Two-Dimensional Discrete Wavelet Transform

Performs a separable two-dimensional discrete wavelet transform (DWT) on a matrix of dyadic dimensions.

dwt.2d(x, wf, J = 4, boundary = "periodic")
idwt.2d(y)

Arguments

x	input matrix (image)
wf	name of the wavelet filter to use in the decomposition
J	depth of the decomposition, must be a number less than or equal to \(\log_2(\min\{M,N\})\)
boundary	only "periodic" is currently implemented
y	an object of class dwt.2d

Value

List structure containing the \(3J+1\) sub-matrices from the decomposition.

Details

See references.

References

Mallat, S. (1998) A Wavelet Tour of Signal Processing, Academic Press.

Vetterli, M. and J. Kovacevic (1995) Wavelets and Subband Coding, Prentice Hall.

See also

modwt.2d.

Author

B. Whitcher

Examples

## Xbox image
data(xbox)
xbox.dwt <- dwt.2d(xbox, "haar", 3)
par(mfrow=c(1,1), pty="s")
plot.dwt.2d(xbox.dwt)

par(mfrow=c(2,2), pty="s")
image(1:dim(xbox)[1], 1:dim(xbox)[2], xbox, xlab="", ylab="",
      main="Original Image")
image(1:dim(xbox)[1], 1:dim(xbox)[2], idwt.2d(xbox.dwt), xlab="", ylab="",
      main="Wavelet Reconstruction")
image(1:dim(xbox)[1], 1:dim(xbox)[2], xbox - idwt.2d(xbox.dwt),
      xlab="", ylab="", main="Difference")

## Daubechies image
data(dau)
par(mfrow=c(1,1), pty="s")

image(dau, col=rainbow(128))

sum(dau^2)
#> [1] 1049732962
dau.dwt <- dwt.2d(dau, "d4", 3)
plot.dwt.2d(dau.dwt)

sum(plot.dwt.2d(dau.dwt, plot=FALSE)^2)
#> [1] 1049732962