Wavelet Toolbox 4 Users Guide
Michel Misiti Yves Misiti Georges Oppenheim Jean-Michel Poggi

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Revision History

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Contents

Acknowledgments

Wavelet Applications

Introduction to Wavelet Analysis. Detecting Discontinuities and Breakdown Points I. Discussion. Detecting Discontinuities and Breakdown Points II. Discussion. Detecting Long-Term Evolution. Discussion. Detecting Self-Similarity. Wavelet Coefficients and Self-Similarity. Discussion. Identifying Pure Frequencies. Discussion. Suppressing Signals. Discussion. De-Noising Signals. Discussion. De-Noising Images. Discussion. 1-2 1-3 1-4 1-6 1-7 1-8 1-9 1-10 1-10 1-11 1-12 1-12 1-15 1-16 1-18 1-18 1-21 1-22
Compressing Images. Discussion. Fast Multiplication of Large Matrices. Example 1: Effective Fast Matrix Multiplication. Example 2: Ineffective Fast Matrix Multiplication.

1-27 1-27 1-29 1-29 1-30

Wavelets in Action: Examples and Case Studies
Illustrated Examples. Advice to the Reader. Example 1: A Sum of Sines. Example 2: A Frequency Breakdown. Example 3: Uniform White Noise. Example 4: Colored AR(3) Noise. Example 5: Polynomial + White Noise. Example 6: A Step Signal. Example 7: Two Proximal Discontinuities. Example 8: A Second-Derivative Discontinuity. Example 9: A Ramp + White Noise. Example 10: A Ramp + Colored Noise. Example 11: A Sine + White Noise. Example 12: A Triangle + A Sine. Example 13: A Triangle + A Sine + Noise. Example 14: A Real Electricity Consumption Signal. Case Study: An Electrical Signal. Data and the External Information. Analysis of the Midday Period. Analysis of the End of the Night Period. Suggestions for Further Analysis. 2-2 2-5 2-7 2-9 2-11 2-13 2-15 2-17 2-19 2-21 2-23 2-25 2-27 2-29 2-31 2-33 2-35 2-35 2-37 2-38 2-41

Using Wavelet Packets

About Wavelet Packet Analysis. One-Dimensional Wavelet Packet Analysis. Compressing a Signal Using Wavelet Packets. De-Noising a Signal Using Wavelet Packets. Two-Dimensional Wavelet Packet Analysis. Compressing an Image Using Wavelet Packets. Importing and Exporting from Graphical Tools. Saving Information to Disk. Loading Information into the Graphical Tools. 3-2 3-7 3-12 3-15 3-22 3-26 3-31 3-31 3-35

Advanced Concepts

Mathematical Conventions. General Concepts. Wavelets: A New Tool for Signal Analysis. Wavelet Decomposition: A Hierarchical Organization. Finer and Coarser Resolutions. Wavelet Shapes. Wavelets and Associated Families. Wavelet Transforms: Continuous and Discrete. Local and Global Analysis. Synthesis: An Inverse Transform. Details and Approximations. Fast Wavelet Transform (FWT) Algorithm. Filters Used to Calculate the DWT and IDWT. Algorithms. Why Does Such an Algorithm Exist?. One-Dimensional Wavelet Capabilities. 4-2 4-5 4-5 4-5 4-6 4-6 4-7 4-12 4-14 4-15 4-15 4-19 4-19 4-23 4-28 4-32

