Single-level discrete 2-D wavelet transform
[cA,cH,cV,cD] = dwt2(X,
[cA,cH,cV,cD] = dwt2(X,Lo_D,Hi_D)
[cA,cH,cV,cD] = dwt2(...,'mode',MODE)
dwt2 command performs
a single-level two-dimensional wavelet decomposition. Compare this
may be more useful for your application. The decomposition is done
with respect to either a particular wavelet (
wfilters for more information)
or particular wavelet decomposition filters (
[cA,cH,cV,cD] = dwt2(X, computes
the approximation coefficients matrix
cA and details
cD (horizontal, vertical, and diagonal, respectively),
obtained by wavelet decomposition of the input matrix
'wname' character vector contains the
[cA,cH,cV,cD] = dwt2(X,Lo_D,Hi_D) computes
the two-dimensional wavelet decomposition as above, based on wavelet
decomposition filters that you specify.
Lo_D is the decomposition low-pass
Hi_D is the decomposition high-pass
Hi_D must be
the same length.
sx = size(X) and
lf = the
length of filters; then
size(cA) = size(cH) = size(cV) = size(cD) = sa where
= ceil(sx/2), if the DWT extension mode is set to periodization.
For the other extension modes,
sa = floor((sx+lf-1)/2).
For information about the different Discrete Wavelet Transform
extension modes, see
[cA,cH,cV,cD] = dwt2(...,'mode',MODE) computes
the wavelet decomposition with the extension mode
MODE is a character vector containing the
desired extension mode.
An example of valid use is
[cA,cH,cV,cD] = dwt2(x,'db1','mode','sym');
This example shows how to obtain the 2-D DWT of an image.
Load the "woman" image and obtain the 2-D DWT using the
sym4 wavelet. Use the periodic extension mode.
load woman wname = 'sym4'; [CA,CH,CV,CD] = dwt2(X,wname,'mode','per');
Display the vertical detail image and the lowpass approximation.
subplot(211) imagesc(CV) title('Vertical Detail Image') colormap gray subplot(212) imagesc(CA) title('Lowpass Approximation')
When X represents an indexed image, then X, as well as the output
arrays cA,cH,cV,cD are
When X represents a truecolor image, it is an
array, where each
represents a red, green, or blue color plane concatenated along the
For images, there exist an algorithm similar to the one-dimensional case for two-dimensional wavelets and scaling functions obtained from one- dimensional ones by tensorial product.
This kind of two-dimensional DWT leads to a decomposition of approximation coefficients at level j in four components: the approximation at level j + 1, and the details in three orientations (horizontal, vertical, and diagonal).
The following chart describes the basic decomposition steps for images:
To deal with signal-end effects involved by a convolution-based
algorithm, a global variable managed by
used. This variable defines the kind of signal extension mode used.
The possible options include zero-padding (used in the previous example)
and symmetric extension, which is the default mode.
Daubechies, I. (1992), Ten lectures on wavelets, CBMS-NSF conference series in applied mathematics. SIAM Ed.
Mallat, S. (1989), “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Pattern Anal. and Machine Intell., vol. 11, no. 7, pp. 674–693.
Meyer, Y. (1990), Ondelettes et opérateurs, Tome 1, Hermann Ed. (English translation: Wavelets and operators, Cambridge Univ. Press. 1993.)