wavethresh (version 2.2-11)

draw.default: Draw a Picture of a Wavelet

Description

Draws pictures of basic (Daubechies') wavelets in the wavethresh package.

Usage

## S3 method for class 'default':
draw(x = filter.number, filter.number = 2,
     family = c("DaubExPhase", "DaubLeAsymm"),
     resolution = 1024, verbose = FALSE, plot = TRUE,
     main = paste("Wavelet Picture", if(enhance) " (Enhanced)", sep=""),
     sub = zwd$filter$name, xlab = "x", ylab = expression(psi(x)), dimension = 1,
     twodplot = persp, enhance = TRUE, efactor = 0.05, ...)

Arguments

x, filter.number

integer; number of the filter in the wavelet family specified. The range of possible numbers depends on the family.

family

character, the family of wavelets to use, defaults to the first of the list shown above.

resolution

integer specifying the resolution to use in drawing pictures. A higher resolution gives a better quality picture but will take longer to draw (especially in two dimensions).

verbose

logical, if TRUE, report the progress of drawing.

plot

logical, if true then plot the appropriate wavelet on the current active graphics device, otherwise the information about the wavelet is returned.

main, sub

main and sub-title for the plot, see title.

xlab, ylab

labels for the x and y axis.

dimension

If this is 1, then a one-dimensional version of the wavelet is drawn, if this is 2, then a two-d picture is drawn with the twodplot function.

twodplot

function such as persp that can do something interesting with a matrix.

enhance

logical; with Daubechies' wavelets their effective support is much less than their actual support. If enhance is true, the function tries to draw the wavelet on its effective support.

efactor

numeric, defining the effective support. Define z0 to be efactor times the maximum absolute value of the wavelet w(x). Define the set $A = { x: |w(x)| > z0 }$, and the effective support is the smallest interval (in 1D, square in 2D) contai

...

further arguments to plot or twodplot.

Value

• If plot is false then no actual drawing is done, however, a list is returned with dimension + 1 components: If dimension is one, the list has components x and y, representing the domain of definition (or effective support) of the wavelet and the value of the wavelet at x. If dimension == 2, the list has three components x,y and z.

Side Effects

a picture of a wavelet is drawn on the currently active graphics device if plot=TRUE.

RELEASE

Release 2.2 Copyright Guy Nason 1993

Details

For Daubechies' compactly supported wavelets there is no known closed form expression. However, it is possible to make use of the inverse wavelet transform to draw pictures of the wavelets. The idea is to set all but one wavelet coefficient equal to zero and that one coefficient equal to one in a wavelet expansion. Then the inverse transform (reconstruction) is applied to the expansion and a picture of one wavelet is produced.

A method similar to the one we present here is presented in Daubechies (1992) in Section~6.5 on the cascade algorithm.

The principle is simple, but the implementation to get good pictures is surprisingly tricky. Ideally you want to put in the non-zero coefficient at the lowest resolution level as possible, this will give you as much detail as possible. However, the support of some of the large-scale wavelets is greater than the union of the supports of all the high-resolution small-scale wavelets and so it may not be possible to draw a complete wavelet.

This function analyses the supports of the wavelets at different levels and finds a coefficient to set to one by choosing the wavelet at the lowest possible resolution and whose support is within the union of supports of the highest resolution wavelets. If there is a choice of such wavelets, then the middle-most one is selected.

See Also

draw, draw.wd etc.

Examples

op <- par(mfrow=c(5,2), oma = c(0,0, 4, 0),
          mgp = c(1.2, .8, 0), mar = .1 + c(4,4, .5,1))
for(fn in 1:10) {
   draw.default(filter.number= fn, col = "blue", main = NULL, xlab= "")
   abline(h=0,v=0, lty=3, lwd=.5, col = "gray")
}
mtext(paste("draw.default(*, family = '",formals(draw.default)$family[[2]],"')"),
      side = 3, line = 1, outer = TRUE,
      cex = par("cex.main"), font = par("font.main"))
par(op)

# Draw a 2-dimensional Daubechies least asymmetric wavelet
draw.default(filter.number=6, family="DaubLeAsymm", dim=2, resolution=128)