wavedec2
Multilevel 2-D discrete wavelet transform
Description
[
returns the wavelet decomposition of the matrix C,S] = wavedec2(X,N,wname)X at level
N using the wavelet wname. The output
decomposition structure consists of the wavelet decomposition vector
C and the bookkeeping matrix S, which
contains the number of coefficients by level and orientation.
Note
For gpuArray inputs, the supported modes are
"symh" ("sym") and
"per". If the input is a gpuArray,
the discrete wavelet transform extension mode used by
wavedec2 defaults to "symh"
unless the current extension mode is "per". See the
example Multilevel 2-D Discrete Wavelet Transform on a GPU.
Examples
This example shows how to extract and display images of wavelet decomposition level details.
Load an image. Perform a level 2 wavelet decomposition of the image using the haar wavelet.
load belmont1 [c,s]=wavedec2(X,2,"haar");
Extract the level 1 approximation and detail coefficients.
[H1,V1,D1] = detcoef2("all",c,s,1); A1 = appcoef2(c,s,"haar",1);
Use wcodemat to rescale the coefficients based on their absolute values. Display the rescaled coefficients.
V1img = wcodemat(V1,255,"mat",1); H1img = wcodemat(H1,255,"mat",1); D1img = wcodemat(D1,255,"mat",1); A1img = wcodemat(A1,255,"mat",1); tiledlayout(2,2) nexttile imagesc(A1img) colormap pink(255) title("Approximation Coef. of Level 1") nexttile imagesc(H1img) title("Horizontal Detail Coef. of Level 1") nexttile imagesc(V1img) title("Vertical Detail Coef. of Level 1") nexttile imagesc(D1img) title("Diagonal Detail Coef. of Level 1")
Extract the level 2 approximation and detail coefficients.
[H2,V2,D2] = detcoef2("all",c,s,2); A2 = appcoef2(c,s,"haar",2);
Use wcodemat to rescale the coefficients based on their absolute values. Display the rescaled coefficients.
V2img = wcodemat(V2,255,"mat",1); H2img = wcodemat(H2,255,"mat",1); D2img = wcodemat(D2,255,"mat",1); A2img = wcodemat(A2,255,"mat",1); figure tiledlayout(2,2) nexttile imagesc(A2img) colormap pink(255) title("Approximation Coef. of Level 2") nexttile imagesc(H2img) title("Horizontal Detail Coef. of Level 2") nexttile imagesc(V2img) title("Vertical Detail Coef. of Level 2") nexttile imagesc(D2img) title("Diagonal Detail Coef. of Level 2")
This example shows the structure of wavedec2 output matrices.
Load and display an image.
load mask
imagesc(X)
colormap(map)
Save the current discrete wavelet transform extension mode.
origMode = dwtmode("status","nodisplay");
Change to periodic boundary handling. The dwtmode function displays a message indicating that the DWT extension mode is changing.
dwtmode("per")
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! WARNING: Change DWT Extension Mode !
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
*****************************************
** DWT Extension Mode: Periodization **
*****************************************
Perform a level 3 decomposition of the image using the db1 (Haar) wavelet.
[c,s] = wavedec2(X,3,"db1");Return the number of elements in the image X and coefficient vector c. Confirm the number of elements in each are equal.
numel(X)
ans = 65536
numel(c)
ans = 65536
Display the bookkeeping matrix s. The first row displays the dimensions of the coarse scale approximation of the image. The last row displays the dimensions of the original image. The intermediate rows display the dimensions of the detail coefficients at the three levels of the decomposition, proceeding from coarse to fine scale.
s
s = 5×2
32 32
32 32
64 64
128 128
256 256
Reset discrete wavelet transform extension mode to its original mode.
dwtmode(origMode,"nodisplay")Refer to GPU Computing Requirements (Parallel Computing Toolbox) to see what GPUs are supported.
