This functions implement the Propagation-Separation approach
(local constant and local polynomial model) for smoothing images.
Function awsaniso uses anisotropic location weights. This is done by evaluating local gradient estimates obtained from the actual estimated color values.
awsimage(object, hmax=4, aws=TRUE, varmodel=NULL, ladjust=1.25,
mask=NULL, xind = NULL, yind = NULL,
wghts=c(1,1,1,1), scorr=TRUE,
lkern="Plateau", plateau=NULL, homogen=TRUE, earlystop=TRUE,
demo=FALSE, graph=FALSE,
max.pixel=4.e2, clip = FALSE, compress=TRUE)
awspimage(object, hmax=12, aws=TRUE, degree=1, varmodel = NULL,
ladjust=1.0, xind = NULL, yind = NULL,
wghts=c(1,1,1,1), scorr= TRUE,
lkern="Plateau", plateau=NULL, homogen=TRUE, earlystop=TRUE,
demo=FALSE, graph=FALSE,
max.pixel= 4.e2, clip = FALSE, compress=TRUE)
awsaniso(object, hmax = 4, g = 3, rho = 0, aws = TRUE, varmodel = NULL,
ladjust = 1, xind = NULL, yind = NULL, wghts = c(1, 1, 1, 1),
scorr = TRUE, lkern = "Triangle", demo = FALSE, graph = FALSE,
satexp = 0.25, max.pixel = 400, clip = FALSE, compress = TRUE)Object of class "adimpro"
Contains the reconstructed image.
Contains the sum of weights, i.e. trace(W_i), in all grid points i.
Contains the maximum sum of weights for an nonadaptive kernel estimate with the same bandwidth.
Bandwidth used in the last iteration.
The arguments of the function call.
Estimated coefficients in the variance model for the
color channels, if varmodel is "Constant" or "Linear".
The weights used for the color channels.
Estimated spatial correlations for each channel, if scorr=TRUE
Estimated correlations between
color channels, if scorr=TRUE
Image object, class "adimpro", as from
read.image, read.raw, or make.image.
Maximum bandwidth to use in the iteration procedure.
Bandwidth for anisotropic smoothing gradient estimates,
preferably \(g \ge 3\) for images with line type texture and small
\(g \approx 1\) for improving edges between homogeneous regions (function awsaniso only).
Regularization parameter for anisotropic smoothing gradient estimates,
preferably \(\rho = 0\) for images with line type texture and large
\(\rho \approx 3\) for improving edges between homogeneous regions. (function awsaniso only)
(logical). If TRUE the propagation - separation
(PS) approach from Polzehl and Spokoiny (2006) is used.
aws=FALSE turns off the statistical penalty resulting in a
nonadaptive kernel estimate using a kernel with bandwidth hmax.
Degree of the local polynomial model for
awspimage. 0, 1, or 2 only.
varmodel specifies how variances are to be
estimated. This can be a homogeneous variance estimate
(varmodel="None") assuming uncorrelated errors (both spatial
and between channels). Alternatives are an adaptive homogeneous or linear
(function of the mean) variance estimate that depends on estimated
correlations and on residuals from the last iteration step.
The default varmodel=NULL corresponds to
varmodel == "Linear" if img$gamma==FALSE and
varmodel == "Constant" otherwise.
adjustment factor for lambda (>=1). Default values for
lambda are selected for Gaussian distributions and default settings of
parameters lkern and plateau. Skewed or heavy
tailed distributions may require slightly larger values for lambda
to meet the propagation condition. ladjust allows to increase
lambda in such situations.
logical array of the same size as the image or
NULL (default). Smoothing is restricted to the smallest rectangle
including all pixel where mask==TRUE and restricts
computations to these pixel. This need not be a
connected area (Typical usage: smooth all bright regions)!
Only used if is.null(xind) &&
is.null(yind). Inactive if
mask==NULL. Can only be used if varmodel="None".
Restrict smoothing to rectangular area defined by pixel
indices xind,yind in x- and y-direction. Full range
if NULL (default).
allows to weight the information from
different (up to 4) color channels. The weights are used in the
statistical penalty of the PS-procedure. Note that lambda-values are selected for wghts==c(1,1,1,1),
please use parameter ladjust to set an appropriate value.
(logical). Specifies whether spatial correlation is to be
estimated. Defaults to TRUE. Is set to FALSE if
mask is not NULL.
Specifies the location kernel. Defaults to "Triangle", other choices are "Quadratic", "Cubic" and "Uniform". The use of "Triangle" corresponds to the Epanechnicov kernel nonparametric kernel regression.
Extension of the plateau in the statistical kernel. Can take
values from (0,1), defaults to 0.25.
If TRUE the algorithm determines, in each design point i, a circle of maximum radius,
such that the statistical penalty s_{ij} for all points j within the
circle is less than the value specified in plateau. In subsequent
iteration steps the statistical penalty for such points is set to zero.
This is only used if plateau>0. This results in more stable intermediate estimates and in a smoother reconstruction. homogen=TRUE
leads to increased memory requirements.
If TRUE the algorithm determines, in each design point i, a circle of minimal radius,
such that the circle includes all point j with positive weights w_{ij}.
if this radius is considerably smaller than the actual bandwidth then the
estimate in point i is fixed. This should considarably reduce computing time
in case of large hmax.earlystop=TRUE
slightly increases memory requirements.
(logical). If demo=TRUE the function pauses after each
iteration. Defaults to FALSE.
(logical). If graph=TRUE intermediate results are
illustrated after each iteration step. Defaults to FALSE.
Maximum dimension of images for display
if graph=TRUE. If the true dimension is larger, the
images are downscaled for display. See also show.image.
exponent used for scaling saturation in anisotropy visualization (function awsaniso only)
(logical). If TRUE a clipping region is selected, see
clip.image, using the information contained in
xind or yind. If both are NULL a clipping
region can be defined by left mouse clicks. The image object
is reduced to the clipping region before smoothing.
logical, determines if image data are stored in raw-format.
Karsten Tabelow tabelow@wias-berlin.de and Joerg Polzehl polzehl@wias-berlin.de
The function implements the Propagation-Separation (PS) approach to nonparametric smoothing (formerly introduced as Adaptive Weights Smoothing) for varying coefficient likelihood (awsimage) and local polynomial (awspimage) models for greyscale and color images.
The distribution of grey (color) values is considered to be Gaussian. Noise can be colored.
The numerical complexity of the procedure is mainly determined by
hmax. The number of iterations is 2*log(hmax)/log(1.25).
Comlexity in each iteration step is Const*hakt*n with hakt
being the actual bandwith in the iteration step and n the number of pixels.
hmax determines the maximal possible variance reduction.
All other parameters of the approach only depend on the specified
values for skern/lkern and are therefore set internally to
meaningful default values.
For a detailed description of the procedure see references below.
The script used to control the values of parameter lambda is stored in
directory inst/adjust.
Polzehl and Spokoiny (2006). Propagation-Separation Approach for Local Likelihood Estimation. Probability Theory and Related Fields. 3 (135) 335 - 362.
Polzehl and Spokoiny (2005). Structural adaptive smoothing adaptive smoothing by Propagation-Separation-methods. WIAS-Preprint No. 1068.
Polzehl, J. and Tabelow, K. (2007). Adaptive smoothing of digital images, Journal of Statistical Software 19 (1).
read.image, read.raw, make.image, show.image, clip.image