Two-Dimensional Maximal Overlap Discrete Wavelet Transform

Performs a separable two-dimensional maximal overlap discrete wavelet transform (MODWT) on a matrix of arbitrary dimensions.

modwt.2d(x, wf, J = 4, boundary = "periodic")
imodwt.2d(y)

Arguments

x	input matrix
wf	name of the wavelet filter to use in the decomposition
J	depth of the decomposition
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

Liang, J. and T. W. Parks (1994) A two-dimensional translation invariant wavelet representation and its applications, Proceedings ICIP-94, Vol. 1, 66-70.

Liang, J. and T. W. Parks (1994) Image coding using translation invariant wavelet transforms with symmetric extensions, IEEE Transactions on Image Processing, 7, No. 5, 762-769.

See also

dwt.2d, shift.2d.

Author

B. Whitcher

Examples

## Xbox image
data(xbox)
xbox.modwt <- modwt.2d(xbox, "haar", 2)
## Level 1 decomposition
par(mfrow=c(2,2), pty="s")
image(xbox.modwt$LH1, col=rainbow(128), axes=FALSE, main="LH1")
image(xbox.modwt$HH1, col=rainbow(128), axes=FALSE, main="HH1")
frame()
image(xbox.modwt$HL1, col=rainbow(128), axes=FALSE, main="HL1")

## Level 2 decomposition
par(mfrow=c(2,2), pty="s")
image(xbox.modwt$LH2, col=rainbow(128), axes=FALSE, main="LH2")
image(xbox.modwt$HH2, col=rainbow(128), axes=FALSE, main="HH2")
image(xbox.modwt$LL2, col=rainbow(128), axes=FALSE, main="LL2")
image(xbox.modwt$HL2, col=rainbow(128), axes=FALSE, main="HL2")

sum((xbox - imodwt.2d(xbox.modwt))^2)
#> [1] 0

data(dau)
par(mfrow=c(1,1), pty="s")
image(dau, col=rainbow(128), axes=FALSE, main="Ingrid Daubechies")

sum(dau^2)
#> [1] 1049732962
dau.modwt <- modwt.2d(dau, "d4", 2)
## Level 1 decomposition
par(mfrow=c(2,2), pty="s")
image(dau.modwt$LH1, col=rainbow(128), axes=FALSE, main="LH1")
image(dau.modwt$HH1, col=rainbow(128), axes=FALSE, main="HH1")
frame()
image(dau.modwt$HL1, col=rainbow(128), axes=FALSE, main="HL1")

## Level 2 decomposition
par(mfrow=c(2,2), pty="s")
image(dau.modwt$LH2, col=rainbow(128), axes=FALSE, main="LH2")
image(dau.modwt$HH2, col=rainbow(128), axes=FALSE, main="HH2")
image(dau.modwt$LL2, col=rainbow(128), axes=FALSE, main="LL2")
image(dau.modwt$HL2, col=rainbow(128), axes=FALSE, main="HL2")

sum((dau - imodwt.2d(dau.modwt))^2)
#> [1] 0