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#' Dual-tree Complex 2D Discrete Wavelet Transform
#'
#' Dual-tree complex 2D discrete wavelet transform (DWT).
#'
#'
#' @usage cplxdual2D(x, J, Faf, af)
#' @usage icplxdual2D(w, J, Fsf, sf)
#' @aliases cplxdual2D icplxdual2D
#' @param x 2D array.
#' @param w wavelet coefficients.
#' @param J number of stages.
#' @param Faf first stage analysis filters for tree i.
#' @param af analysis filters for the remaining stages on tree i.
#' @param Fsf last stage synthesis filters for tree i.
#' @param sf synthesis filters for the preceeding stages.
#' @return For the analysis of \code{x}, the output is \item{w}{wavelet
#' coefficients indexed by \code{[[j]][[i]][[d1]][[d2]]}, where
#' \eqn{j=1,\ldots,J} (scale), \eqn{i=1} (real part) or \eqn{i=2} (imag part),
#' \eqn{d1=1,2} and \eqn{d2=1,2,3} (orientations).} For the synthesis of
#' \code{w}, the output is \item{y}{output signal.}
#' @author Matlab: S. Cai, K. Li and I. Selesnick; R port: B. Whitcher
#' @seealso \code{\link{FSfarras}}, \code{\link{farras}}, \code{\link{afb2D}},
#' \code{\link{sfb2D}}.
#' @keywords ts
#' @examples
#'
#' \dontrun{
#' ## EXAMPLE: cplxdual2D
#' x = matrix(rnorm(32*32), 32, 32)
#' J = 5
#' Faf = FSfarras()$af
#' Fsf = FSfarras()$sf
#' af = dualfilt1()$af
#' sf = dualfilt1()$sf
#' w = cplxdual2D(x, J, Faf, af)
#' y = icplxdual2D(w, J, Fsf, sf)
#' err = x - y
#' max(abs(err))
#' }
#'
cplxdual2D <- function(x, J, Faf, af) {
## Dual-Tree Complex 2D Discrete Wavelet Transform
##
## USAGE:
## w = cplxdual2D(x, J, Faf, af)
## INPUT:
## x - 2-D array
## J - number of stages
## Faf{i}: first stage filters for tree i
## af{i}: filters for remaining stages on tree i
## OUTPUT:
## w{j}{i}{d1}{d2} - wavelet coefficients
## j = 1..J (scale)
## i = 1 (real part); i = 2 (imag part)
## d1 = 1,2; d2 = 1,2,3 (orientations)
## w{J+1}{m}{n} - lowpass coefficients
## d1 = 1,2; d2 = 1,2
## EXAMPLE:
## x = rand(256);
## J = 5;
## [Faf, Fsf] = FSfarras;
## [af, sf] = dualfilt1;
## w = cplxdual2D(x, J, Faf, af);
## y = icplxdual2D(w, J, Fsf, sf);
## err = x - y;
## max(max(abs(err)))
##
## WAVELET SOFTWARE AT POLYTECHNIC UNIVERSITY, BROOKLYN, NY
## http://eeweb.poly.edu/iselesni/WaveletSoftware/
## normalization
x <- x/2
w <- vector("list", J+1)
for (m in 1:2) {
w[[1]][[m]] <- vector("list", 2)
for (n in 1:2) {
w[[1]][[m]][[n]] <- vector("list", 2)
temp <- afb2D(x, Faf[[m]], Faf[[n]])
lo <- temp$lo
w[[1]][[m]][[n]] <- temp$hi
if (J > 1) {
for (j in 2:J) {
temp <- afb2D(lo, af[[m]], af[[n]])
lo <- temp$lo
w[[j]][[m]][[n]] <- temp$hi
}
w[[J+1]][[m]][[n]] <- lo
}
}
}
for (j in 1:J) {
for (m in 1:3) {
w[[j]][[1]][[1]][[m]] <- pm(w[[j]][[1]][[1]][[m]])
w[[j]][[2]][[2]][[m]] <- pm(w[[j]][[2]][[2]][[m]])
w[[j]][[1]][[2]][[m]] <- pm(w[[j]][[1]][[2]][[m]])
w[[j]][[2]][[1]][[m]] <- pm(w[[j]][[2]][[1]][[m]])
}
}
return(w)
}
icplxdual2D <- function(w, J, Fsf, sf) {
## Inverse Dual-Tree Complex 2D Discrete Wavelet Transform
##
## USAGE:
## y = icplxdual2D(w, J, Fsf, sf)
## INPUT:
## w - wavelet coefficients
## J - number of stages
## Fsf - synthesis filters for final stage
## sf - synthesis filters for preceeding stages
## OUTPUT:
## y - output array
## See cplxdual2D
##
## WAVELET SOFTWARE AT POLYTECHNIC UNIVERSITY, BROOKLYN, NY
## http://eeweb.poly.edu/iselesni/WaveletSoftware/
for (j in 1:J) {
for (m in 1:3) {
w[[j]][[1]][[1]][[m]] <- pm(w[[j]][[1]][[1]][[m]])
w[[j]][[2]][[2]][[m]] <- pm(w[[j]][[2]][[2]][[m]])
w[[j]][[1]][[2]][[m]] <- pm(w[[j]][[1]][[2]][[m]])
w[[j]][[2]][[1]][[m]] <- pm(w[[j]][[2]][[1]][[m]])
}
}
y <- matrix(0, 2*nrow(w[[1]][[1]][[1]][[1]]), 2*ncol(w[[1]][[1]][[1]][[1]]))
for (m in 1:2) {
for (n in 1:2) {
lo <- w[[J+1]][[m]][[n]]
if (J > 1) {
for (j in J:2) {
lo <- sfb2D(lo, w[[j]][[m]][[n]], sf[[m]], sf[[n]])
}
lo <- sfb2D(lo, w[[1]][[m]][[n]], Fsf[[m]], Fsf[[n]])
y <- y + lo
}
}
}
## normalization
return(y/2)
}
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