Wavelet Toolbox™

Discrete stationary wavelet transform 2-D

Syntax

• SWC = swt2(X,N,'wname')
  [A,H,V,D] = swt2(X,N,'wname')
  SWC = swt2(X,N,Lo_D,Hi_D)
  [A,H,V,D] = swt2(X,N,Lo_D,Hi_D)
  

Description

swt2 performs a multilevel 2-D stationary wavelet decomposition using either a specific orthogonal wavelet ('wname'--see wfilters for more information) or specific orthogonal wavelet decomposition filters.

SWC = swt2(X,N,'wname') or [A,H,V,D] = swt2(X,N,'wname') compute the stationary wavelet decomposition of the matrix X at level N, using 'wname'.

N must be a strictly positive integer (see wmaxlev for more information), and 2^N must divide size(X,1) and size(X,2).

Outputs [A,H,V,D] are 3-D arrays, which contain the coefficients:

• For 1 i N, the output matrix A(:,:,i) contains the coefficients of approximation of level i.
• The output matrices H(:,:,i), V(:,:,i) and D(:,:,i) contain the coefficients of details of level i (horizontal, vertical, and diagonal):

• SWC = [H(:,:,1:N) ; V(:,:,1:N) ; D(:,:,1:N) ; A(:,:,N)]
  

SWC = swt2(X,N,Lo_D,Hi_D) or [A,H,V,D] = swt2(X,N,Lo_D,Hi_D), computes the stationary wavelet decomposition as in the previous syntax, given these filters as input:

• Lo_D is the decomposition low-pass filter.
• Hi_D is the decomposition high-pass filter.

Lo_D and Hi_D must be the same length.

Note    swt2 is defined using dwt with periodic extension.

Remarks

When X represents an indexed image, X is an m-by-n matrix and the output arrays SWC or cA,cH,cV, and cD are m-by-n-by-p arrays.

When X represents a truecolor image, it becomes an m-by-n-by-3 array. This array 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 output arrays SWC or cA,cH,cV, and cD are m-by-n-by-p-by-3 arrays.

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

Examples

• % Load original image.
  load nbarb1;
  
  % Image coding.
  nbcol = size(map,1);
  cod_X = wcodemat(X,nbcol);
  
  % Visualize the original image.
  subplot(221)
  image(cod_X)
  title('Original image');
  colormap(map)
  
  % Perform SWT decomposition
  % of X at level 3 using sym4.
  [ca,chd,cvd,cdd] = swt2(X,3,'sym4');
  
  % Visualize the decomposition.
  
  for k = 1:3
      % Images coding for level k.
      cod_ca  = wcodemat(ca(:,:,k),nbcol);
      cod_chd = wcodemat(chd(:,:,k),nbcol);
      cod_cvd = wcodemat(cvd(:,:,k),nbcol);
      cod_cdd = wcodemat(cdd(:,:,k),nbcol);
      decl = [cod_ca,cod_chd;cod_cvd,cod_cdd];
  
      % Visualize the coefficients of the decomposition
      % at level k.
      subplot(2,2,k+1)
      image(decl)
  
      title(['SWT dec.: approx. ', ...
     'and det. coefs (lev. ',num2str(k),')']);
      colormap(map)
  end
  % Editing some graphical properties,
  % the following figure is generated.
  
  
  

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 SWT 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:

References

Nason, G.P.; B.W. Silverman (1995), "The stationary wavelet transform and some statistical applications," Lecture Notes in Statistics, 103, pp. 281-299.

Coifman, R.R.; Donoho, D.L. (1995), "Translation invariant de-noising," Lecture Notes in Statistics, 103, pp. 125-150.

Pesquet, J.C.; H. Krim, H. Carfatan (1996), "Time-invariant orthonormal wavelet representations," IEEE Trans. Sign. Proc., vol. 44, 8, pp. 1964-1970.

See Also

dwt2, wavedec2

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