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wst2D objects

(Packet ordered) Two-dimensional nondecimated wavelet transform decomposition objects.

DESCRIPTION

These are objects of class


wst2D

They represent a decomposition of a function with respect to a set of (all possible) shifted two-dimensional wavelets. They are a 2D extension of the wst objects.

GENERATION

This class of objects is returned from the wst2D function which computes the packets-ordered two-dimensional non-decimated wavelet transform (effectively all possible shifts of the standard two-dimensional discrete wavelet transform).

Many other functions return an object of class wst2D.

METHODS

The wst2D class of objects has methods for the following generic functions: AvBasis, getpacket. plot, print, putpacket, summary,

INHERITANCE

None (yet).

STRUCTURE

The following components must be included in a legitimate wst2D object.

wst2D

This a three-dimensional array. Suppose that the original image that created the wst2D object is n x n. Then the dimension of the wst2D array is [ nlevels, 2n, 2n]. The first index of the array refers to the resolution level of the array with "resolution level = index - 1" (so, e.g. the coarsest scale detailed is stored at index 1 and the finest at index nlevels). For a given resolution level (selected first index) the associated 2n x 2n matrix contains the two-dimensional non-decimated wavelet coefficients for that level packed as follows.

At the finest resolution level the 2n x 2n coefficient image may be broken up into four n x n subimages. Each of the four images corresponds to data shifts in the horizontal and vertical directions. The top left image corresponds to ``no shift'' and indeed the top left image corresponds to the coefficients obtained using the decimated 2D wavelet transform as obtained using the imwd function. The top right image corresponds to a horizontal data shift; the bottom left to a vertical data shift and the bottom right corresponds to both horizontal and vertical data shift.

Within each of the four n x n images named in the previous paragraph are again 4 subimages each of dimension n/2 x n/2. These correspond to (starting at the top left and moving clockwise) the smooth (CC), horizontal detail (DC), diagonal detail (DD) and vertical detail (CD).

At coarser resolution levels the coefficients are smaller submatrices corresponding to various levels of data shifts and types of detail (smooth, horizontal, vertical, diagonal).

We strongly recommend the use of the getpacket.wst2D and putpacket.wst2D functions to remove and replace coefficients from wst2D objects.

nlevels

The number of levels in the decomposition. If you raise 2 to the power of 2 nlevels you get the number of data points used in the decomposition.

filter

a list containing the details of the filter that did the decomposition (equivalent to the return value from the filter.select function).

date

The date that the transform was performed or the wst2D was modified.

DETAILS

To retain your sanity we recommend that the coefficients from a wst2D object be extracted or replaced using

getpacket.wst2D to obtain individual packets of either father or mother wavelet coefficients.
putpacket.wst2D to insert coefficients.

You can obtain the coefficients directly from the wst2D$wst2D component but you have to understand their organization described above.