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threshold.imwd

Threshold two-dimensional wavelet decomposition object

DESCRIPTION

This function provides various ways to threshold a imwd class object.

USAGE

threshold.imwd(imwd, levels = 3:(imwd$nlevels - 1), type = "hard", policy = 
        "universal", by.level = F, value = 0, dev = var, verbose = F, 
        return.threshold = F, compression = T, Q = 0.05)

REQUIRED ARGUMENTS

imwd: The two-dimensional wavelet decomposition object that you wish to threshold.

OPTIONAL ARGUMENTS

levels: a vector of integers which determines which scale levels are thresholded in the decomposition. Each integer in the vector must refer to a valid level in the imwd object supplied. This is usually any integer from 0 to nlevels(wd)-1 inclusive. Only the levels in this vector contribute to the computation of the threshold and its application. (except for the fdr policy).
type: determines the type of thresholding this can be "hard" or "soft".
policy: selects the technique by which the threshold value is selected. Each policy corresponds to a method in the literature. At present the different policies are: "universal", "manual", "fdr", "probability". The policies are described in detail below.
by.level: If FALSE then a global threshold is computed on and applied to all scale levels defined in levels. If TRUE a threshold is computed and applied separately to each scale level.
value: This argument conveys the user supplied threshold. If the policy="manual" then value is the actual threshold value; if policy="probability" then value conveys the the user supplied quantile level.
dev: this argument supplies the function to be used to compute the spread of the absolute values coefficients. The function supplied must return a value of spread on the variance scale (i.e. not standard deviation) such as the var() function. A popular, useful and robust alternative is the madmad function.
verbose: if TRUE then the function prints out informative messages as it progresses.
return.threshold: If this option is TRUE then the actual value of the threshold is returned. If this option is FALSE then a thresholded version of the input is returned.
compression: If this option is TRUE then this function returns a comressed two-dimensional wavelet transform object of class imwdc. This can be useful as the resulting object will be smaller than if it was not compressed. The compression makes use of the fact that many coefficients in a thresholded object will be exactly zero. If this option is FALSE then a larger imwd object will be returned.
Q: Parameter for the false discovery rate "fdr" policy.

VALUE

An object of class imwdc if the compression option above is TRUE, otherwise a imwd object is returned. In either case the returned object contains the thresholded coefficients. Note that if the return.threshold option is set to TRUE then the threshold values will be returned rather than the thresholded object.

SIDE EFFECTS

None

DETAILS

This function thresholds or shrinks wavelet coefficients stored in a imwd object and by default returns the coefficients in a modified imwdc object. See the seminal papers by Donoho and Johnstone for explanations about thresholding. For a gentle introduction to wavelet thresholding (or shrinkage as it is sometimes called) see Nason and Silverman, 1994. For more details on each technique see the descriptions of each method below

The basic idea of thresholding is very simple. In a signal plus noise model the wavelet transform of an image is very sparse, the wavelet transform of noise is not (in particular, if the noise is iid Gaussian then so if the noise contained in the wavelet coefficients). Thus since the image gets concentrated in few wavelet coefficients and the noise remains "spread" out it is "easy" to separate the signal from noise by keeping large coefficients (which correspond to true image) and delete the small ones (which correspond to noise). However, one has to have some idea of the noise level (computed using the dev option in threshold functions). If the noise level is very large then it is possible, as usual, that no image coefficients "stick up" above the noise.

There are many components to a successful thresholding procedure. Some components have a larger effect than others but the effect is not the same in all practical data situations. Here we give some rough practical guidance, although you must refer to the papers below when using a particular technique. You cannot expect to get excellent performance on all signals unless you fully understand the rationale and limitations of each method below. I am not in favour of the "black-box" approach. The thresholding functions of WaveThresh3 are not a black box: experience and judgement are required!

