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# Documentation

## Non-Decimated Wavelet Analysis

This example shows you how to decompose a signal or an image using non-decimated wavelet analysis.

This kind of redundant, translation-invariant transform is especially useful for denoising, which is one of the most important wavelet applications. This type of analysis has already been developed for stationary wavelet transform (SWT) functions (1-D stationary wavelet functions) and the SWT GUI. This transform however has a serious limitation: the signal length must be a power of 2 and the periodized extension mode must be used for the underlying DWT.

Wavelet Toolbox™ provides a new tool for handling such transforms for signals (or images) of arbitrary size and using different extension modes.

```load noisbloc;
L = length(noisbloc)
plot(noisbloc,'r')
axis tight
title('Original Signal')
```
```L =

1024

```

The loaded signal is of length 1024, so you can use the SWT functions to decompose it. We now truncate the loaded signal and keep only 979 samples, which is not a power of 2 (blue line in the plot below).

```nkeep = 979;
sig = noisbloc(1:nkeep);
hold on
plot(sig,'b')
title('Original and Truncated Signals')
```

Multilevel 1-D Non-Decimated Wavelet Decomposition

Perform non-decimated wavelet decomposition of sig at level 5 using the haar wavelet.

```n = 5;                  % Decomposition level
w = 'db1';              % Haar wavelet
WT = ndwt(sig,n,w)      % Multilevel 1-D wavelet decomposition.
```
```WT =

rowvect: 1
level: 5
mode: 'sym'
filters: [1x1 struct]
dec: {1x6 cell}
longs: [984 984 983 982 981 980 979]

```

The function NDWT returns a structure containing the parameters of the non-decimated transform.

Filters associated with the wavelet:

```WT.filters
```
```ans =

LoD: [0.7071 0.7071]
HiD: [-0.7071 0.7071]
LoR: [0.7071 0.7071]
HiR: [0.7071 -0.7071]

```

The decomposition is organized in sequences of coefficients according to the level and nature of the signal (approximation or detail). The coefficients are concatenated and are given by WT.dec which is equal to [Ca(N) Cd(N) Cd(N-1) ... Cd(1)]:

```WT.dec
```
```ans =

Columns 1 through 4

[1x984 double]    [1x984 double]    [1x983 double]    [1x982 double]

Columns 5 through 6

[1x981 double]    [1x980 double]

```

The lengths of the coefficient sequences and of the original signal are given by WT.longs:

```WT.longs
```
```ans =

984   984   983   982   981   980   979

```

And finally, the discrete wavelet transform extension mode is

```WT.mode
```
```ans =

sym

```

Multilevel 1-D Non-Decimated Wavelet Reconstruction

Starting from a decomposition, we can construct all the components (approximations and details) useful for analysis:

```A = cell(1,n);
D = cell(1,n);
for k = 1:n
A{k} = indwt(WT,'a',k);   % Approximations (low-pass components)
D{k} = indwt(WT,'d',k);   % Details (high-pass components)
end
```

To denoise or process the signal, you can modify the coefficients before reconstruction, for example, by thresholding it.

We now concentrate on the capabilities of the non-decimated transform. We first check that if the coefficients are unchanged, the reconstructions are perfect, i.e. sig = A(k) + D(k) + ... + D(1).

```err = zeros(1,n);
for k = 1:n
E = sig-A{k};
for j = 1:k
E = E-D{j};
end
err(k) = max(abs(E(:)));
end
disp(err)
```
```   1.0e-13 *

0.1688    0.1538    0.1925    0.1764    0.1749

```

Next we illustrate graphically two different decompositions of the original signal: approximation and detail at level 1 and approximation at level 5, plus the sum of the details of level 1 to 5.

```for k = [1 5]
figure('Color','w')
subplot(2,1,1);
plot(A{k},'b'); axis tight; xlabel(['A' int2str(k)])
subplot(2,1,2);
plot(sig-A{k},'g'); axis tight;
xlabel(['Sum of Details from 1 to ' int2str(k)])
end
clear A D E err
```

Starting from a different initial signal, we can show the effect of changing the extension mode.

```load noissin;
% Truncate the signal
x = noissin(1:512);
wname = 'sym4';
level = 5;
% Consider the decompositions obtained using two different extension modes:
W1 = ndwt(x,level,wname,'mode','zpd'),  % Zero padding
W2 = ndwt(x,level,wname,'mode','per'),  % Periodization

A5_W1 = indwt(W1,'a',level);
A5_W2 = indwt(W2,'a',level);
```
```W1 =

rowvect: 1
level: 5
mode: 'zpd'
filters: [1x1 struct]
dec: {1x6 cell}
longs: [547 547 540 533 526 519 512]

W2 =

rowvect: 1
level: 5
mode: 'per'
filters: [1x1 struct]
dec: {1x6 cell}
longs: [547 547 540 533 526 519 512]

