 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>. |
| License: | GPL (>= 2) |
| Copyright: | This package is Copyright (C) 2005-2024 Weierstrass Institute for Applied Analysis and Stochastics. |
| URL: | https://www.wias-berlin.de/people/polzehl/ |
| RoxygenNote: | 5.0.1 |
| NeedsCompilation: | yes |
| Packaged: | 2024-09-29 20:05:48 UTC; polzehl |
| Repository: | CRAN |
| Date/Publication: | 2024-09-30 17:40:01 UTC |
| Config/pak/sysreqs: | libgsl0-dev |
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
Further information is available in the following vignettes:
aws-Example |
A very short inroduction into the aws package (source, pdf) |
Joerg Polzehl [aut, cre], Felix Anker [ctb]
Maintainer: Joerg Polzehl <[email protected]>
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.
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.
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)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)
x |
lower bound of search interval |
y |
upper bound of search interval |
lkern |
code for location kernel |
value |
target sum of location weights |
wght |
relative size of voxel dimensions |
eps |
attempted precision for bandwidth search |
bw |
vector of bandwidths, length equal to 1,2 or 3 depending on the dimensionality of the problem. |
L |
number of effective coils, |
to |
upper interval bound. |
delta |
discretization width. |
residuals |
array of residuals, if |
mask |
mask of active voxel (e.g. brain masks) |
resscale |
scale for residuals (residuals may be scaled for optimal integer*2 storage) |
compact |
logical, determines if only information for voxel within mask or full for full data cubes is given. |
lags |
positive integer vector of length 3, maximum lags for spatial correlations for each coordinate direction to be computed |
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.
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.
These functions are for internal use only. They are only exported to be available in other packages.
Joerg Polzehl [email protected]
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(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)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)
y |
array |
hmax |
|
aws |
logical: if TRUE structural adaptation (AWS) is used. |
mask |
optional logical mask, same dimensionality as |
memory |
logical: if TRUE stagewise aggregation is used as an additional adaptation scheme. |
family |
|
lkern |
character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian". The default "Triangle" is equivalent to using an Epanechnikov kernel, "Quadratic" and "Cubic" refer to a Bi-weight and Tri-weight kernel, see Fan and Gijbels (1996). "Gaussian" is a truncated (compact support) Gaussian kernel. This is included for comparisons only and should be avoided due to its large computational costs. |
aggkern |
character: kernel used in stagewise aggregation, either "Triangle" or "Uniform" |
sigma2 |
|
shape |
Allows to specify an additional shape parameter for certain family models. Currently only used for family="Variance", that is |
scorr |
The vector |
spmin |
Determines the form (size of the plateau) in the adaptation kernel. Not to be changed by the user. |
ladjust |
factor to increase the default value of lambda |
wghts |
|
u |
a "true" value of the regression function, may be provided to
report risks at each iteration. This can be used to test the propagation condition with |
graph |
If |
demo |
If |
testprop |
If set this provides diagnostics for testing the propagation condition. The values of |
maxni |
If TRUE use |
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
where is the aws-estimate in step k and
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.
returns anobject of class aws with slots
y = "numeric" |
y |
dy = "numeric" |
dim(y) |
x = "numeric" |
numeric(0) |
ni = "integer" |
integer(0) |
mask = "logical" |
logical(0) |
theta = "numeric" |
Estimates of regression function, |
mae = "numeric" |
Mean absolute error for each iteration step if u was specified, numeric(0) else |
var = "numeric" |
approx. variance of the estimates of the regression function. Please note that this does not reflect variability due to randomness of weights. |
xmin = "numeric" |
numeric(0) |
xmax = "numeric" |
numeric(0) |
wghts = "numeric" |
numeric(0), ratio of distances |
degree = "integer" |
0 |
hmax = "numeric" |
effective hmax |
sigma2 = "numeric" |
provided or estimated error variance |
scorr = "numeric" |
scorr |
family = "character" |
family |
shape = "numeric" |
shape |
lkern = "integer" |
integer code for lkern, 1="Plateau", 2="Triangle", 3="Quadratic", 4="Cubic", 5="Gaussian" |
lambda = "numeric" |
effective value of lambda |
ladjust = "numeric" |
effective value of ladjust |
aws = "logical" |
aws |
memory = "logical" |
memory |
homogen = "logical" |
homogen |
earlystop = "logical" |
FALSE |
varmodel = "character" |
"Constant" |
vcoef = "numeric" |
numeric(0) |
call = "function" |
the arguments of the call to |
use setCores='number of threads' to enable parallel execution.
Joerg Polzehl, [email protected], https://www.wias-berlin.de/people/polzehl/
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 paws, lpaws, vaws,link{awsdata}, aws.irreg, aws.gaussian
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)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)
"aws"
The "aws" class is
used for objects obtained by functions aws, lpaws, aws.irreg and aws.gaussian.
Objects are created by calls to functions aws, lpaws, aws.irreg and aws.gaussian.
.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.
signature(x = "aws"): ...
signature(y = "aws"): ...
Method for Function ‘plot’ in Package ‘aws’.
Method for Function ‘show’ in Package ‘aws’.
Method for Function ‘print’ in Package ‘aws’.
Method for Function ‘summary’ in Package ‘aws’.
Joerg Polzehl, [email protected]
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.
aws, lpaws, aws.irreg, aws.gaussian
showClass("aws")showClass("aws")
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.
