AvBasis.wst2D(wst2D)
2^J = n where n is the dimension of
the input matrix. As such the packet-ordered
2D non-decimated wavelet transform contains a library of all possible shifted
wavelet bases.
Basis averaging. Rather than select a basis it is often useful to preserve information from all of the bases. For example, in curve estimation, after thresholding, the coefficients are coefficients of an estimate of the truth with respect to all of the shifted basis functions. Rather than select one of them we can average over all estimates. This sometimes gives a better curve estimate and can, for example, get rid of Gibbs effects. See Coifman and Donoho (1995) for more information on how to do curve estimation using the packet ordered non-decimated wavelet transform, thresholding and basis averaging. See Lang et al. (1995) for further details of surface/image estimation using the 2D non-decimated DWT.
# # Generate some test data # test.data <- matrix(rnorm(16), 4,4) # # Now take the 2D packet ordered DWT # tdwst2D <- wst2D(test.data) # # Now "invert" it using basis averaging # tdwstAB <- AvBasis(tdwst2D) # # Let's compare it to the original # sum( (tdwstAB - test.data)^2) # # [1] 1.61215e-17 # # Very small. They're essentially same. #