Two-Dimensional Analysis Using the Command Line :: Using Wavelets (Wavelet Toolbox)

Wavelet Toolbox

Two-Dimensional Analysis Using the Command Line

In this example we'll show how you can use two-dimensional 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 reference page).

Load an image.

From the MATLAB prompt, type
- ```
load wbarb; 
whos
```
  Name
  Size
  Bytes
  Class
  
  X
  256x256
  524288
  double array
  
  map
  192x3
  4608
  double array
```
 


 
```

Name	Size	Bytes	Class
`X`	`256x256`	`524288`	`double array`
`map`	`192x3`	`4608`	`double array`

Display the image. Type

```
image(X); colormap(map); colorbar;
```

Convert an indexed image to a grayscale 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.

To perform a single-level decomposition of the image using the bior3.7 wavelet, type
- ```
[cA1,cH1,cV1,cD1] = dwt2(X,'bior3.7');
```
This generates the coefficient matrices of the level-one approximation (cA1) and horizontal, vertical and diagonal details (cH1,cV1,cD1, respectively).

Construct and display approximations and details from the coefficients.

To construct the level-one approximation and details (A₁, H₁, V₁, and D₁) from the coefficients cA₁, cH₁, cV₁, and cD₁, type

A1 = upcoef2('a',cA1,'bior3.7',1); 
H1 = upcoef2('h',cH1,'bior3.7',1);
V1 = upcoef2('v',cV1,'bior3.7',1);
D1 = upcoef2('d',cD1,'bior3.7',1);

sx = size(X);
A1 = idwt2(cA1,[],[],[],'bior3.7',sx); 
H1 = idwt2([],cH1,[],[],'bior3.7',sx); 
V1 = idwt2([],[],cV1,[],'bior3.7',sx); 
D1 = idwt2([],[],[],cD1,'bior3.7',sx);

To display the results of the level 1 decomposition, type

colormap(map);
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')

Regenerate an image by single-level Inverse Wavelet Transform.

To find the inverse transform, type
- ```
Xsyn = idwt2(cA1,cH1,cV1,cD1,'bior3.7');
```
This reconstructs or synthesizes the original image from the coefficients of the level 1 approximation and details.

Perform a multilevel wavelet decomposition.

To perform a level 2 decomposition of the image (again using the bior3.7 wavelet), type
- ```
[C,S] = wavedec2(X,2,'bior3.7');
```
where X is the original image matrix, and 2 is the level of decomposition.

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.

Extract approximation and detail coefficients.

To extract the level 2 approximation coefficients from C, type
- ```
cA2 = appcoef2(C,S,'bior3.7',2);
```
To extract the first- and second-level detail coefficients from C, type
- ```
cH2 = detcoef2('h',C,S,2);
cV2 = detcoef2('v',C,S,2); 
cD2 = detcoef2('d',C,S,2); 
cH1 = detcoef2('h',C,S,1);
cV1 = detcoef2('v',C,S,1); 
cD1 = detcoef2('d',C,S,1);
```
or
- ```
[cH2,cV2,cD2] = detcoef2('all',C,S,2); 
[cH1,cV1,cD1] = detcoef2('all',C,S,1); 
```
where the first argument ('h', 'v', or 'd') determines the type of detail (horizontal, vertical, diagonal) extracted, and the last argument determines the level.

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

To reconstruct the level 2 approximation from C, type

```
A2 = wrcoef2('a',C,S,'bior3.7',2);
```

To reconstruct the level 1 and 2 details from C, type

H1 = wrcoef2('h',C,S,'bior3.7',1); 
V1 = wrcoef2('v',C,S,'bior3.7',1); 
D1 = wrcoef2('d',C,S,'bior3.7',1); 
H2 = wrcoef2('h',C,S,'bior3.7',2);
V2 = wrcoef2('v',C,S,'bior3.7',2); 
D2 = wrcoef2('d',C,S,'bior3.7',2);

Display the results of a multilevel decomposition.

Note With all the details involved in a multilevel image decomposition, it makes sense to import the decomposition into the Wavelet 2-D graphical tool in order to more easily display it. For information on how to do this, see Loading Decompositions.

To display the results of the level 2 decomposition, type

colormap(map);
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')

Reconstruct the original image from the multilevel decomposition.

To reconstruct the original image from the wavelet decomposition structure, type
- ```
X0 = waverec2(C,S,'bior3.7');
```
This reconstructs or synthesizes the original image from the coefficients C of the multilevel decomposition.

Compress the image and display it.

To compress the original image X, use the ddencmp command to calculate the default parameters and the wdencmp command to perform the actual compression. Type
- ```
[thr,sorh,keepapp] = ddencmp('cmp','wv',X);
[Xcomp,CXC,LXC,PERF0,PERFL2] = 
wdencmp('gbl',C,S,'bior3.7',2,thr,sorh,keepapp);
```
Note that we pass in to wdencmp the results of the decomposition (C and S) we calculated in Step 7. We also specify the bior3.7 wavelets, because we used this wavelet to perform the original analysis. Finally, we specify the global thresholding option 'gbl'. See ddencmp and wdencmp reference pages for more information about the use of these commands.

To view the compressed image side by side with the original, type
- ```
colormap(map);
subplot(121); image(X); title('Original Image');
axis square
subplot(122); image(Xcomp); title('Compressed Image');
axis square


PERF0

PERF0 =
    49.8076

PERFL2

PERFL2 =
    99.9817
```
These returned values tell, respectively, what percentage of the wavelet coefficients was set to zero and what percentage of the image's energy was preserved in the compression process.

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.

Two-Dimensional Discrete Wavelet Analysis Two-Dimensional Analysis Using the Graphical Interface