Dual-tree Complex Discrete Wavelet Transform
One- and two-dimensional dual-tree complex discrete wavelet transforms developed by Kingsbury and Selesnick et al.
dualtree(x, J, Faf, af) idualtree(w, J, Fsf, sf) dualtree2D(x, J, Faf, af) idualtree2D(w, J, Fsf, sf)
x |
N-point vector or MxN matrix. |
w |
DWT coefficients. |
J |
number of stages. |
Faf |
analysis filters for the first stage. |
af |
analysis filters for the remaining stages. |
Fsf |
synthesis filters for the last stage. |
sf |
synthesis filters for the preceeding stages. |
In one dimension N is divisible by 2^J and N≥2^{J-1}\cdot\mbox{length}(\mbox{\code{af}}).
In two dimensions, these two conditions must hold for both M and N.
For the analysis of x, the output is
w |
DWT coefficients. Each wavelet scale is a list containing the real and imaginary parts. The final scale (J+1) contains the low-pass filter coefficients. |
For the synthesis of w, the output is
y |
output signal |
Matlab: S. Cai, K. Li and I. Selesnick; R port: B. Whitcher
WAVELET SOFTWARE AT POLYTECHNIC UNIVERSITY, BROOKLYN, NY
http://eeweb.poly.edu/iselesni/WaveletSoftware/
## EXAMPLE: dualtree x = rnorm(512) J = 4 Faf = FSfarras()$af Fsf = FSfarras()$sf af = dualfilt1()$af sf = dualfilt1()$sf w = dualtree(x, J, Faf, af) y = idualtree(w, J, Fsf, sf) err = x - y max(abs(err)) ## Example: dualtree2D x = matrix(rnorm(64*64), 64, 64) J = 3 Faf = FSfarras()$af Fsf = FSfarras()$sf af = dualfilt1()$af sf = dualfilt1()$sf w = dualtree2D(x, J, Faf, af) y = idualtree2D(w, J, Fsf, sf) err = x - y max(abs(err)) ## Display 2D wavelets of dualtree2D.m J <- 4 L <- 3 * 2^(J+1) N <- L / 2^J Faf <- FSfarras()$af Fsf <- FSfarras()$sf af <- dualfilt1()$af sf <- dualfilt1()$sf x <- matrix(0, 2*L, 3*L) w <- dualtree2D(x, J, Faf, af) w[[J]][[1]][[1]][N/2, N/2+0*N] <- 1 w[[J]][[1]][[2]][N/2, N/2+1*N] <- 1 w[[J]][[1]][[3]][N/2, N/2+2*N] <- 1 w[[J]][[2]][[1]][N/2+N, N/2+0*N] <- 1 w[[J]][[2]][[2]][N/2+N, N/2+1*N] <- 1 w[[J]][[2]][[3]][N/2+N, N/2+2*N] <- 1 y <- idualtree2D(w, J, Fsf, sf) image(t(y), col=grey(0:64/64), axes=FALSE)
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