awsimage {adimpro}

Propagation-Separation approach for smoothing of 2D images

Description

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.

Usage

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)

Arguments

object

Image object, class "adimpro", as from read.image, read.raw, or make.image.

hmax

Maximum bandwidth to use in the iteration procedure.

g

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).

rho

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)

aws

(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

Degree of the local polynomial model for awspimage. 0, 1, or 2 only.

varmodel

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.

ladjust

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.

mask

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".

xind, yind

Restrict smoothing to rectangular area defined by pixel indices xind,yind in x- and y-direction. Full range if NULL (default).

wghts

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.

scorr

(logical). Specifies whether spatial correlation is to be estimated. Defaults to TRUE. Is set to FALSE if mask is not NULL.

lkern

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.

plateau

Extension of the plateau in the statistical kernel. Can take values from (0,1), defaults to 0.25.

homogen

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.

earlystop

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.

demo

(logical). If demo=TRUE the function pauses after each iteration. Defaults to FALSE.

graph

(logical). If graph=TRUE intermediate results are illustrated after each iteration step. Defaults to FALSE.

max.pixel

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.

satexp

exponent used for scaling saturation in anisotropy visualization (function awsaniso only)

clip

(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.

compress

logical, determines if image data are stored in raw-format.

Details

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.

Value

Object of class "adimpro"

img

Contains the reconstructed image.

ni

Contains the sum of weights, i.e. trace(W_i), in all grid points i.

ni0

Contains the maximum sum of weights for an nonadaptive kernel estimate with the same bandwidth.

hmax

Bandwidth used in the last iteration.

call

The arguments of the function call.

varcoef

Estimated coefficients in the variance model for the color channels, if varmodel is "Constant" or "Linear".

wghts

The weights used for the color channels.

scorr

Estimated spatial correlations for each channel, if scorr=TRUE

chcorr

Estimated correlations between color channels, if scorr=TRUE

Author(s)

Karsten Tabelow [e-mail] and Joerg Polzehl [e-mail]

References

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).

See Also

read.image, read.raw, make.image, show.image, clip.image

Examples

## Not run: demo(awsimage)