Many thanks to those who tested and helped to refine the software and the printed matter and at last to the MathWorks group and specially to Roy Lurie, Jim Tung, Bruce Sesnovich, Jad Succari, Jane Carmody, and Paul Costa. And finally, apologies to those we may have omitted. About the Authors Michel Misiti, Georges Oppenheim, and Jean-Michel Poggi are mathematics professors at Ecole Centrale de Lyon, University of Marne-La-Valle and Paris 5 University. Yves Misiti is a research engineer specializing in Computer Sciences at Paris 11 University. The authors are members of the Laboratoire de Mathmatique at Orsay-Paris 11 University France. Their fields of interest are statistical signal processing, stochastic processes, adaptive control, and wavelets. The authors group, established more than 15 years ago, has published numerous theoretical papers and carried out applications in close collaboration with industrial teams. For instance: Robustness of the piloting law for a civilian space launcher for which an expert system was developed Forecasting of the electricity consumption by nonlinear methods Forecasting of air pollution Notes by Yves Meyer The history of wavelets is not very old, at most 10 to 15 years. The field experienced a fast and impressive start, characterized by a close-knit international community of researchers who freely circulated scientific information and were driven by the researchers youthful enthusiasm. Even as the commercial rewards promised to be significant, the ideas were shared, the trials were pooled together, and the successes were shared by the community. There are lots of successes for the community to share. Why? Probably because the time is ripe. Fourier techniques were liberated by the appearance of windowed Fourier methods that operate locally on a time-frequency approach. In another direction, Burt-Adelsons pyramidal algorithms, the quadrature mirror filters, and filter banks and subband coding are available.
The mathematics underlying those algorithms existed earlier, but new computing techniques enabled researchers to try out new ideas rapidly. The numerical image and signal processing areas are blooming. The wavelets bring their own strong benefits to that environment: a local outlook, a multiscaled outlook, cooperation between scales, and a time-scale analysis. They demonstrate that sines and cosines are not the only useful functions and that other bases made of weird functions serve to look at new foreign signals, as strange as most fractals or some transient signals. Recently, wavelets were determined to be the best way to compress a huge library of fingerprints. This is not only a milestone that highlights the practical value of wavelets, but it has also proven to be an instructive process for the researchers involved in the project. Our initial intuition generally was that the proper way to tackle this problem of interweaving lines and textures was to use wavelet packets, a flexible technique endowed with quite a subtle sharpness of analysis and a substantial compression capability. However, it was a biorthogonal wavelet that emerged victorious and at this time represents the best method in terms of cost as well as speed. Our intuitions led one way, but implementing the methods settled the issue by pointing us in the right direction. For wavelets, the period of growth and intuition is becoming a time of consolidation and implementation. In this context, a toolbox is not only possible, but valuable. It provides a working environment that permits experimentation and enables implementation. Since the field still grows, it has to be vast and open. The Wavelet Toolbox product addresses this need, offering an array of tools that can be organized according to several criteria: Synthesis and analysis tools Wavelet and wavelet packets approaches Signal and image processing Discrete and continuous analyses Orthogonal and redundant approaches Coding, de-noising and compression approaches

Detecting Self-Similarity
The purpose of this example is to show how analysis by wavelets can detect a self-similar, or fractal, signal. The signal here is the Koch curve a synthetic signal that is built recursively.
This analysis was performed with the Continuous Wavelet 1-D graphical tool. A repeating pattern in the wavelet coefficients plot is characteristic of a signal that looks similar on many scales.
Wavelet Coefficients and Self-Similarity
From an intuitive point of view, the wavelet decomposition consists of calculating a resemblance index between the signal and the wavelet. If the index is large, the resemblance is strong, otherwise it is slight. The indices are the wavelet coefficients.
If a signal is similar to itself at different scales, then the resemblance index or wavelet coefficients also will be similar at different scales. In the coefficients plot, which shows scale on the vertical axis, this self-similarity generates a characteristic pattern.
The work of many authors and the trials that they have carried out suggest that wavelet decomposition is very well adapted to the study of the fractal properties of signals and images. When the characteristics of a fractal evolve with time and become local, the signal is called a multifractal. The wavelets then are an especially suitable tool for practical analysis and generation.
Identifying Pure Frequencies
The purpose of this example is to show how analysis by wavelets can effectively perform what is thought of as a Fourier-type function that is, resolving a signal into constituent sinusoids of different frequencies. The signal is a sum of three pure sine waves.
The signal is a sum of three sines: slow, medium, and rapid, which have periods (relative to the sampling period of 1) of 200, 20, and 2, respectively. The slow, medium, and rapid sinusoids appear most clearly in approximation A4, detail D4, and detail D1, respectively. The slight differences that can be observed on the decompositions can be attributed to the sampling period.
Detail D1 contains primarily the signal components whose period is between 1 and 2 (i.e., the rapid sine), but this period is not visible at the scale that is used for the graph. Zooming in on detail D1 (see below) reveals that each belly is composed of 10 oscillations, and this can be used to estimate the period. We indeed find that it is close to 2.

One-Dimensional Wavelet Packet Analysis
We now turn to the Wavelet Packet 1-D tool to analyze a synthetic signal that is the sum of two linear chirps. Starting the Wavelet Packet 1-D Tool.
1 From the MATLAB prompt, type wavemenu.
The Wavelet Toolbox Main Menu appears.
Click the Wavelet Packet 1-D menu item.
The tool appears on the desktop.

Loading a Signal.