```

The approximations are very similar, with the differences appearing at the beginning and the end of these approximations.

```figure('Color','w')
subplot(3,1,1); plot(A5_W1,'r'); axis tight; title('A5 - Zero padding');
subplot(3,1,2); plot(A5_W2,'b'); axis tight; title('A5 - Periodization');
subplot(3,1,3); plot(A5_W1-A5_W2,'m','LineWidth',2); axis tight;
xlabel('Diffence between the approximations');
```

Load Original Image and Select a ROI

Now we illustrate the 2-D counterparts of the previous capabilities, with fewer details.

```load noiswom;
S = size(X)
```
```S =

96    96

```

Select a ROI, the size of which is not a power of 2.

```X = X(2:94,2:94);
sX = size(X)
```
```sX =

93    93

```

Display the original image.

```figure('Color','w')
map = pink(255);
image(X)
colormap(map)
axis tight
title('Original Image')
```

Multilevel 2-D Non-Decimated Wavelet Decomposition

Perform non-decimated wavelet decomposition of signal X at level 4 using the haar wavelet.

```n = 4;                   % Decomposition Level
w = 'db1';               % Haar wavelet
WT = ndwt2(X,n,w)        % Multilevel 2-D wavelet decomposition.
```
```WT =

sizeINI: [93 93]
level: 4
filters: [1x1 struct]
mode: 'sym'
dec: {13x1 cell}
sizes: [5x2 double]

```

Multilevel 2-D Non-Decimated Wavelet Coefficients

Extract the resulting family of coefficients from the wavelet decomposition

```cAA = cell(1,n);
cDA = cell(1,n);
cDD = cell(1,n);
for k = 1:n
cAA{k} = indwt2(WT,'caa',k);   % Coefficients of approximations
cDA{k} = indwt2(WT,'cda',k);   % Coefficients of vertical details
cDD{k} = indwt2(WT,'cdd',k);   % Coefficients of diagonal details
end
```

Display the resulting family of coefficients

```figure('DefaultAxesXTick',[],'DefaultAxesYTick',[],'Color','w')
colormap(map)
for k = 1:n
subplot(4,n,k);
imagesc(cAA{k}); xlabel(['cAA' int2str(k)])
subplot(4,n,k+n);
subplot(4,n,k+2*n);
imagesc(abs(cDA{k})); xlabel(['cDA' int2str(k)])
subplot(4,n,k+3*n);
imagesc(abs(cDD{k})); xlabel(['cDD' int2str(k)])
end
```

Multilevel 2-D Non-Decimated Wavelet Reconstruction

```A = cell(1,n);
D = cell(1,n);
for k = 1:n
A{k} = indwt2(WT,'a',k);   % Approximations (low-pass components)
D{k} = indwt2(WT,'d',k);   % Details (high-pass components)
end

% We now check that without changing the coefficients, the various
% reconstructions are perfect.
err = zeros(1,n);
for k = 1:n
E = X-A{k}-D{k};
err(k) = max(abs(E(:)));
end
disp(err)
```
```   1.0e-12 *

0.3197    0.3375    0.3464    0.3295

```
```figure('DefaultAxesXTick',[],'DefaultAxesYTick',[],'Color','w')
colormap(map)
for k = 1:n
subplot(2,n,k);
imagesc(A{k}); xlabel(['A' int2str(k)])
subplot(2,n,k+n);
imagesc(abs(D{k})); xlabel(['Det' int2str(k)])
end
clear D E err
```

```figure('DefaultAxesXTick',[],'DefaultAxesYTick',[],'Color','w')
colormap(map)
subplot(1,2,1);
image(X)
title('Original Image')
subplot(1,2,2);
image(A{1})
title('Denoised Image - A1')
```

Consider the approximations at level 1 obtained using three different extension modes. These approximations are rough denoised images.

```WT = ndwt2(X,1,w,'mode','per');   % Multilevel 2-D wavelet decomposition.
A{1} = indwt2(WT,'a',1);
WT = ndwt2(X,1,w,'mode','sym');   % Multilevel 2-D wavelet decomposition.
A{2} = indwt2(WT,'a',1);
WT = ndwt2(X,1,w,'mode','zpd');   % Multilevel 2-D wavelet decomposition.
A{3} = indwt2(WT,'a',1);
```

As expected, the approximations are very similar, with the differences concentrated on the edge.

```figure('DefaultAxesXTick',[],'DefaultAxesYTick',[],'Color','w')
colormap(map)
subplot(2,3,2);
image(X)
title('Original Image')
subplot(2,3,4);
image(A{1})
xlabel('A1 - PER')
subplot(2,3,5);
image(A{2})
title('Denoised Images')
xlabel('A1 - SYM')
subplot(2,3,6);
image(A{3})
xlabel('A1 - ZPD')
```

Summary

This example showed how the 1-D and 2-D Non-Decimated Wavelet transform allows you to process signals or images of any size using any mode of extension for the decomposition.