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)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)
y |
|
hmax |
|
hpre |
Describe |
aws |
logical: if TRUE structural adaptation (AWS) is used. |
memory |
logical: if TRUE stagewise aggregation is used as an additional adaptation scheme. |
varmodel |
Implemented are "Constant", "Linear" and "Quadratic" refering to a polynomial model of degree 0 to 2. |
lkern |
character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian". The default "Triangle" is equivalent to using an Epanechnikov kernel, "Quadratic" and "Cubic" refer to a Bi-weight and Tri-weight kernel, see Fan and Gijbels (1996). "Gaussian" is a truncated (compact support) Gaussian kernel. This is included for comparisons only and should be avoided due to its large computational costs. |
aggkern |
character: kernel used in stagewise aggregation, either "Triangle" or "Uniform" |
scorr |
The vector |
mask |
Restrict smoothing to points where |
ladjust |
factor to increase the default value of lambda |
wghts |
|
u |
a "true" value of the regression function, may be provided to
report risks at each iteration. This can be used to test the propagation condition with |
varprop |
Small variance estimates are replaced by |
graph |
If |
demo |
If |
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
where is the aws-estimate in step k and
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.
returns anobject of class aws with slots
y = "numeric" |
y |
dy = "numeric" |
dim(y) |
x = "numeric" |
numeric(0) |
ni = "integer" |
integer(0) |
mask = "logical" |
logical(0) |
theta = "numeric" |
Estimates of regression function, |
mae = "numeric" |
Mean absolute error for each iteration step if u was specified, numeric(0) else |
var = "numeric" |
approx. variance of the estimates of the regression function. Please note that this does not reflect variability due to randomness of weights. |
xmin = "numeric" |
numeric(0) |
xmax = "numeric" |
numeric(0) |
wghts = "numeric" |
numeric(0), ratio of distances |
degree = "integer" |
0 |
hmax = "numeric" |
effective hmax |
sigma2 = "numeric" |
provided or estimated error variance |
scorr = "numeric" |
scorr |
family = "character" |
"Gaussian" |
shape = "numeric" |
NULL |
lkern = "integer" |
integer code for lkern, 1="Plateau", 2="Triangle", 3="Quadratic", 4="Cubic", 5="Gaussian" |
lambda = "numeric" |
effective value of lambda |
ladjust = "numeric" |
effective value of ladjust |
aws = "logical" |
aws |
memory = "logical" |
memory |
homogen = "logical" |
homogen |
earlystop = "logical" |
FALSE |
varmodel = "character" |
varmodel |
vcoef = "numeric" |
estimated parameters of the variance model |
call = "function" |
the arguments of the call to |
Joerg Polzehl, [email protected], https://www.wias-berlin.de/people/polzehl/
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 aws, link{awsdata}, aws.irreg
require(aws)require(aws)
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.
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)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)
y |
The observed response vector (length n) |
x |
Design matrix, dimension n x d, |
hmax |
|
aws |
logical: if TRUE structural adaptation (AWS) is used. |
memory |
logical: if TRUE stagewise aggregation is used as an additional adaptation scheme. |
varmodel |
determines the model that relates variance to mean. Either "Constant", "Linear" or "Quadratic". |
lkern |
character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian" |
aggkern |
character: kernel used in stagewise aggregation, either "Triangle" or "Uniform" |
sigma2 |
|
nbins |
numer of bins, can be NULL, a positive integer or a vector of positive integers (length d) |
hpre |
smoothing bandwidth for initial variance estimate |
henv |
radius of balls around each observed design point where estimates will be calculated |
ladjust |
factor to increase the default value of lambda |
varprop |
exclude the largest 100*varprop% squared residuals when estimating the error variance |
graph |
If |
Data are first binned (1D/2D), then aws is performed on all datapoints within distance <= henv of nonempty bins.
returns anobject of class aws with slots
y = "numeric" |
y |
dy = "numeric" |
dim(y) |
x = "numeric" |
x |
ni = "integer" |
number of observations per bin |
mask = "logical" |
bins where parameters have been estimated |
theta = "numeric" |
Estimates of regression function, |
mae = "numeric" |
numeric(0) |
var = "numeric" |
approx. variance of the estimates of the regression function. Please note that this does not reflect variability due to randomness of weights. |
xmin = "numeric" |
vector of minimal x-values (bins) |
xmax = "numeric" |
vector of maximal x-values (bins) |
wghts = "numeric" |
relative binwidths |
degree = "integer" |
0 |
hmax = "numeric" |
effective hmax |
sigma2 = "numeric" |
provided or estimated error variance |
scorr = "numeric" |
0 |
family = "character" |
"Gaussian" |
shape = "numeric" |
numeric(0) |
lkern = "integer" |
integer code for lkern, 1="Plateau", 2="Triangle", 3="Quadratic", 4="Cubic", 5="Gaussian" |
lambda = "numeric" |
effective value of lambda |
ladjust = "numeric" |
effective value of ladjust |
aws = "logical" |
aws |
memory = "logical" |
memory |
homogen = "logical" |
FALSE |
earlystop = "logical" |
FALSE |
varmodel = "character" |
varmodel |
vcoef = "numeric" |
estimated coefficients in variance model |
call = "function" |
the arguments of the call to |
Joerg Polzehl, [email protected]
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 lpaws, link{awsdata}, lpaws
require(aws) # 1D local constant smoothing ## Not run: demo(irreg_ex1) # 2D local constant smoothing ## Not run: demo(irreg_ex2)require(aws) # 1D local constant smoothing ## Not run: demo(irreg_ex1) # 2D local constant smoothing ## Not run: demo(irreg_ex2)
The function implements a modification of the adaptive weights smoothing algorithm for segmentation into three classes. The
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)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)
y |
|
level |
center of second class |
delta |
half width of second class |
hmax |
|
hpre |
Describe |
mask |
optional logical mask, same dimensionality as |
varmodel |
Implemented are "Constant", "Linear" and "Quadratic" refering to a polynomial model of degree 0 to 2. |
lkern |
character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian". The default "Triangle" is equivalent to using an Epanechnikov kernel, "Quadratic" and "Cubic" refer to a Bi-weight and Tri-weight kernel, see Fan and Gijbels (1996). "Gaussian" is a truncated (compact support) Gaussian kernel. This is included for comparisons only and should be avoided due to its large computational costs. |
scorr |
The vector |
ladjust |
factor to increase the default value of lambda |
wghts |
|
u |
a "true" value of the regression function, may be provided to
report risks at each iteration. This can be used to test the propagation condition with |
varprop |
Small variance estimates are replaced by |
ext |
Intermediate results are fixed if the test statistics exceeds the critical value by |
graph |
If |
demo |
If |
fov |
Field of view. Size of region (sample size) to adjust for in multiscale testing. |
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.
