AvBasis.wst2D           package:wavethresh           R Documentation

_P_e_r_f_o_r_m _b_a_s_i_s _a_v_e_r_a_g_i_n_g _f_o_r (_p_a_c_k_e_t-_o_r_d_e_r_e_d) _2_D _n_o_n-_d_e_c_i_m_a_t_e_d _w_a_v_e_l_e_t _t_r_a_n_s_f_o_r_m.

_D_e_s_c_r_i_p_t_i_o_n:

     Perform basis averaging for (packet-ordered) 2D non-decimated
     wavelet transform.

_U_s_a_g_e:

     AvBasis.wst2D(wst2D, ...)

_A_r_g_u_m_e_n_t_s:

   wst2D: An object of class 'wst2D' that contains coefficients of a
          packet ordered 2D non-decimated wavelet transform (e.g.
          produced by the 'wst2D' function.

     ...: any other arguments

_D_e_t_a_i_l_s:

     The packet-ordered 2D non-decimated wavelet transform computed by
     'wst2D' computes the coefficients of an input matrix with respect
     to a library of all shifts of wavelet basis functions at all
     scales. Here "all shifts" means all integral shifts with respect
     to the finest scale coefficients with shifts in both the
     horizontal and vertical directions, and "all scales" means all
     dyadic scales from 0 (the coarsest) to J-1 (the finest) where '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.

_V_a_l_u_e:

     A square matrix of dimension $2^nlevels$ containing the
     average-basis ``reconstruction'' of the 'wst2D' object.

_R_E_L_E_A_S_E:

     Version 3.9 Copyright Guy Nason 1998

_S_e_e _A_l_s_o:

     'wst2D', 'wst2D.object'

_E_x_a_m_p_l_e_s:

     #
     # 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.
     #

