In the orthogonal wavelet decomposition procedure, the generic step splits the approximation coefficients into two parts. After splitting we obtain a vector of approximation coefficients and a vector of detail coefficients, both at a coarser scale. The information lost between two successive approximations is captured in the detail coefficients. Then the next step consists of splitting the new approximation coefficient vector, successive details are never reanalyzed.

In the corresponding wavelet packet situation, each detail coefficient vector is also decomposed into two parts using the same approach as in approximation vector splitting. This offers the richest analysis. The complete binary tree is produced in the one-dimensional case or a quaternary tree in the two-dimensional case.

**Wavelet Packets Decomposition**

A single decomposition using wavelet packets generates a large number of bases. Decompose a signal at depth 3 with db1 wavelet, using default entropy (shannon).

load noisdopp; wpt = wpdec(noisdopp,3,'db1'); plot(wpt)

Decompose the packet [3 0] and plot the modified wavelet packet tree

wpt = wpsplt(wpt,[3 0]); plot(wpt)

You can then look for the best representation with respect to a design objective, using the function **besttree** with an entropy function. The resulting tree may be much smaller than the initial one.

bt = besttree(wpt); plot(bt)

You can change the node labels. Instead of **Depth_Position** labels you can use **Index** labels. Using the **Node Label** submenu, you can select the displayed labels.

```
openfig('wpdemo_fig_1.fig');
```

You can examine the data associated to each node clicking with the mouse on this node. In the next window, the data associated to the **node 2** is displayed. You can find the **coefficients of detail at level 1** in wavelet decomposition of the original signal.

```
openfig('wpdemo_fig_2.fig');
```

Using the **Node Action** submenu, you can select the type of action associated to each node. The default action is **Visualize**, you can choose **Split-Merge** to split or merge selected nodes.

```
openfig('wpdemo_fig_3.fig');
```

**Wavelet Packets for De-Noising**

In the wavelet packet framework, compression and de-noising ideas are identical to those developed in the wavelet framework. The only new feature is a more complete analysis that provides increased flexibility. Next you will de-noise a noisy chirp signal using of Stein's Unbiased Estimate of Risk (SURE) criterion threshold and you will compare wavelet packets-based and de-noising wavelet-based de-noising results.

load noischir; x = noischir; n = length(x); thr = sqrt(2*log(n*log(n)/log(2))); % SURE criterion threshold xwpd = wpdencmp(x,'s',4,'sym4','sure',thr,1); % Wavelet-packets-based de-noising xwd = wden(x,'rigrsure','s','one',4,'sym4'); % Wavelet-based de-noising

Plot original and de-noised signals

subplot(311),plot(x), xlim([1 n]); title('Original Signal'); subplot(312),plot(xwpd), xlim([1 n]); title('De-noised Signal using Wavelet Packets'); subplot(313),plot(xwd), xlim([1 n]); title('De-noised Signal using Wavelets');

**Wavelet Packets for Compression**

In this section, you employ wavelet packets to analyze and compress an image of a fingerprint. This is a real-world problem: the Federal Bureau of Investigation (FBI) maintains a large database of fingerprints. The FBI uses eight bits per pixel to define the shade of gray and stores 500 pixels per inch, which works out to about 700 000 pixels and 0.7 megabytes per finger to store finger prints in electronic form. By turning to wavelets, the FBI has achieved a 15:1 compression ratio. In this application, wavelet compression is better than the more traditional JPEG compression, as it avoids small square artifacts and is particularly well suited to detect discontinuities (lines) in the fingerprint.

Note that the international standard JPEG 2000 includes the wavelets as a part of the compression and quantization process. This points out the present strength of the wavelets.

load detfingr wname = 'bior6.8'; lev = 3; sorh = 'h'; crit = 'shannon'; thr = 30; keepapp = 1; [xd,t,perf0,perfl2] = wpdencmp(X,sorh,lev,wname,crit,thr,keepapp);

Plot the wavelet packet best tree decomposition

plot(t);

Plot original and compressed image

sm = size(map,1); figure('Color','white'); colormap(pink(sm)); image(wcodemat(X,sm)); title('Original Image') axis('square'); ax = gca; ax.XTick = []; ax.YTick = []; figure('Color','white'); colormap(pink(sm)); image(wcodemat(xd,sm)); title('Compressed Image') N2Str = ['Retained Energy: ',num2str(perfl2,'%5.2f'),' %']; N0Str = ['Number of zeros: ',num2str(perf0,'%5.2f'),' %']; xlabel({N2Str,N0Str}); axis('square'); ax = gca; ax.XTick = []; ax.YTick = [];

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