returns anobject of class aws with slots
y = "numeric" |
y |
dy = "numeric" |
dim(y) |
x = "numeric" |
numeric(0) |
ni = "integer" |
integer(0) |
mask = "logical" |
logical(0) |
segment = "integer" |
Segmentation results, class numbers 1-3 |
theta = "numeric" |
Estimates of regression function, |
mae = "numeric" |
Mean absolute error for each iteration step if u was specified, numeric(0) else |
var = "numeric" |
approx. variance of the estimates of the regression function. Please note that this does not reflect variability due to randomness of weights. |
xmin = "numeric" |
numeric(0) |
xmax = "numeric" |
numeric(0) |
wghts = "numeric" |
numeric(0), ratio of distances |
degree = "integer" |
0 |
hmax = "numeric" |
effective hmax |
sigma2 = "numeric" |
provided or estimated error variance |
scorr = "numeric" |
scorr |
family = "character" |
"Gaussian" |
shape = "numeric" |
NULL |
lkern = "integer" |
integer code for lkern, 1="Plateau", 2="Triangle", 3="Quadratic", 4="Cubic", 5="Gaussian" |
lambda = "numeric" |
effective value of lambda |
ladjust = "numeric" |
effective value of ladjust |
aws = "logical" |
aws |
memory = "logical" |
memory |
homogen = "logical" |
FALSE |
earlystop = "logical" |
FALSE |
varmodel = "character" |
varmodel |
vcoef = "numeric" |
estimated parameters of the variance model |
call = "function" |
the arguments of the call to |
This function is still experimental and may be changes considerably in future.
Joerg Polzehl, [email protected], https://www.wias-berlin.de/people/polzehl/
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
require(aws)require(aws)
Extract data and estimates from an object of class aws
awsdata(awsobj, what)awsdata(awsobj, what)
awsobj |
an object of class |
what |
can be "data" (extracts observed response), "theta" (estimated parameters), "est" (estimated regression function), "var" (approx. variance of estimated regression function), "sd" (approx. standard deviation of estimated regression function), "sigma2" (error variance), "mae" (mean absolute error for each iteration step, if available), "ni" (number of observations per bin), "mask" (logical indicator for bins where the regression function is estimated). "bi" (array of sum of weights or NULL) "bi2" (array of sum of squared weights or NULL) |
The returned object is formatted as an array if appropriate. The returned object may be NULL if the information is not available.
an vector or array containing the specified information.
Joerg Polzehl [email protected]
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
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)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)
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).
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)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)
y |
3D array of image intensities. |
steps |
number of steps in adapive weights smoothing, used to reveal the unerlying mean structure. |
mask |
restrict computations to voxel in mask, if |
ncoils |
effective number of coils, or equivalently number of effective degrees of freedom of non-central chi distribution divided by 2. |
vext |
voxel extentions or relative voxel extensions |
lambda |
scale parameter in adaptive weights smoothing |
minni |
minimal bandwidth for calculating local variance estimates |
hsig |
bandwwidth for median filter |
sigma |
optional initial global variance estimate |
family |
type of distribution, either noncentral Chi ("NCchi") or Gaussian ("Gauss") |
verbose |
if |
trace |
if |
u |
if |
hmax |
maximal bandwidth |
hpre |
minimal bandwidth |
h0 |
bandwidth vector characterizing to spatial correlation as correlation induced by convolution with a Gaussian kernel |
ladjust |
correction factor for lambda |
wghts |
relative voxel extensions |
varprop |
defines a lower bound for the estimated variance as |
A0 |
select voxel with |
A1 |
select voxel with |
level |
threshold for mask definition |
h |
bandwidth for local variance estimates. |
hinit |
minimal bandwidth for local variance estimates with |
hadj |
bandwidth for mode estimation |
q |
Quantile for interquantile estimate of standard deviation |
sequence |
logical, return sequence of estimated variances for iterative methods. |
method |
determines variance estimation method |
magnitude |
magnitude of complex 3D image |
phase |
phase of complex 3D image |
kstar |
number of steps in adapive weights smoothing, used to reveal the unerlying mean structure. |
kmin |
iteration to start adaptation |
all functions return lists with variance estimates in component sigma
J\"org Polzehl [email protected]
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.
