wavedec2 (Wavelet Toolbox)

Multi-level 2-D wavelet decomposition.

[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).

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

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

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. The output wavelet two-dimensional decomposition structure [C,S] contains the wavelet decomposition vector C and the corresponding bookkeeping matrix S.

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 columnwise 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).

Examples

% 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: