wavedec2 :: Functions (Wavelet Toolbox™)

wavedec2

Purpose

Multilevel 2-D wavelet decomposition

Syntax

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

Description

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 string '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

C = [ 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, D, are row vectors such that

A = approximation coefficients
H = horizontal detail coefficients
V = vertical detail coefficients
D = diagonal detail coefficients
Each vector is the vector column-wise storage of a matrix.

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).

Remarks

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.

Examples

% The current extension mode is zero-padding (see dwtmode).

% Load original image. 
load woman; 
% X contains the loaded image.

% Perform decomposition at level 2 
% of X using db1. 
[c,s] = wavedec2(X,2,'db1');

% Decomposition structure organization. 
sizex = size(X)

sizex =
    256  256
sizec = size(c)

sizec =
    1  65536
    val_s = s 

val_s =
    64  64 
    64  64 
    128  128 
    256  256

Algorithm

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

References

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.)