"awssegment"
The "aws" class is
used for objects obtained by functions aws.segment
Objects are created by calls to functions aws.segment
.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.
signature(x = "awssegment"): ...
signature(x = "awssegment"): ...
signature(x = "awssegment"): ...
signature(y = "awssegment"): ...
signature(object = "awssegment"): ...
signature(object = "awssegment"): ...
Joerg Polzehl, [email protected]
showClass("awssegment")showClass("awssegment")
The function enables testing of the propagation condition in order to select
appropriate values for the parameter lambda in function aws.
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)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)
dy |
Dimension of grid used in 1D, 2D or 3D. May also be specified as an array of values.
In this case data are generated with parameters |
hmax |
Maximum bandwidth. |
theta |
Parameter determining the distribution in case of
|
family |
|
lkern |
character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian" |
aws |
logical: if TRUE structural adaptation (AWS) is used. |
patchsize |
patchsize in case of paws. |
memory |
logical: if TRUE stagewise aggregation is used as an additional adaptation scheme. |
shape |
Allows to specify an additional shape parameter for certain family models. Currently only used for family="Variance", that is |
homogeneous |
if |
varadapt |
if |
ladjust |
Factor to increase the default value of lambda |
spmin |
Determines the form (size of the plateau) in the adaptation kernel. Not to be changed by the user. |
seed |
Seed value for random generator. |
minlevel |
Minimum exceedence probability to use in contour plots. |
maxz |
Maximum of z-scale in plots. |
diffz |
Gridlength in z |
maxni |
If TRUE use |
verbose |
If TRUE provide additional information. |
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.
A list with components
h |
Sequence of bandwidths used |
z |
|
prob |
the matrix of exceedence probabilities, columns corresponding to |
probna |
the matrix of exceedence probabilities for corresponding nonadaptive estimates, columns corresponding to |
Joerg Polzehl [email protected]
S. Becker, P. Mathe, Electron. J. Statist. (2013), 2702-2736, doi:10.1214/13-EJS860
Utility function to create a weighting scheme for an additional aws step. Inteded to be used for illustrations only.
awsweights(awsobj, spmin = 0.25, inx = NULL)awsweights(awsobj, spmin = 0.25, inx = NULL)
awsobj |
object obtained by a call to function |
spmin |
Size of the plateau in the adaptation kernel. |
inx |
either a matrix of dimension length(awsobj@dy) x number of points containing the integer coordinates of points of interest or NULL. In the latter case the weight scheme for all points is generated. |
an array of either dimension awsobj@dy x number of points or awsobj@dy x awsobj@dy
Joerg Polzehl, [email protected], https://www.wias-berlin.de/people/polzehl/
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 aws
The function performs a binning in 1D, 2D or 3D.
binning(x, y, nbins, xrange = NULL)binning(x, y, nbins, xrange = NULL)
x |
design matrix, dimension n x d, |
y |
either a response vector of length n or NULL |
nbins |
vector of length d containing number of bins for each dimension, may be set to NULL |
xrange |
range for endpoints of bins for each dimension, either matrix
of dimension 2 x d or NULL. |
A list with components
x |
matrix of coordinates of non-empty bin centers |
x.freq |
number of observations in nonempty bins |
midpoints.x1 |
Bin centers in dimension 1 |
midpoints.x2 |
if |
midpoints.x3 |
if |
breaks.x1 |
Break points dimension 1 |
breaks.x2 |
if |
breaks.x3 |
if |
table.freq |
number of observations per bin |
means |
if |
devs |
if |
This function has been adapted from the code of function binning in package sm.
Joerg Polzehl, [email protected]
See Also as aws.irreg
extract in Package aws
The method extract and/or compute specified statistics from object of class
"aws", "awssegment", ICIsmooth and "kernsm".
## 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")## 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")
x |
object |
what |
Statistics to extract, defaults to |
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).
The function performs adaptive smoothing by Intersection of Confidence Intervals (ICI) using multiple windows as described in Katkovnik et al (2006)
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"))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"))
y |
Object of class |
hmax |
maximum bandwidth |
hinc |
factor used to increase the bandwidth from scale to scale |
thresh |
threshold used in tests to determine the best scale |
kern |
Determines the kernel function.
Object of class |
m |
Object of class |
sigma |
error standard deviation |
nsector |
number of sectors to use. |
symmetric |
Object of class |
presmooth |
Object of class |
combine |
Either |
unit |
How should the bandwidth be interpreted in case of a Gaussian kernel.
For |
This mainly follows Chapter 6.2 in Katkovnik et al (2006).
An object of class ICIsmooth
Joerg Polzehl [email protected]
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
ICIsmooth, ICIsmooth-class, kernsm
The function performs adaptive smoothing by Intersection of Confidence Intervals (ICI) as described in Katkovnik et al (2006)
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"))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"))
y |
Object of class |
hmax |
maximum bandwidth |
hinc |
factor used to increase the bandwidth from scale to scale |
thresh |
threshold used in tests to determine the best scale |
kern |
Determines the kernel function.
Object of class |
m |
Object of class |
sigma |
error standard deviation |
nsector |
number of sectors to use. Positive weights are restricted to the sector
selected by |
sector |
Object of class |
symmetric |
Object of class |
presmooth |
Object of class |
unit |
How should the bandwidth be interpreted in case of a Gaussian kernel.
