swt2
Discrete stationary 2-D wavelet transform
Description
[
returns the approximation coefficients A
,H,V,D
] = swt2(X
,N
,wname
)A
and the horizontal,
vertical, and diagonal detail coefficients H
, V
,
and D
, respectively, of the stationary 2-D wavelet decomposition of
the image X
at level N
using the wavelet
wname
.
Note
swt2
is uses periodic extension.swt2
uses double-precision arithmetic internally and returns double-precision coefficient matrices.swt2
warns if there is a loss of precision when converting to double.
returns the
approximation and detail coefficients in swc
= swt2(___)swc
.
Examples
Extract and Display 2-D Stationary Wavelet Decomposition
Load and display an image.
load woman imagesc(X) colormap(map) title('Original')
Perform the stationary wavelet decomposition of the image at level 2 using db6
.
[ca,chd,cvd,cdd] = swt2(X,2,'db6');
Extract the level 1 and level 2 approximation and detail coefficients from the decomposition.
A1 = wcodemat(ca(:,:,1),255); H1 = wcodemat(chd(:,:,1),255); V1 = wcodemat(cvd(:,:,1),255); D1 = wcodemat(cdd(:,:,1),255); A2 = wcodemat(ca(:,:,2),255); H2 = wcodemat(chd(:,:,2),255); V2 = wcodemat(cvd(:,:,2),255); D2 = wcodemat(cdd(:,:,2),255);
Display the approximation and detail coefficients from the two levels.
subplot(2,2,1) imagesc(A1) title('Approximation Coef. of Level 1') subplot(2,2,2) imagesc(H1) title('Horizontal Detail Coef. of Level 1') subplot(2,2,3) imagesc(V1) title('Vertical Detail Coef. of Level 1') subplot(2,2,4) imagesc(D1) title('Diagonal Detail Coef. of Level 1')
subplot(2,2,1) imagesc(A2) title('Approximation Coef. of Level 2') subplot(2,2,2) imagesc(H2) title('Horizontal Detail Coef. of Level 2') subplot(2,2,3) imagesc(V2) title('Vertical Detail Coef. of Level 2') subplot(2,2,4) imagesc(D2) title('Diagonal Detail Coef. of Level 2')
Stationary Wavelet Transform of RGB Image
This example shows how to obtain single-level and multilevel stationary wavelet decompositions of an RGB image.
Load and view an RGB image. The image is a 3-D array of type uint8
. Since swt2
requires that the first and second dimensions both be divisible by a power of 2, extract a portion of the image.
imdata = imread('ngc6543a.jpg'); x = imdata(1:512,1:512,:); image(x) title('RGB Image')
Obtain the level 4 stationary wavelet decomposition of the image using the db4
wavelet. Return the approximation coefficients. Note the dimensions of the coefficients array.
[a,~,~,~] = swt2(x,4,'db4');
size(a)
ans = 1×4
512 512 3 4
The coefficients are all of type double
. In an RGB array of type double
, each color component is a value between 0 and 1. Rescale the level 2 approximation coefficients to values between 0 and 1 and view the result.
a2 = a(:,:,:,2);
a2 = (a2-min(a2(:)))/(max(a2(:))-min(a2(:)));
image(a2)
title('Level 2 Approximation')
Obtain the single-level stationary wavelet decomposition of the image using the db4
wavelet. Return the approximation coefficients. In a single-level decomposition of an RGB image, the third dimension is singleton.
[a,~,~,~] = swt2(x,1,'db4');
size(a)
ans = 1×4
512 512 1 3
View the approximation coefficients. To prevent an error when using image
, squeeze the approximation coefficients array to remove the singleton dimension.
a2 = squeeze(a);
a2 = (a2-min(a(:)))/(max(a(:))-min(a(:)));
image(a2)
title('Approximation')
Input Arguments
X
— Input image
2-D matrix | 3-D array
Input image, specified as a real-valued 2-D matrix or real-valued 3-D array. If
X
is 3-D, X
is assumed to be an RGB image,
also referred to as a truecolor image, and the third dimension of
X
must equal 3. For more information on truecolor images, see
Working with Image Types in MATLAB.
