This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.


Multilevel 2-D wavelet decomposition


[C,S] = wavedec2(X,N,'wname')
[C,S] = wavedec2(X,N,Lo_D,Hi_D)


wavedec2 is a two-dimensional wavelet analysis function.

[C,S] = wavedec2(X,N,'wname') returns the wavelet decomposition of the matrix X at level N, using the wavelet named in character vector 'wname' (see wfilters for more information).

Outputs are the decomposition vector C and the corresponding bookkeeping matrix S.

N must be a strictly positive integer (see wmaxlev for more information).

Instead of giving the wavelet name, you can give the filters.

For [C,S] = wavedec2(X,N,Lo_D,Hi_D), Lo_D is the decomposition low-pass filter and Hi_D is the decomposition high-pass filter.

Vector C is organized as a vector with 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 vector 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

Matrix S is such that

  • S(1,:) = size of approximation coefficients(N).

  • S(i,:) = size of detail coefficients(N-i+2) for i = 2, ...N+1 and S(N+2,:) = size(X).


Decomposition Structure

This example shows the structure of wavedec2 output matrices.

Load original image from the woman.mat file, which contains variables named X and map.

load woman;

Get current discrete wavelet transform extension mode.

origMode = dwtmode('status','nodisplay');

Change to periodic boundary handling.

!  WARNING: Change DWT Extension Mode  !
**  DWT Extension Mode: Periodization  **

Perform decomposition at level 2 of X using db1.

[c,s] = wavedec2(X,2,'db1');

Get the decomposition structure organization.

sizex = size(X)
sizex = 

   256   256

sizec = size(c)
sizec = 

           1       65536

Reset discrete wavelet transform extension mode to its original mode.

!  WARNING: Change DWT Extension Mode  !
**  DWT Extension Mode: Symmetrization (half-point)  **

Extract and Display Image Decomposition Level

Extract and display images of wavelet decomposition level details. The resulting images are similar to the visualizations in the At level 2, with haar --> woman indexed image example in the Wavelet 2-D interactive tool. Use waveletAnalyzer to launch this tool.

load woman; 
[H1,V1,D1] = detcoef2('all',c,s,1);
A1 = appcoef2(c,s,'haar',1); 
V1img = wcodemat(V1,255,'mat',1);
H1img = wcodemat(H1,255,'mat',1);
D1img = wcodemat(D1,255,'mat',1);
A1img = wcodemat(A1,255,'mat',1);
[H2,V2,D2] = detcoef2('all',c,s,2);
A2 = appcoef2(c,s,'haar',2); 
V2img = wcodemat(V2,255,'mat',1);
H2img = wcodemat(H2,255,'mat',1);
D2img = wcodemat(D2,255,'mat',1);
A2img = wcodemat(A2,255,'mat',1);
colormap pink(255);
title('Approximation Coef. of Level 1');

title('Horizontal detail Coef. of Level 1');

title('Vertical detail Coef. of Level 1');

title('Diagonal detail Coef. of Level 1');

colormap pink(255);
title('Approximation Coef. of Level 2');

title('Horizontal detail Coef. of Level 2');

title('Vertical detail Coef. of Level 2');

title('Diagonal detail Coef. of Level 2');


When X represents an indexed image, X, as well as 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 follows.

For more information on image formats, see the image and imfinfo reference pages.


For images, an algorithm similar to the one-dimensional case is possible for two-dimensional wavelets and scaling functions obtained from one-dimensional ones 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 following chart describes the basic decomposition step for images:

So, for J=2, the two-dimensional wavelet tree has the form


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.)

Extended Capabilities

Introduced before R2006a

Was this topic helpful?