1 From the File menu, choose the Load Signal option.
2 When the Load Signal dialog box appears, select the demo
MAT-file sumlichr.mat, which should reside in the MATLAB folder toolbox/wavelet/wavedemo. Click the OK button.
The sumlichr signal is loaded into the Wavelet Packet 1-D tool.

Analyzing a Signal.

1 Make the appropriate settings for the analysis. Select the db2 wavelet,
level 4, entropy threshold, and for the threshold parameter type 1. Click the Analyze button.
The available entropy types are listed below. Type Shannon Description Nonnormalized entropy involving the logarithm of the squared value of each signal sample or, more formally,

si log(si ).

Threshold Norm Log Energy
The number of samples for which the absolute value of the signal exceeds a threshold. The concentration in l p norm with 1 p. The logarithm of energy, defined as the sum over all samples:

log(si2 ).

Type SURE (Steins Unbiased Risk Estimate)
Description A threshold-based method in which the threshold equals

2 log e ( n log 2 (n) )

where n is the number of samples in the signal. User An entropy type criterion you define in a file.
For more information about the available entropy types, user-defined entropy, and threshold parameters, see the wentropy reference page and Choosing the Optimal Decomposition on page 4-158. Note Many capabilities are available using the command area on the right of the Wavelet Packet 1-D window. Some of them are used in the sequel. For a more complete description, see Appendix A, Wavelet Packet Tool Features (1-D and 2-D) on page A-21.

Continuous Versus Discrete Transform
Using a redundant representation close to the so-called continuous analysis, instead of a nonredundant discrete time-scale representation, can be useful for analysis purposes. The nonredundant representation is associated with an orthonormal basis, whereas the redundant representation uses much more scale and position values than a basis. For a classical fractal signal, the redundant methods are quite accurate. Graphic representation of discrete analysis: (in the middle of the figure Continuous Versus Discrete Transform on page 4-13) time is on the abscissa and on the ordinate the scale a is dyadic: 21, 22, 23, 24, and 25 (from the bottom to the top), levels are 1, 2, 3, 4, and 5. Each coefficient of level k is repeated 2k times. Graphic representation of continuous analysis: (at the bottom of the figure Continuous Versus Discrete Transform on page 4-13) time is on the abscissa and on the ordinate the scale varies almost continuously between 21 and 25 by step 1 (from the bottom to the top). Keep in mind that when a scale is small, only small details are analyzed, as in a geographical map.
Local and Global Analysis
A small scale value permits us to perform a local analysis; a large scale value is used for a global analysis. Combining local and global is a useful feature of the method. Let us be a bit more precise about the local part and glance at the frequency domain counterpart. Imagine that the analyzing function or is zero outside of a domain U, which is contained in a disk of radius : (u) = 0, u U. The wavelet is localized. The signal s and the function are then compared in the disk, taking into account only the t values in the disk. The signal values, which are located outside of the domain U, do not influence the value of the coefficient
R s(t) (t)dt and we get R s(t) (t)dt = U s(t) (t)dt
The same argument holds when is translated to position b and the corresponding coefficient analyzes s around b. So this analysis is local. The wavelets having a compact support are used in local analysis. This is the case for Haar and Daubechies wavelets, for example. The wavelets whose values are considered as very small outside a domain U can be used with caution, as if they were in fact actually zero outside U. Not every wavelet has a compact support. This is the case, for instance, of the Meyer wavelet. The previous localization is temporal, and is useful in analyzing a temporal signal (or spatial signal if analyzing an image). The good spectral domain localization is a second type of a useful property. A result (linked to the Heisenberg uncertainty principle) links the dispersion of the signal f and the

dispersion of its Fourier transform f , and therefore of the dispersion of and . The product of these dispersions is always greater than a constant c (which does not depend on the signal, but only on the dimension of the space). So it is impossible to reduce arbitrarily both time and frequency localization.
In the Fourier and spectral analysis, the basic function is f(x) = exp(ix). This function is not a time localized function. The support is R. Its Fourier
transform f is a generalized function concentrated at point.
The function f is very poorly localized in time, but f is perfectly localized in frequency. The wavelets generate an interesting compromise on the supports, and this compromise differs from that of complex exponentials, sine, or cosine.
Synthesis: An Inverse Transform
In order to be efficient and useful, a method designed for analysis also has to be able to perform synthesis. The wavelet method achieves this. The analysis starts from s and results in the coefficients C(a,b). The synthesis starts from the coefficients C(a,b) and reconstructs s. Synthesis is the reciprocal operation of analysis. For signals of finite energy, there are two formulas to perform the inverse wavelet transform: Continuous synthesis:

s(t) =

R R C(a, b)

t b da db a a a2

where K is a constant depending on. Discrete synthesis:

C( j, k) j,k (t).