Data Types: double
N
— Level of decomposition
positive integer
Level of decomposition, specified as a positive integer.
2N must divide size(X,1)
and
size(X,2)
. Use wmaxlev
to determine the maximum level of decomposition.
wname
— Analyzing wavelet
character vector | string scalar
Analyzing wavelet, specified as a character vector or string scalar.
swt2
supports only Type 1 (orthogonal) or Type 2 (biorthogonal)
wavelets. See wfilters
for a list of orthogonal and
biorthogonal wavelets.
LoD,HiD
— Wavelet decomposition filters
even-length real-valued vectors
Wavelet decomposition filters, specified as a pair of even-length real-valued
vectors. LoD
is the lowpass decomposition filter, and
HiD
is the highpass decomposition filter. The lengths of
LoD
and HiD
must be equal. See wfilters
for additional information.
Output Arguments
A
— Approximation coefficients
2-D matrix | 3-D array | 4-D array
Approximation coefficients, returned as a multidimensional array. The dimensions of
A
depend on the dimensions of the input X
and the level of decomposition N
.
If
X
is m-by-n:If
N
is greater than 1, thenA
is m-by-n-by-N
. For 1 ≤ i ≤N
,A(:,:,i)
contains the approximation coefficients at level i.If
N
is equal to 1, thenA
is m-by-n.
If
X
is m-by-n-by-3:If
N
is greater than 1, thenA
is m-by-n-by-3-by-N
. For 1 ≤ i ≤N
andj = 1, 2, 3
,A(:,:,j,i)
contains approximation coefficients at level i.If
N
is equal to 1, thenA
is m-by-n-by-1-by-3. Since MATLAB® removes singleton last dimensions by default, the third dimension is singleton.
Data Types: double
H,V,D
— Detail coefficients
2-D matrix | 3-D array | 4-D array
Detail coefficients, returned as multidimensional arrays of equal size.
H
, V
, and D
contain the
horizontal, vertical, and diagonal detail coefficients, respectively. The dimensions of
the arrays depend on the dimensions of the input X
and the level of
decomposition N
.
If
X
is m-by-n:If
N
is greater than 1, the arrays are m-by-n-by-N
. For 1 ≤ i ≤N
,H(:,:,i)
,V(:,:,i)
, andD(:,:,i)
contain the detail coefficients at level i.If
N
is equal to 1, the arrays are m-by-n.
If
X
is m-by-n-by-3:If
N
is greater than 1, the arrays are m-by-n-by-3-by-N
. For 1 ≤ i ≤N
andj = 1, 2, 3
,H(:,:,j,i)
,V(:,:,j,i)
, andD(:,:,j,i)
contain the detail coefficients at level i.If
N
is equal to 1, the arrays are m-by-n-by-1-by-3. Forj = 1, 2, 3
,H(:,:,1,j)
,V(:,:,1,j)
andD(:,:,1,j)
contain the detail coefficients. Since MATLAB removes singleton last dimensions by default, the third dimension is singleton.
Data Types: double
swc
— Stationary wavelet decomposition
3-D array | 4-D array
Stationary wavelet decomposition, returned as a multidimensional array.
swc
is the concatenation of the approximation coefficients
A
and detail coefficients H
,
V
, and D
.
Algorithms
2-D Discrete Stationary Wavelet Transform
For images, a stationary wavelet transform (SWT) algorithm similar to the one-dimensional case is possible for two-dimensional wavelets and scaling functions obtained from one-dimensional functions by tensor product. This kind of two-dimensional SWT leads to a decomposition of approximation coefficients at level j into four components: the approximation at level j+1, and the details in three orientations (horizontal, vertical, and diagonal).
This chart describes the basic decomposition step for images.
where
— Convolve the rows of the entry with filter X.
— Convolve the columns of the entry with filter X.
Initialization
cA0 = s
F0 =
LoD
G0 =
HiD
-
where denotes upsample.
Note that size(cAj) =
size(cDj(h)) =
size(cDj(v)) =
size(cDj(d)) =
s
, where s equals the size of the
analyzed image.
Truecolor Image Coefficient Arrays
To distinguish a single-level decomposition of a truecolor image from a multilevel decomposition of an indexed image, the approximation and detail coefficient arrays of truecolor images are 4-D arrays.