Some issues to watch for:

levels

The default of levels = 3:(wd$nlevels - 1) for the levels option most certainly does not work globally for all data problems and situations. The level at which thresholding begins (i.e. the given threshold and finer scale wavelets) is called the primary resolution and is unique to a particular problem. In some ways choice of the primary resolution is very similar to choosing the bandwidth in kernel regression albeit on a logarithmic scale. See Hall and Patil, (1995) and Hall and Nason (1997) for more information. For each data problem you need to work out which is the best primary resolution. This can be done by gaining experience at what works best, or using prior knowledge. It is possible to "automatically" choose a "best" primary resolution using cross-validation (but not in WaveThresh).

Secondly the levels argument computes and applies the threshold at the levels specified in the levels argument. It does this for all the levels specified. Sometimes, in wavelet shrinkage, the threshold is computed using only the finest scale coefficients (or more precisely the estimate of the overall noise level). If you want your threshold variance estimate only to use the finest scale coefficients (e.g. with universal thresholding) then you will have to apply the threshold.imwd function twice. Once (with levels set equal to nlevels(wd)-1 and with return.threshold=TRUE to return the threshold computed on the finest scale and then apply the threshold function with the manual option supplying the value of the previously computed threshold as the value options.

Note that the fdr policy does its own thing.

by.level

for a wd object which has come from data with noise that is correlated then you should have a threshold computed for each resolution level. See the paper by Johnstone and Silverman, 1997.

POLICIES

This section gives a brief description of the different thresholding policies available. For further details see the associated papers. If there is no paper available then a small description is provided here. More than one policy may be good for problem, so experiment! They are arranged here in alphabetical order:

fdr: See Abramovich and Benjamini, 1996. Contributed by Felix Abramovich.
manual: specify a user supplied threshold using value to pass the value of the threshold. The value argument should be a vector. If it is of length 1 then it is replicated to be the same length as the levels vector, otherwise it is repeated as many times as is necessary to be the levels vector's length. In this way, different thresholds can be supplied for different levels. Note that the by.level option has no effect with this policy.
probability: The probability policy works as follows. All coefficients that are smaller than the valueth quantile of the coefficients are set to zero. If by.level is false, then the quantile is computed for all coefficients in the levels specified by the "levels" vector; if by.level is true, then each level's quantile is estimated separately. The probability policy is pretty stupid - do not use it.
universal: See Donoho and Johnstone, 1995.

Acknowledgement

The FDR code segments were kindly donated by Felix Abramovich.

RELEASE

EXAMPLES

#
# Let's use the lennon test image. Lennon has 256x256 pixels with the
# range of pixel values being 0 to 249. 
#
image(lennon)
# 
#
# Now let's do the 2D discrete wavelet transform with the default arguments.
#
lwd <- imwd(lennon)
#
# Let's look at the coefficients
#
plot(lwd)

#
# Let's explain the plot. The plot consists of a number of subimages.
# Each subimage corresponds to wavelet coefficients at different scales
# and different orientations. See the help for imwd
# but here is a brief description. Each scale contains three orientations:
# horizontal, vertical and diagonal. The plot contains all the scales and
# all the orientations. The three large subimages at the top left, top right
# and bottom right are the finest scale detail in the vertical, diagonal and
# horizontal directions. The next largest three subimages tucked under the
# previous mentioned are the same  orientations but at the next coarser scale.
# As one moves to the bottom left of the whole image the scale of the images
# gets coarser and coarser. For 256^2 pixels there are log_2(256)=8 scales
# of subimages although it is harded to see some of the smaller, coarser
# scales.
# 
# You can get to see coarser scales by extracting the coefficients out
# using the lt.to.name function. 
#
#
# Now let's threshold the coefficients
#
lwdT <- threshold(lwd)
#
# And let's plot those the thresholded coefficients
#
plot(lwdT)

#
# Note that the only remaining coefficients are down in the bottom
# left hand corner of the plot. All the others (black) have been set
# to zero (i.e. thresholded).