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2-D Discrete Wavelet Analysis

This section takes you through the features of 2-D discrete wavelet analysis using the Wavelet Toolbox™ software. The toolbox provides these functions for image analysis. For more information, see the function reference pages.

Note

In this section the presentation and examples use 2-D arrays corresponding to indexed image representations. However, the functions described are also available when using truecolor images, which are represented by m-by-n-by-3 arrays of uint8. For more information on image formats, see Wavelets: Working with Images.

Analysis-Decomposition Functions

Function Name

Purpose

dwt2

Single-level decomposition

wavedec2

Decomposition

wmaxlev

Maximum wavelet decomposition level

Synthesis-Reconstruction Functions

Function Name

Purpose

idwt2

Single-level reconstruction

waverec2

Full reconstruction

wrcoef2

Selective reconstruction

upcoef2

Single reconstruction

Decomposition Structure Utilities

Function Name

Purpose

detcoef2

Extraction of detail coefficients

appcoef2

Extraction of approximation coefficients

upwlev2

Recomposition of decomposition structure

Denoising and Compression

Function Name

Purpose

wdenoise2

Wavelet image denoising

ddencmp

Provide default values for denoising and compression

wbmpen

Penalized threshold for wavelet 1-D or 2-D denoising

wdcbm2

Thresholds for wavelet 2-D using Birgé-Massart strategy

wdencmp

Wavelet denoising and compression

wthrmngr

Threshold settings manager

In this section, you'll learn

  • How to load an image

  • How to analyze an image

  • How to compress an image

Wavelet Image Analysis and Compression

This example shows how you can use 2-D wavelet analysis to compress an image efficiently without sacrificing its clarity.

Note: Instead of directly using image(I) to visualize the image I, we use image(wcodemat(I)), which displays a rescaled version of I leading to a clearer presentation of the details and approximations (see wcodemat).

Load an image.

load wbarb
whos X map
  Name        Size              Bytes  Class     Attributes

  X         256x256            524288  double              
  map       192x3                4608  double              

Display the image.

image(X)
colormap(map)
colorbar

Figure contains an axes object. The axes object contains an object of type image.

If the colormap is smooth, the wavelet transform can be directly applied to the indexed image; otherwise the indexed image should be converted to grayscale format. For more information, see Wavelets: Working with Images. Since the colormap is smooth in this image, you can now perform the decomposition.

Perform a single-level wavelet decomposition of the image using the bior3.7 wavelet. The coefficient matrix cA1 are the approximation coefficients. The horizontal, vertical, and diagonal details are in the matrices cH1, cV1, and cD1, respectively.

wv = 'bior3.7';
[cA1,cH1,cV1,cD1] = dwt2(X,wv);

Use idwt1 to construct the approximations and details from the coefficients. (Note: You can also use upcoef2.

sx = size(X);
A1 = idwt2(cA1,[],[],[],wv,sx);
H1 = idwt2([],cH1,[],[],wv,sx);
V1 = idwt2([],[],cV1,[],wv,sx);
D1 = idwt2([],[],[],cD1,wv,sx);

Display the approximations and details.

figure
subplot(2,2,1)
image(wcodemat(A1,192))
title('Approximation A1')
subplot(2,2,2)
image(wcodemat(H1,192))
title('Horizontal Detail H1')
subplot(2,2,3)
image(wcodemat(V1,192))
title('Vertical Detail V1')
subplot(2,2,4)
image(wcodemat(D1,192))
title('Diagonal Detail D1')
colormap(map)

Figure contains 4 axes objects. Axes object 1 with title Approximation A1 contains an object of type image. Axes object 2 with title Horizontal Detail H1 contains an object of type image. Axes object 3 with title Vertical Detail V1 contains an object of type image. Axes object 4 with title Diagonal Detail D1 contains an object of type image.

Regenerate the image by the single-level inverse discrete wavelet transform. Confirm the difference between the regenerated and original images are small.

Xrec = idwt2(cA1,cH1,cV1,cD1,wv);
max(abs(X(:)-Xrec(:)))
ans = 1.4211e-13

Perform a level-2 wavelet decomposition of the image using the same bior3.7 wavelet. The coefficients of all the components of a second-level decomposition (that is, the second-level approximation and the first two levels of detail) are returned concatenated into one vector, C. Argument S is a bookkeeping matrix that keeps track of the sizes of each component.

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

Extract the level 2 approximation coefficients. Extract the first- and second-level detail coefficients.

cA2 = appcoef2(c,s,wv,2);
[cH2,cV2,cD2] = detcoef2('all',c,s,2);
[cH1,cV1,cD1] = detcoef2('all',c,s,1);

Reconstruct the level 2 approximation and the level 1 and level 2 details.

A2 = wrcoef2('a',c,s,wv,2);
H1 = wrcoef2('h',c,s,wv,1);
V1 = wrcoef2('v',c,s,wv,1);
D1 = wrcoef2('d',c,s,wv,1);
H2 = wrcoef2('h',c,s,wv,2);
V2 = wrcoef2('v',c,s,wv,2);
D2 = wrcoef2('d',c,s,wv,2);

Display the approximation and details.

figure
subplot(2,4,1)
image(wcodemat(A1,192))
title('Approximation A1')
subplot(2,4,2)
image(wcodemat(H1,192))
title('Horizontal Detail H1')
subplot(2,4,3)
image(wcodemat(V1,192))
title('Vertical Detail V1')
subplot(2,4,4)
image(wcodemat(D1,192))
title('Diagonal Detail D1')
subplot(2,4,5)
image(wcodemat(A2,192))
title('Approximation A2')
subplot(2,4,6)
image(wcodemat(H2,192))
title('Horizontal Detail H2')
subplot(2,4,7)
image(wcodemat(V2,192))
title('Vertical Detail V2')
subplot(2,4,8)
image(wcodemat(D2,192))
title('Diagonal Detail D2')
colormap(map)

Figure contains 8 axes objects. Axes object 1 with title Approximation A1 contains an object of type image. Axes object 2 with title Horizontal Detail H1 contains an object of type image. Axes object 3 with title Vertical Detail V1 contains an object of type image. Axes object 4 with title Diagonal Detail D1 contains an object of type image. Axes object 5 with title Approximation A2 contains an object of type image. Axes object 6 with title Horizontal Detail H2 contains an object of type image. Axes object 7 with title Vertical Detail V2 contains an object of type image. Axes object 8 with title Diagonal Detail D2 contains an object of type image.

Compress the image. Use ddencmp to calculate the default parameters and wdencmp to perform the actual compression.

[thr,sorh,keepapp] = ddencmp('cmp','wv',X);
[Xcomp,CXC,LXC,PERF0,PERFL2] = ... 
wdencmp('gbl',c,s,wv,2,thr,sorh,keepapp);

Compare the compressed image with the original image.

fprintf('Percentage of wavelet coefficients set to zero: %.4f\nPercentage of energy preserved: %.4f\n',...
    PERF0,PERFL2);
Percentage of wavelet coefficients set to zero: 49.8011
Percentage of energy preserved: 99.9817
figure
subplot(121)
image(X)
title('Original Image') 
axis square
subplot(122)
image(Xcomp)
title('Compressed Image') 
axis square
colormap(map)

Figure contains 2 axes objects. Axes object 1 with title Original Image contains an object of type image. Axes object 2 with title Compressed Image contains an object of type image.

Note that, even though the compressed image is constructed from only about half as many nonzero wavelet coefficients as the original, there is almost no detectable deterioration in the image quality.