cddews {LS2W}

Compute the local wavelet spectrum estimate

Description

This function computes the local wavelet spectrum (LWS) estimate of an image (or non-decimated wavelet transform of a time series). The estimate is computed by taking the non-decimated wavelet transform of the image, squaring the detail coefficients, smoothing using wavelet shrinkage and then correcting the redundancy caused by use of the non-decimated wavelet transform.

Usage

cddews(data, filter.number = 1, family = "DaubExPhase", switch = "direction", 
correct = TRUE, verbose = FALSE, smooth = TRUE, 
sm.filter.number = 4, sm.family = "DaubExPhase", levels = 3:6, type = "hard", 
policy = "LSuniversal", by.level = FALSE, value = 0, dev = var)

Arguments

data

The image you want to analyse.

filter.number

This selects the index of the wavelet used in the analysis of the time series (i.e.\ the wavelet basis functions used to model the time series). For Daubechies compactly supported wavelets the filter number is the number of vanishing moments.

family

This selects the wavelet family to use in the analysis of the time series (i.e.\ which wavelet family to use to model the time series). Only use the Daubechies compactly supported wavelets DaubExPhase and DaubLeAsymm.

switch

This allows one to order the corrected spectrum by scale or decomposition direction. Two options are available switch = "direction": structures the matrix by scale within each decomposition direction. Thus, the ordering goes as follows $(-1, V), (-2, V), (-3, V)\ ...$. switch = "level" structures the matrix by direction within each scale. Thus the ordering is as follows $(-1,V), (-1, H), (-1, D), (-2, V), (-2, H), (-2, D), ...$. For further details, see Eckley Nason and Treloar (2010).

correct

Eckley, Nason and Treloar (2009) have demonstrated that, as a consequence of the inherent redundancy of the non-decimated wavelet transform, the raw wavelet spectrum is biased. However, an asymptotically unbiased estimator may be obtained by applying the inverse of the inner product matrix of discrete autocorrelation wavelets. This argument permits the user to decide whether or not to correct for this inherent bias.

verbose

Allows certain informative messages to be printed on screen.

smooth

This binary argument allows the user to specify whether or not the resulting local wavelet periodogram should be smoothed to obtain. It is advised that this option be set to TRUE in order that consistent estimates be obtained.

sm.filter.number

Selects the index number of the wavelet that smooths each scale of the wavelet periodogram.

sm.family

Selects the wavelet family that smooths each scale of the wavelet periodogram.

levels

This specifies the levels which are smoothed when performing the wavelet shrinkage.

type

The type of shrinkage: either "hard" or "soft".

policy

This dictates the threshold selection method used for smoothing. For LWS estimation LSuniversal is recommended for thi Chi-squared nature of the periodogram coefficients.

by.level

If TRUE then the wavelet shrinkage is performed by computing and applying a separate threshold to each level in the transform of each scale. Note that each scale in the LWS is smoothed separately and independently. Each smooth consists of taking the (second-stage) non-decimated wavelet transform and applying a threshold to each level of a wavelet transformed scale.

If FALSE then the same threshold is applied to the discrete wavelet transform of a scale. Different thresholds may be computed for different scales but the threshold will be the same for each level arising from the non-decimated transform of a scale.

value

This argument supplies the threshold value used when a manual policy is adopted.

dev

The method for estimating the variance of the empirical wavelet coefficients for smoothing purposes.

Details

This function computes an estimate of the directionally dependent wavelet spectrum of an image according to the work of Eckley, Nason and Treloar (2010). The function works as follows:

1. The non-decimated wavelet transform of the series is computed.

2. The squared modulus of the non-decimated wavelet transform is computed (this is the raw wavelet periodogram, which is returned).

3. The squared modulus is smoothed using wavelet shrinkage.

4. The smoothed coefficients are corrected using the inverse of the inner product matrix of the autocorrelation wavelets. To display the LWS use the specplot function on the S component (see the examples below).

Value

A list with the following components:

S:

The directionally dependent wavelet spectral estimate of the input data. This is a large array, the first dimension refers to a specific scale-direction pair (see Eckley et al. (2009) for further details). The next dimension refers to the rows of the spectral image, whilst the third element refers to the columns of the image.

datadim

The dimension of the original image.

filter.number:

This gives the index of the wavelet used in the analysis of the image (i.e. the wavelet basis functions used in the modelling). For Daubechies compactly supported wavelets the filter number is the number of vanishing moments.

family:

This contains the wavelet family used in the analysis of the image (i.e. the wavelet family used in the modelling).

structure:

Explains the structure of the inner product matrix and S. It can only take two values, direction and scale.

nlevels:

The number of levels in the decomposition.

correct:

TRUE or FALSE, depending on whether the user corrected for the bias.

smooth:

TRUE or FALSE, depending on whether the LWP has been smoothed.

date:

The date when the analysis was perfromed.

Author(s)

Idris Eckley

References

Eckley, I.A., Nason, G.P. and Treloar, R.L. (2010) Locally stationary wavelet fields with application to the modelling and analysis of image texture. Journal of the Royal Statistical Society (Series C), 59, 595 - 616.

Eckley, I.A. and Nason, G.P. (2011). LS2W: Implementing the Locally Stationary 2D Wavelet Process Approach in R, Journal of Statistical Software, 43(3), 1-23. URL http://www.jstatsoft.org/v43/i03/.

See Also

D2Amat, specplot

Examples

# Apply the cddews estimate function to a HaarMontage realisation
#
monty <- HaarMontage(direction="diagonal")
monty.cddews <- cddews(monty, filter.number=1, family="DaubExPhase")
monty.cddews