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)

Arguments

x

\(N\)-point vector or \(M{\times}N\) 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.

Value

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

Details

In one dimension \(N\) is divisible by \(2^J\) and \(N\ge2^{J-1}\cdot\mbox{length}(\mbox{\code{af}})\).

In two dimensions, these two conditions must hold for both \(M\) and \(N\).

References

WAVELET SOFTWARE AT POLYTECHNIC UNIVERSITY, BROOKLYN, NY
http://taco.poly.edu/WaveletSoftware/

See also

FSfarras, farras, convolve, cshift, afb, sfb.

Author

Matlab: S. Cai, K. Li and I. Selesnick; R port: B. Whitcher

Examples

## 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))
#> [1] 1.895423e-08

## 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))
#> [1] 1.930035e-08

## 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)