If you perform a multilevel decomposition, the dimensions of
A
,H
,V
, andD
are m-by-n-by-3-by-k, where k is the level of decomposition.If you perform a single-level decomposition, the dimensions of
A
,H
,V
, andD
are m-by-n-by-1-by-3. Since MATLAB removes singleton last dimensions by default, the third dimension of the arrays is singleton.
References
[1] Nason, G. P., and B. W. Silverman. “The Stationary Wavelet Transform and Some Statistical Applications.” In Wavelets and Statistics, edited by Anestis Antoniadis and Georges Oppenheim, 103:281–99. New York, NY: Springer New York, 1995. https://doi.org/10.1007/978-1-4612-2544-7_17.
[2] Coifman, R. R., and D. L. Donoho. “Translation-Invariant De-Noising.” In Wavelets and Statistics, edited by Anestis Antoniadis and Georges Oppenheim, 103:125–50. New York, NY: Springer New York, 1995. https://doi.org/10.1007/978-1-4612-2544-7_9.
[3] Pesquet, J.-C., H. Krim, and H. Carfantan. “Time-Invariant Orthonormal Wavelet Representations.” IEEE Transactions on Signal Processing 44, no. 8 (August 1996): 1964–70. https://doi.org/10.1109/78.533717.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
The input wavelet name must be constant.
The input level of decomposition must be defined as a scalar during compilation.
Version History
Introduced before R2006aR2017b: Distinguish Single-Level Truecolor Image from Multilevel Indexed Image Decompositions
To distinguish a single-level decomposition of a truecolor image from a multilevel decomposition of an indexed image, the approximation and detail coefficient arrays of truecolor images are 4-D arrays.
Migrate from Previous Releases to R2017b
Depending on the original input data type and level of wavelet decomposition, you might have to take different steps to make
swt2
coefficient arrays from previous releases compatible with R2017b coefficient arrays. The steps depend on whether you have a single coefficient array or separate approximation and detail coefficient arrays.Single Coefficient Array Multiple Coefficient Arrays Input: Index image
Single-level: No compatibility issues
Multi-level: No compatibility issues
Input: Index image
Single-level: No compatibility issues
Multi-level: No compatibility issues
Input: Truecolor image
Single-level: If
swc
is the output ofswt2
from a previous release, execute:swc1 = double(swc);
Multi-level: If
swc
is the output ofswt2
from a previous release, execute:swc1 = double(swc);
Input: Truecolor image
Single-level: If
ca
,chd
,cvd
, andcdd
are outputs ofswt2
from a previous release, execute:ca1 = double(ca); chd1 = double(chd); cvd1 = double(cvd); cdd1 = double(cdd); ca2 = reshape(ca1,[m,n,1,3]); chd2 = reshape(chd1,[m,n,1,3]); cvd2 = reshape(cvd1,[m,n,1,3]); cdd2 = reshape(cdd1,[m,n,1,3]);
Multi-level: If
ca
,chd
,cvd
, andcdd
are outputs ofswt2
from a previous release, execute:ca1 = double(ca); chd1 = double(chd); cvd1 = double(cvd); cdd1 = double(cdd);
Migrate from R2017b to Previous Releases
Depending on the original input data type and level of wavelet decomposition, you might have to take different steps to make R2017b
swt2
coefficient arrays compatible with the coefficient arrays from previous releases. The steps depend on whether you have a single coefficient array or separate approximation and detail coefficient arrays.Single Coefficient Array Multiple Coefficient Arrays Input: Index image
Single-level: No compatibility issues
Multi-level: No compatibility issues
Input: Index image
Single-level: No compatibility issues
Multi-level: No compatibility issues
Input: Truecolor image
Single-level: No compatibility issues
Multi-level: No compatibility issues
Input: Truecolor image
Single-level: If
ca
,chd
,cvd
, andcdd
are outputs ofswt2
from R2017b, execute:ca1 = single(squeeze(ca)); chd1 = single(squeeze(chd)); cvd1 = single(squeeze(cvd)); cdd1 = single(squeeze(cdd));
Multi-level: No compatibility issues
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