image.smooth: Kernel smoother for irregular 2-d data

Description

Takes an image matrix and applies a kernel smoother to it. Missing values are handled using the Nadaraya/Watson normalization of the kernel.

Usage

# S3 method for smooth
image(x, wght = NULL, dx = 1, dy = 1,
    kernel.function = double.exp,
    aRange = 1, grid = NULL, tol = 1e-08, xwidth = NULL, ywidth = NULL,
    weights = NULL, theta=NULL, ...)
setup.image.smooth(nrow = 64, ncol = 64, dx = 1, dy = 1,
                   kernel.function = double.exp,
                   aRange = 1, xwidth = nrow * dx, ywidth = ncol * dx,
		   lambda=NULL,theta=NULL, ...)

Value

The smoothed image in R image format. ( A list with components x, y and z.) setup.image.smooth returns a list with components W a matrix being the FFT of the kernel, dx, dy, xwidth and ywidth.

Arguments

x: A matrix image. Missing values can be indicated by NAs.
wght: FFT of smoothing kernel. If this is NULL the default is to compute this object.
grid: A list with x and y components. Each are equally spaced and define the rectangular. ( see grid.list)
dx: Grid spacing in x direction
dy: Grid spacing in x direction
kernel.function: An R function that takes as its argument the squared distance between two points divided by the bandwidth. The default is exp( -abs(x)) yielding a normal kernel
aRange: the bandwidth or scale parameter.
theta: Same as aRange.
xwidth: Amount of zero padding in horizontal dimension in units of the grid spacing. If NULL the default value is equal to the width of the image the most conservative value but possibly inefficient for computation. Set this equal to zero to get periodic wrapping of the smoother. This is useful to smooth a Mercator map projection.
ywidth: Same as xwidth but for the vertical dimension.
weights: Weights to apply when smoothing.
tol: Tolerance for the weights of the N-W kernel. This avoids kernel estimates that are "far" away from data. Grid points with weights less than tol are set to NA.
nrow: X dimension of image in setting up smoother weights
ncol: Y dimension of image
lambda: Smoothing parameter if smoother is interpreted in a spline-like way.
...: Other arguments to be passed to the kernel function

Details

The function works by taking convolutions using an FFT. The missing pixels are taken into account and the kernel smoothing is correctly normalized for the edge effects following the classical Nadaraya-Watson estimator. For this reason the kernel doe snot have to be a desity as it is automatically normalized when the kernel weight function is found for the data. If the kernel has limited support then the width arguments can be set to reduce the amount of computation. (See example below.) For multiple smoothing compute the fft of the kernel just once using setup.image.smooth and pass this as the wght argument to image.smooth. this will save an FFT in computations.

Examples

Run this code

# first convert precip data to the 128X128 discretized image format ( with 
# missing  values to indicate where data is not observed) 
# 
out<- as.image( RMprecip$y, x= RMprecip$x, nx=128, ny=128) 
# out$z is the image matrix 

dx<- out$x[2]- out$x[1] 
dy<-  out$y[2] - out$y[1] 

#  
# grid scale in degrees and choose kernel bandwidth to be .25 degrees. 

look<- image.smooth( out, aRange= .25)

# pass in a tophat kernel
topHat<- function( dd, h ){ ifelse( dd <= h^2, 1, 0)} 
## dd is the distance squared
look2<- image.smooth( out, kernel.function=topHat, h=.8)

image.plot(look) 
points( RMprecip$x)
US( add=TRUE, col="grey", lwd=2)

# to save on computation, decrease the padding with zeroes 
# only pad 32 grid points around the margins ofthe image. 

look<- image.smooth(out$z, dx=dx, dy=dy, aRange= .25, xwidth=32*dx,ywidth=32*dy) 

# the range of these data is ~ 10 degrees  and so 
# with a padding of 32 grid points  32*( 10/128) =  2.5 
# about 10 standard deviations of the normal kernel so there is still 
# lots of room for padding  
# a minimal choice might be  xwidth = 4*(.25)= 1  4 SD for the normal kernel
# creating weighting object outside the call  
# this is useful when one wants to smooth different data sets but on the 
# same grid with the same kernel function 
# 

#
#  random fields from smoothing white noise with this filter.
#
set.seed(123)
test.image<- matrix( rnorm(128**2),128,128)
dx<- .1
dy<- .8

wght<- setup.image.smooth( nrow=128, ncol=128,  dx=dx, dy=dy,
             aRange=.25, xwidth=2.5, ywidth=2.5)
#
look<- image.smooth( test.image, dx=dx, dy=dy, wght)

# NOTE:   this is the same as using 
#
#     image.smooth( test.image , 128,128), xwidth=2.5,
#                        ywidth=2.5, dx=dx,dy=dy, aRange=.25)
#
#   but the call to image.smooth is faster because the fft of kernel
#   has been precomputed.



# periodic smoothing in the horizontal dimension

look<- image.smooth( test.image , xwidth=1.5,
                        ywidth=2.5, dx=dx,dy=dy, aRange=1.5)
look2<- image.smooth( test.image , xwidth=0,
                        ywidth=2.5, dx=dx,dy=dy, aRange=1.5)
# compare these two
set.panel( 1,2)
image.plot( look, legend.mar=7.1)
title("free boundaries")
image.plot( look2, legend.mar=7.1) # look for periodic continuity at edges!
title("periodic boundary in horizontal")
set.panel(1,1)