Of course, the previous formulas need some hypotheses on the function. More precisely, see What Functions Are Candidates to Be a Wavelet? on page 4-65 for the continuous synthesis formula and Why Does Such an Algorithm Exist? on page 4-28 for the discrete one.
Details and Approximations
The equations for continuous and discrete synthesis are of considerable interest and can be read in order to define the detail at level j:
1 Let us fix j and sum on k. A detail Dj is nothing more than the function

D j (t) =

C( j, k) j,k (t)
2 Now, let us sum on j. The signal is the sum of all the details:

Dealing with Border Distortion
Classically, the DWT is defined for sequences with length of some power of 2, and different ways of extending samples of other sizes are needed. Methods for extending the signal include zero-padding, smooth padding, periodic extension, and boundary value replication (symmetrization). The basic algorithm for the DWT is not limited to dyadic length and is based on a simple scheme: convolution and downsampling. As usual, when a convolution is performed on finite-length signals, border distortions arise.
Signal Extensions: Zero-Padding, Symmetrization, and Smooth Padding
To deal with border distortions, the border should be treated differently from the other parts of the signal. Various methods are available to deal with this problem, referred to as wavelets on the interval (see [CohDJV93] in References on page 4-168). These interesting constructions are effective in theory but are not entirely satisfactory from a practical viewpoint. Often it is preferable to use simple schemes based on signal extension on the boundaries. This involves the computation of a few extra coefficients at each stage of the decomposition process to get a perfect reconstruction. It should be noted that extension is needed at each stage of the decomposition process. Details on the rationale of these schemes are in Chapter 8 of the book Wavelets and Filter Banks, by Strang and Nguyen (see [StrN96] in References on page 4-168). The available signal extension modes are as follows (see dwtmode): Zero-padding ('zpd'): This method is used in the version of the DWT given in the previous sections and assumes that the signal is zero outside the original support. The disadvantage of zero-padding is that discontinuities are artificially created at the border.
Symmetrization ('sym'): This method assumes that signals or images can be recovered outside their original support by symmetric boundary value replication. It is the default mode of the wavelet transform in the toolbox. Symmetrization has the disadvantage of artificially creating discontinuities of the first derivative at the border, but this method works well in general for images. Smooth padding of order 1 ('spd'or 'sp1'): This method assumes that signals or images can be recovered outside their original support by a simple first-order derivative extrapolation: padding using a linear extension fit to the first two and last two values. Smooth padding works well in general for smooth signals. Smooth padding of order 0 ('sp0'): This method assumes that signals or images can be recovered outside their original support by a simple constant extrapolation. For a signal extension this is the repetition of the first value on the left and last value on the right. Periodic-padding (1) ('ppd'): This method assumes that signals or images can be recovered outside their original support by periodic extension. The disadvantage of periodic padding is that discontinuities are artificially created at the border. The DWT associated with these five modes is slightly redundant. But IDWT ensures a perfect reconstruction for any of the five previous modes whatever the extension mode used for DWT. Periodic-padding (2) ('per'): If the signal length is odd, the signal is first extended by adding an extra-sample equal to the last value on the right. Then a minimal periodic extension is performed on each side. The same kind of rule exists for images. This extension mode is used for SWT (1-D & 2-D). This last mode produces the smallest length wavelet decomposition. But the extension mode used for IDWT must be the same to ensure a perfect reconstruction.