For |
This mainly follows Chapter 6.1 in Katkovnik et al (2006).
An object of class ICIsmooth
Joerg Polzehl [email protected]
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
ICIcombined, ICIsmooth-class, kernsm
"ICIsmooth"
The "ICIsmooth" class is
used for objects obtained by functions ICIsmooth and ICIcombined.
Objects can be created by calls of the form new("ICIsmooth", ...)
or by functions ICIsmooth and ICIcombined.
.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.
signature(x = "ICIsmooth"): ...
signature(y = "ICIsmooth"): ...
Method for Function ‘plot’ in Package ‘aws’.
Method for Function ‘show’ in Package ‘aws’.
Method for Function ‘print’ in Package ‘aws’.
Method for Function ‘summary’ in Package ‘aws’.
Joerg Polzehl [email protected]
V. Katkovnik, K. Egiazarian and J. Astola, Local Approximation Techniques in Signal And Image Processing, SPIE Society of Photo-Optical Instrumentation Engin., 2006, PM157
ICIsmooth, ICIcombined, kernsm, aws
showClass("ICIsmooth")showClass("ICIsmooth")
Performs Kernel smoothing on a 1D, 2D or 3D grid by fft
kernsm(y, h = 1, kern = "Gaussian", m = 0, nsector = 1, sector = 1, symmetric = FALSE, unit = c("SD","FWHM"))kernsm(y, h = 1, kern = "Gaussian", m = 0, nsector = 1, sector = 1, symmetric = FALSE, unit = c("SD","FWHM"))
y |
Object of class |
h |
bandwidth |
kern |
Determines the kernel function.
Object of class |
m |
Object of class |
nsector |
number of sectors to use. Positive weights are restricted to the sector
selected by |
sector |
Object of class |
symmetric |
Object of class |
unit |
How should the bandwidth be interpreted in case of a Gaussian kernel.
For |
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.
An object of class kernsm
Joerg Polzehl [email protected]
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
kernsm-class, ICIsmooth,ICIcombined
"kernsm"
This class refers to objects created by function kernsm. These objects contain
Objects can be created by calls of the form new("kernsm", ...). they are usually
created by a call to function{kernsm}.
.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.
signature(x = "aws"): ...
signature(y = "aws"): ...
Method for Function ‘plot’ in Package ‘aws’.
Method for Function ‘show’ in Package ‘aws’.
Method for Function ‘print’ in Package ‘aws’.
Method for Function ‘summary’ in Package ‘aws’.
J\"org Polzehl [email protected]
kernsm,
ICIsmooth,
ICIcombined,
ICIsmooth
showClass("kernsm")showClass("kernsm")
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.
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)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)
y |
Response, either a vector (1D) or matrix (2D). The corresponding design is assumed to be a regular grid in 1D or 2D, respectively. |
degree |
Polynomial degree of the local model |
hmax |
maximal bandwidth |
aws |
logical: if TRUE structural adaptation (AWS) is used. |
memory |
logical: if TRUE stagewise aggregation is used as an additional adaptation scheme. |
lkern |
character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian". The default "Triangle" is equivalent to using an Epanechnikov kernel, "Quadratic" and "Cubic" refer to a Bi-weight and Tri-weight kernel, see Fan and Gijbels (1996). "Gaussian" is a truncated (compact support) Gaussian kernel. This is included for comparisons only and should be avoided due to its large computational costs. |
homogen |
logical: if TRUE the function tries to determine regions where weights can be fixed to 1. This may increase speed. |
earlystop |
logical: if TRUE the function tries to determine points where the homogeneous region is unlikely to change in further steps. This may increase speed. |
aggkern |
character: kernel used in stagewise aggregation, either "Triangle" or "Uniform" |
sigma2 |
Error variance, the value is estimated if not provided. |
hw |
Regularisation bandwidth, used to prevent from unidentifiability of local estimates for small bandwidths. |
ladjust |
factor to increase the default value of lambda |
u |
a "true" value of the regression function, may be provided to
report risks at each iteration. This can be used to test the propagation condition with |
graph |
logical: If TRUE intermediate results are illustrated graphically. May significantly slow down the computations in 2D. Please
avoid using the default |
demo |
logical: if TRUE wait after each iteration |
returns anobject of class aws with slots
y = "numeric" |
y |
dy = "numeric" |
dim(y) |
x = "numeric" |
numeric(0) |
ni = "integer" |
integer(0) |
mask = "logical" |
logical(0) |
theta = "numeric" |
Estimates of regression function and derivatives, |
mae = "numeric" |
Mean absolute error for each iteration step if u was specified, numeric(0) else |
var = "numeric" |
approx. variance of the estimates of the regression function. Please note that this does not reflect variability due to randomness of weights. |
xmin = "numeric" |
numeric(0) |
xmax = "numeric" |
numeric(0) |
wghts = "numeric" |
numeric(0), ratio of distances |
degree = "integer" |
degree |
hmax = "numeric" |
effective hmax |
sigma2 = "numeric" |
provided or estimated error variance |
scorr = "numeric" |
0 |
family = "character" |
"Gaussian" |
shape = "numeric" |
numeric(0) |
lkern = "integer" |
integer code for lkern, 1="Plateau", 2="Triangle", 3="Quadratic", 4="Cubic", 5="Gaussian" |
lambda = "numeric" |
effective value of lambda |
ladjust = "numeric" |
effective value of ladjust |
aws = "logical" |
aws |
memory = "logical" |
memory |
homogen = "logical" |
homogen |
earlystop = "logical" |
eralustop |
varmodel = "character" |
"Constant" |
vcoef = "numeric" |
numeric(0) |
call = "function" |
the arguments of the call to |
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).