Load an image. Put the image on the GPU using gpuArray. Save the current extension mode.
load mask imgg = gpuArray(X); origMode = dwtmode("status","nodisp");
Use dwtmode to change the extension mode to zero-padding. Obtain the three-level DWT of the image on the GPU using the db4 wavelet.
dwtmode("zpd","nodisp") [c,s] = wavedec2(imgg,3,"db4");
The current extension mode "zpd" is not supported for gpuArray input. Therefore, the DWT is instead performed using the "sym" extension mode. To confirm this, set the extension mode to "sym" and take DWT of noisdoppg, then compare with the previous result.
dwtmode("sym","nodisp") [csym,ssym] = wavedec2(imgg,3,"db4"); [max(abs(c-csym)) max(abs(s-ssym))]
ans =
0 0 0
Set the current extension mode to "per" and obtain the three-level DWT of imgg. The extension mode "per" is supported for gpuArray input. Confirm the result is different from the "sym" results.
dwtmode("per","nodisp") [cper,sper] = wavedec2(imgg,3,"db4"); [length(csym) ; length(cper)]
ans = 2×1
71542
65536
ssym
ssym = 5×2
38 38
38 38
69 69
131 131
256 256
sper
sper = 5×2
32 32
32 32
64 64
128 128
256 256
Restore the extension mode to the original setting.
dwtmode(origMode,"nodisp")Input Arguments
Output Arguments
Wavelet decomposition vector. The vector C contains
the approximation and detail coefficients organized by level. The
bookkeeping matrix S is used to parse
C.
The vector C is organized as
A(N),
H(N),
V(N),
D(N),
H(N-1),
V(N-1),
D(N-1), …,
H(1), V(1), D(1),
where A, H, V, and
D are each a row vector. Each vector is the
column-wise storage of a matrix.
A contains the approximation coefficients.
H contains the horizontal detail coefficients.
V contains the vertical detail coefficients.
D contains the diagonal detail coefficients.
Data Types: double
Bookkeeping matrix. The matrix S contains the
dimensions of the wavelet coefficients by level and is used to parse the
wavelet decomposition vector C.
S(1,:)= size of approximation coefficients(N).S(i,:)= size of detail coefficients(N-i+2) fori= 2, ...N+1andS(N+2,:) = size(X).
The following diagram shows the relationship between
C and S in the wavelet
decomposition of a 512-by-512 matrix.
When X represents an indexed image, the output arrays
cA, cH, cV, and
cD are m-by-n
matrices. When X represents a truecolor image, it is an
m-by-n-by-3 array, where each
m-by-n matrix represents a red,
green, or blue color plane concatenated along the third dimension. The size
of vector C and the size of matrix
S depend on the type of analyzed image.
For a truecolor image, the decomposition vector C and
the corresponding bookkeeping matrix S can be
represented as shown.
Algorithms
For images, an algorithm similar to the one-dimensional case is possible for two-dimensional wavelets and scaling functions obtained from one-dimensional vectors by tensor 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 chart describes the basic decomposition step for images:
where
— Downsample columns: keep the even-indexed
columns.
— Downsample rows: keep the even-indexed rows.
— Convolve with filter X the rows of
the entry.
— Convolve with filter X the columns of
the entry.
and
Initialization: cA0 = s.
So, for J = 2, the two-dimensional wavelet tree has the form
References
[1] Daubechies, Ingrid. Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61. Philadelphia, Pa: Society for Industrial and Applied Mathematics, 1992.
[2] Mallat, S.G. “A Theory for Multiresolution Signal Decomposition: The Wavelet Representation.” IEEE Transactions on Pattern Analysis and Machine Intelligence 11, no. 7 (July 1989): 674–93. https://doi.org/10.1109/34.192463.
[3] Meyer, Y. Wavelets and Operators. Translated by D. H. Salinger. Cambridge, UK: Cambridge University Press, 1995.
Extended Capabilities
Version History
Introduced before R2006a