Coiflet Wavelets: coifN

In coifN, N is the order. Some authors use 2N instead of N. For the coiflet construction, see [Dau92] pages 258259. By typing waveinfo('coif') at the
MATLAB command prompt, you can obtain a survey of the main properties of this family.
Coiflets coif3 on the Left and coif5 on the Right
Built by Daubechies at the request of Coifman, the function has 2N moments equal to 0 and, what is more unusual, the function has 2N1 moments equal to 0. The two functions have a support of length 6N1. The coifN and are much more symmetrical than the dbNs. With respect to the support length, coifN has to be compared to db3N or sym3N. With respect to the number of vanishing moments of , coifN has to be compared to db2N or sym2N. If s is a sufficiently regular continuous time signal, for large j the coefficient
s, j,k is approximated by 2 j / 2 s(2 j k).
If s is a polynomial of degree d, d N 1, then the approximation becomes an equality. This property is used, connected with sampling problems, when calculating the difference between an expansion over the j,l of a given signal and its sampled version.
Biorthogonal Wavelet Pairs: biorNr.Nd
More about biorthogonal wavelets can be found in [Dau92] pages 259, 26285 and in [Coh92]. By typing waveinfo('bior') at the MATLAB command
prompt, you can obtain a survey of the main properties of this family, as well as information about Nr and Nd orders and associated filter lengths.
Biorthogonal Wavelets bior2.4 on the Left and bior4.4 on the Right
The new family extends the wavelet family. It is well known in the subband filtering community that symmetry and exact reconstruction are incompatible (except for the Haar wavelet) if the same FIR filters are used for reconstruction and decomposition. Two wavelets, instead of just one, are introduced:
One, , is used in the analysis, and the coefficients of a signal s are c j,k = s( x) j,k ( x) dx
The other, , is used in the synthesis

c j ,k j ,k j ,k

In addition, the wavelets and are related by duality in the following sense:
j,k ( x) j,k ( x)dx = 0 as soon as j j or k k and even 0,k ( x)0,k ( x)dx = 0 as soon as k k
It becomes apparent, as Cohen pointed out in his thesis, that the useful properties for analysis (e.g., oscillations, zero moments) can be concentrated on the function whereas the interesting properties for synthesis (regularity) are assigned to the function. The separation of these two tasks proves very useful (see [Coh92] page 110).

fs = 1000; % sampling rate t = 0:1/fs:2; % 2 secs at 1kHz sample rate y = sin(256*pi*t); % sine of period 128 level = 6; wpt = wpdec(y,level,'sym8'); [Spec,Time,Freq] = wpspectrum(wpt,fs,'plot');
If you have the Signal Processing Toolbox software, you can compute the short-time Fourier transform.
figure; windowsize = 128; window = hanning(windowsize); nfft = windowsize; noverlap = windowsize-1; [S,F,T] = spectrogram(y,window,noverlap,nfft,fs); imagesc(T,F,log10(abs(S))) set(gca,'YDir','Normal')
xlabel('Time (secs)') ylabel('Freq (Hz)') title('Short-time Fourier Transform spectrum')
Sum of two sine waves with frequencies of 64 and 128 hertz.
fs = 1000; t = 0:1/fs:2; y = sin(128*pi*t) + sin(256*pi*t); % sine of periods 64 and 128. level = 6; wpt = wpdec(y,level,'sym8'); [Spec,Time,Freq] = wpspectrum(wpt,fs,'plot');
figure; windowsize = 128; window = hanning(windowsize); nfft = windowsize; noverlap = windowsize-1; [S,F,T] = spectrogram(y,window,noverlap,nfft,fs); imagesc(T,F,log10(abs(S))) set(gca,'YDir','Normal') xlabel('Time (secs)') ylabel('Freq (Hz)') title('Short-time Fourier Transform spectrum')
Signal with an abrupt change in frequency from 16 to 64 hertz at two seconds.
fs = 500; t = 0:1/fs:4; y = sin(32*pi*t).*(t<2) + sin(128*pi*t).*(t>=2); level = 6; wpt = wpdec(y,level,'sym8'); [Spec,Time,Freq] = wpspectrum(wpt,fs,'plot');
Wavelet packet spectrum of a linear chirp.
fs = 1000; t = 0:1/fs:2; y = sin(256*pi*t.^2); level = 6; wpt = wpdec(y,level,'sym8'); [Spec,Time,Freq] = wpspectrum(wpt,fs,'plot');
Wavelet packet spectrum of quadratic chirp.
y = wnoise('quadchirp',10); len = length(y); t = linspace(0,5,len); fs = 1/t(2); level = 6; wpt = wpdec(y,level,'sym8'); [Spec,Time,Freq] = wpspectrum(wpt,fs,'plot');
windowsize = 128; window = hanning(windowsize); nfft = windowsize; noverlap = windowsize-1; imagesc(T,F,log10(abs(S))) set(gca,'YDir','Normal') xlabel('Time (secs)') ylabel('Freq (Hz)') title('Short-time Fourier Transform spectrum')

Building Wavelet Packets

The computation scheme for wavelet packets generation is easy when using an orthogonal wavelet. We start with the two filters of length 2N, where h(n) and g(n), corresponding to the wavelet. Now by induction let us define the following sequence of functions: (Wn(x), n = 0, 1, 2,.) by