Joerg Polzehl [email protected]
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.
library(aws) # 1D local polynomial smoothing ## Not run: demo(lpaws_ex1) # 2D local polynomial smoothing ## Not run: demo(lpaws_ex2)library(aws) # 1D local polynomial smoothing ## Not run: demo(lpaws_ex1) # 2D local polynomial smoothing ## Not run: demo(lpaws_ex2)
Implements the Non-Local-Means Filter of Buades et al 2005
nlmeans(x, lambda, sigma, patchhw = 1, searchhw = 7, pd = NULL)nlmeans(x, lambda, sigma, patchhw = 1, searchhw = 7, pd = NULL)
x |
1, 2 or 3-dimensional array of obseved response (image intensity) data. |
lambda |
scale factor for kernel in image space. |
sigma |
error standard deviation (for additive Gaussian errors). |
patchhw |
Half width of patches in each dimension (patchsize is |
searchhw |
Half width of search area (size of search area is |
pd |
If |
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.
A list of class "nlmeans" with components
theta |
Denoised array |
lambda |
Scale parameter used |
sigma |
The error standard deviation |
patchhw |
Half width of patches |
pd |
Effective patchsize used |
searchhw |
Half width of search area |
use setCores='number of threads' to enable parallel execution.
Joerg Polzehl, [email protected], https://www.wias-berlin.de/people/polzehl/
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
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.
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)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)
y |
array |
mask |
logical array defining a mask. All computations are restricted to the mask. |
hmax |
|
onestep |
apply the last step only (use for test purposes only) |
aws |
logical: if TRUE structural adaptation (AWS) is used. |
family |
|
lkern |
character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian". The default "Triangle" is equivalent to using an Epanechnikov kernel, "Quadratic" and "Cubic" refer to a Bi-weight and Tri-weight kernel, see Fan and Gijbels (1996). "Gaussian" is a truncated (compact support) Gaussian kernel. This is included for comparisons only and should be avoided due to its large computational costs. |
aggkern |
character: kernel used in stagewise aggregation, either "Triangle" or "Uniform" |
sigma2 |
|
shape |
Allows to specify an additional shape parameter for certain family models. Currently only used for family="Variance", that is |
scorr |
The vector |
spmin |
Determines the form (size of the plateau) in the adaptation kernel. Not to be changed by the user. |
ladjust |
factor to increase the default value of lambda |
wghts |
|
u |
a "true" value of the regression function, may be provided to
report risks at each iteration. This can be used to test the propagation condition with |
graph |
If |
demo |
If |
patchsize |
positive integer defining the size of patches. Number of grid points within the patch is |
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.
returns an object of class aws with slots
y = "numeric" |
y |
dy = "numeric" |
dim(y) |
x = "numeric" |
numeric(0) |
ni = "integer" |
integer(0) |
mask = "logical" |
logical(0) |
theta = "numeric" |
Estimates of regression function, |
hseq = "numeric" |
sequence of bandwidths employed |
mae = "numeric" |
Mean absolute error for each iteration step if u was specified, numeric(0) else |
psnr = "numeric" |
Peak signal-to-noise ratio for each iteration step if u was specified, numeric(0) else |
var = "numeric" |
approx. variance of the estimates of the regression function. Please note that this does not reflect variability due to randomness of weights. |
xmin = "numeric" |
numeric(0) |
xmax = "numeric" |
numeric(0) |
wghts = "numeric" |
numeric(0), ratio of distances |
degree = "integer" |
0 |
hmax = "numeric" |
effective hmax |
sigma2 = "numeric" |
provided or estimated error variance |
scorr = "numeric" |
scorr |
family = "character" |
family |
shape = "numeric" |
shape |
lkern = "integer" |
integer code for lkern, 1="Plateau", 2="Triangle", 3="Quadratic", 4="Cubic", 5="Gaussian" |
lambda = "numeric" |
effective value of lambda |
ladjust = "numeric" |
effective value of ladjust |
aws = "logical" |
aws |
memory = "logical" |
memory |
homogen = "logical" |
homogen |
earlystop = "logical" |
FALSE |
varmodel = "character" |
"Constant" |
vcoef = "numeric" |
numeric(0) |
call = "function" |
the arguments of the call to |
use setCores='number of threads' to enable parallel execution.
Joerg Polzehl, [email protected], https://www.wias-berlin.de/people/polzehl/
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 aws, lpaws, vpaws,link{awsdata}
## Not run: setCores(2) y <- array(rnorm(64^3),c(64,64,64)) yhat <- paws(y,hmax=6) ## End(Not run)## Not run: setCores(2) y <- array(rnorm(64^3),c(64,64,64)) yhat <- paws(y,hmax=6) ## End(Not run)
Visualization of objects of class "aws", "awsswgment",
"kernsm" and "ICIsmooth"
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"
J\"org Polzehl [email protected]
aws,
awssegment,
ICIsmooth
kernsm
The function provides information on data dimensions, creation of the object and
existing slot-names for objects of class "aws", "awssegment",
"ICIsmooth" and "kernsm"
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"
J\"org Polzehl [email protected]
aws,
awssegment,
ICIsmooth
kernsm
Computes selected criteria for quality assessments of
qmeasures(img, ref, which = c("PSNR", "MAE", "MSE", "RMSE", "SSIM", "MAGE", "RMSGE"), mask = FALSE)qmeasures(img, ref, which = c("PSNR", "MAE", "MSE", "RMSE", "SSIM", "MAGE", "RMSGE"), mask = FALSE)
img |
2D/3D image, object of class |
ref |
Reference image (array, matrix or vector) for comparison. |
which |
Criterion to use for Quality assessment. Please specify a subset of
|
mask |
Logical of same dimension as img/ref. Calculation can be restricted to mask. |
Calculates specified quality indices.
A vector with names as specified in which.
Joerg Polzehl, [email protected], https://www.wias-berlin.de/people/polzehl/
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)).
## 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)## 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)
y |
object |
u |
array of dimension |
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
Joerg Polzehl [email protected]
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.
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
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".
Karsten Tabelow [email protected]
J\"org Polzehl [email protected]
aws,
awssegment,
ICIsmooth
kernsm
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.
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)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)
y |
3D array of data in case of functions |
lkern |
character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian". The default "Triangle" is equivalent to using an Epanechnikov kernel, "Quadratic" and "Cubic" refer to a Bi-weight and Tri-weight kernel, see Fan and Gijbels (1996). "Gaussian" is a truncated (compact support) Gaussian kernel. This is included for comparisons only and should be avoided due to its large computational costs. |
weighted |
logical: use inverse variances as weights. |
sigma2 |
|
mask |
optional logical mask, same dimensionality as |
h |
bandwidth to use. In case of |
wghts |
voxel dimensions. Defaults to |
lambda |
kritical scale parameter in hypothesis testing (adaptive weights smoothing) |
hmax |
maximum bandwidth for adaptive weights smoothing |
res |
array of residuals with dimension |
skern |
|
u |
For test purposes in simulations: noisless 3D data. |
h0 |
Vector of 3 bandwidths corresponding to a Gaussian kernel that would produce a comparable spatial correlation by convoluting iid data. |
testprop |
logical: test the validity of a propagation condition for the specified value of |
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.
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.
Joerg Polzehl [email protected], Karsten Tabelow [email protected]
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.
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)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)
sb |
2D array of diffion weighted data, first dimension refers to index ov voxel within the mask, second dimension to the number diffusion weighted images. |
s0 |
vector of length |
bv |
vector of b-values. |
grad |
matrix of gradient directions with |
kstar |
number of steps in adaptive weights smoothing. |
lambda |
Scale parameter in adaptation |
kappa0 |
determines amount of smoothing on the sphere. Larger values correspond to stronger smoothing
on the sphere. If |
mask |
3D image defining a mask (logical) |
sigma |
Error standard deviation. Assumed to be known and homogeneous in the current implementation.
A reasonable estimate may be defined
as the modal value of standard deviations obtained using method |
ns0 |
Actual number of non-diffusion-weigthed images used to obtain |
ws0 |
Weight for non-diffusion-weigthed images in statistical penalty. |
vext |
Voxel extensions. |
ncoils |
Effective number of receiver coils (in case of e.g. GRAPPA reconstructions),
should be 1 in case of SENSE reconstructions. |
verbose |
If |
usemaxni |
If |
vred |
Used if |
model |
Determines which quantities are smoothed. Possible values are
|
dist |
Distance in SE3. Reasonable values are 1 (default, see Becker et.al. 2012), 2 ( a slight modification of 1: with k6^2 instead of abs(k6)) and 3 (using a 'naive' distance on the sphere) |
The functions return lists with main results in components
th and th0 containing the smoothed data.
These functions are intended to be used internally in package dti only.
J\"org Polzehl [email protected]
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.
The method provides summary information for objects of class "aws".
object |
Object of class |
... |
Additional arguments in |
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"
J\"org Polzehl [email protected]
aws,
awssegment,
ICIsmooth
kernsm
Total variation and total generalized variation are classical energy minimizing methods for image denoising.
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)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)
datanoisy |
matrix of noisy 2D image data. In case of |
alpha |
TV regularization parameter. |
beta |
additional TGV regularization parameter. |
iter |
max. number of iterations |
tolmean |
requested accuracy for mean image correction |
tolsup |
requested accuracy for max (over pixel) image correction |
scale |
image scale |
verbose |
report convergence diagnostics. |
Reimplementation of original matlab code by Kostas Papafitsoros (WIAS).
TV/TGV reconstructed image data (2D array)
Joerg Polzehl, [email protected], https://www.wias-berlin.de/people/polzehl/
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.
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.
The function implements a version the propagation separation approach that uses vector valued instead of scalar responses.
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)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)
y |
|
kstar |
maximal number of steps to employ. Determines maximal bandwidth. |
sigma2 |
specifies a homogeneous error variance. |
invcov |
array of voxelwise inverse covariance matrixes, first index corresponds to upper diagonal inverse covariance matrix. |
mask |
logical mask. All computations are restrikted to design poins within the mask. |
scorr |
The vector |
spmin |
determines the form (size of the plateau) in the adaptation kernel. Not to be changed by the user. |
ladjust |
factor to increase the default value of lambda |
wghts |
|
u |
a "true" value of the regression function, may be provided to
report risks at each iteration. This can be used to test the propagation condition with |
maxni |
If TRUE use |
see aws. Expets vector valued responses. Currently only implements the case of additive Gaussian errors.
returns anobject of class aws with slots
y = "numeric" |
y |
dy = "numeric" |
dim(y) |
x = "numeric" |
numeric(0) |
ni = "integer" |
integer(0) |
mask = "logical" |
logical(0) |
theta = "numeric" |
Estimates of regression function, |
hseq = "numeric" |
sequence of bandwidths employed |
mae = "numeric" |
Mean absolute error for each iteration step if u was specified, numeric(0) else |
psnr = "numeric" |
Peak signal-to-noise ratio for each iteration step if u was specified, numeric(0) else |
var = "numeric" |
approx. variance of the estimates of the regression function. Please note that this does not reflect variability due to randomness of weights. |
xmin = "numeric" |
numeric(0) |
xmax = "numeric" |
numeric(0) |
wghts = "numeric" |
numeric(0), ratio of distances |
degree = "integer" |
0 |
hmax = "numeric" |
effective hmax |
sigma2 = "numeric" |
provided or estimated (inverse) error variance |
scorr = "numeric" |
scorr |
family = "character" |
family |
shape = "numeric" |
shape |
lkern = "integer" |
integer code for lkern, 1="Plateau", 2="Triangle", 3="Quadratic", 4="Cubic", 5="Gaussian" |
lambda = "numeric" |
effective value of lambda |
ladjust = "numeric" |
effective value of ladjust |
aws = "logical" |
aws |
memory = "logical" |
memory |
homogen = "logical" |
homogen |
earlystop = "logical" |
FALSE |
varmodel = "character" |
"Constant" |
vcoef = "numeric" |
numeric(0) |
call = "function" |
the arguments of the call to |
use setCores='number of threads' to enable parallel execution.
Joerg Polzehl, [email protected], https://www.wias-berlin.de/people/polzehl/
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 aws, vpaws,link{awsdata}
## Not run: setCores(2) y <- array(rnorm(4*64^3),c(4,64,64,64)) yhat <- vaws(y,kstar=20) ## End(Not run)## Not run: setCores(2) y <- array(rnorm(4*64^3),c(4,64,64,64)) yhat <- vaws(y,kstar=20) ## End(Not run)
paws with homogeneous covariance structure
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.
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)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)
y |
|
kstar |
maximal number of steps to employ. Determines maximal bandwidth. |
sigma2 |
specifies a homogeneous error variance. |
invcov |
array (or matrix) of voxelwise inverse covariance matrixes, first index corresponds to upper diagonal inverse covariance matrix. |
mask |
logical mask. All computations are restrikted to design poins within the mask. |
scorr |
The vector |
spmin |
determines the form (size of the plateau) in the adaptation kernel. Not to be changed by the user. |
ladjust |
factor to increase the default value of lambda |
wghts |
|
u |
a "true" value of the regression function, may be provided to
report risks at each iteration. This can be used to test the propagation condition with |
patchsize |
positive integer defining the size of patches. Number of grid points within the patch is |
maxni |
require growing sum of weights |
lambda |
explicit value of lambda |
data |
optional vector-valued images to be smoothed using the weighting scheme of the last step |
verbose |
logical: provide information on progress. |
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.
function vpaws returns
returns an object of class aws with slots
y = "numeric" |
y |
dy = "numeric" |
dim(y) |
x = "numeric" |
numeric(0) |
ni = "integer" |
integer(0) |
mask = "logical" |
logical(0) |
theta = "numeric" |
Estimates of regression function, |
hseq = "numeric" |
sequence of bandwidths employed |
mae = "numeric" |
Mean absolute error for each iteration step if u was specified, numeric(0) else |
psnr = "numeric" |
Peak signal-to-noise ratio for each iteration step if u was specified, numeric(0) else |
var = "numeric" |
approx. variance of the estimates of the regression function. Please note that this does not reflect variability due to randomness of weights.Currently also uses factor |
xmin = "numeric" |
numeric(0) |
xmax = "numeric" |
numeric(0) |
wghts = "numeric" |
numeric(0), ratio of distances |
degree = "integer" |
0 |
hmax = "numeric" |
effective hmax |
sigma2 = "numeric" |
provided or estimated error variance |
scorr = "numeric" |
scorr |
family = "character" |
family |
shape = "numeric" |
shape |
lkern = "integer" |
integer code for lkern, 1="Plateau", 2="Triangle", 3="Quadratic", 4="Cubic", 5="Gaussian" |
lambda = "numeric" |
effective value of lambda |
ladjust = "numeric" |
effective value of ladjust |
aws = "logical" |
aws |
memory = "logical" |
memory |
homogen = "logical" |
homogen |
earlystop = "logical" |
FALSE |
varmodel = "character" |
"Constant" |
vcoef = "numeric" |
numeric(0) |
call = "function" |
the arguments of the call to |
If y contained only information (condensed data) for positions within a mask, then the returned object only contains
results for these positions.
use setCores='number of threads' to enable parallel execution.
Joerg Polzehl, [email protected], https://www.wias-berlin.de/people/polzehl/
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 vaws, lpaws, vawscov,link{awsdata}
## Not run: setCores(2) y <- array(rnorm(4*64^3),c(4,64,64,64)) yhat <- vpaws(y,kstar=20) ## End(Not run)## Not run: setCores(2) y <- array(rnorm(4*64^3),c(4,64,64,64)) yhat <- vpaws(y,kstar=20) ## End(Not run)