W2n ( x) = 2

2 N 1 k =0

h(k)Wn (2 x k)

W2n+1 ( x) = 2

g(k)Wn (2 x k)

where W0(x) = (x) is the scaling function and W1(x) = (x) is the wavelet function. For example for the Haar wavelet we have

N = 1, h(0) = h(1) =

g(0) = g(1) =

The equations become

W2n ( x) = Wn (2 x) + Wn (2 x 1)
W2n+1 ( x) = Wn (2 x) Wn (2 x 1)
W0(x) = (x) is the Haar scaling function and W1(x) = (x) is the Haar wavelet, both supported in [0, 1]. Then we can obtain W2n by adding two 1/2-scaled versions of Wn with distinct supports [0,1/2] and [1/2,1] and obtain W2n+1 by subtracting the same versions of Wn. For n = 0 to 7, we have the W-functions shown in the figure Haar Wavelet Packets on page 4-153.

Controlling the Colormap

The Colormap selection box, located at the lower right of the window, allows you to adjust the colormap that is used to plot images or coefficients (wavelet or wavelet packet).
This is more than an aesthetic adjustment because you are likely to see different features depending on your colormap selection. Consider these images of the Mandelbrot set generated in the Wavelet Packet 2-D tool, shown here using the bone and 1 bone colormaps.
Controlling the Number of Colors
The Nb. Colors slider, located at the bottom right of the window, allows you to adjust how many colors the tool uses to plot images or coefficients (wavelet or wavelet packet). You can also use the edit control to adjust the number of colors. Adjusting the number of colors can highlight different features of the plot. Consider the coefficients plot of the Koch curve generated in the Continuous Wavelet tool, shown here using 129 colors.
and here using 68 colors.
Controlling the Coloration Mode
In the Continuous Wavelet tools, the coloration of coefficients can be done in several different ways.
In the Wavelet 1-D tool, you access coefficients coloration with the More Display Options button, and then select the desired Coloration Mode option. The More Display Options button appears only when the Display mode is one of the following Show and Scroll, Show and Scroll (Stem Cfs), Superimposed, and Separate). In this case, scales are replaced by levels in all options of the Coloration Mode menu.

Using Menus

General Menu Bar
At the top of most windows you find the same kind of structure. The menu bar of each figure in Wavelet Toolbox software is very similar to the menu bar of the default MATLAB figures. You can use many of the tools that are offered in the menus and associated toolbar of the standard MATLAB figures.
One of the main differences is the View menu, which depends on the current tool used. View Dynamical Visualization Tool Option. The View > Dynamical Visualization Tool option lets you enable or disable the Dynamical Visualization Tool located at the bottom of each window.
Before using Zoom In, Zoom Out, or Rotate 3D options (or the equivalent icons from the toolbar), you must disable the Dynamical Visualization Tool to avoid possible conflicts. Default Display Mode Option. The Default Display Mode option is specific to the Wavelet 1-D tool and lets you set a default Display Mode for all the different analyses you perform inside the same tool.

File Menu Options

Depending on the tool you are using, the File menu contains customized options. For example, for the Wavelet 1-D tool, the following options are added:
Many windows have a FileExample Analysis menu option, which allows you to select an analysis example using predefined parameters.

From the GUI, you can modify the tree. For example, change Node label from Depth_Position to Index, change Node Action from Visualize to Split_Merge and merge the node 2. You get the following figure.
% From the command line, you can get the new tree.
newt = readtree(fig); % From the command line you can modify the new tree; % then plot it in the same figure. newt = wpjoin(newt,3); drawtree(newt,fig);
You can mix previous commands. The GUI associated with the plot command is simpler and quicker, but more actions and information are available using the full GUI tools related to wavelet packets. The methods associated with WPTREE objects let you do more complicated actions. Namely, using read and write methods, you can change terminal node coefficients. Lets illustrate this point with the following funny example.

Example 3: A Funny One

load gatlin2 t = wpdec2(X,1,'haar'); plot(t);
% Change Node Label from Depth_position to Index and % click the node (0). You get the following figure.
% Now modify the coefficients of the four terminal nodes. newt = t; NBcols = 40; for node = 1:4 cfs = read(t,'data',node); tmp = cfs(1:end,1:NBcols); cfs(1:end,1:NBcols) = cfs(1:end,end-NBcols+1:end); cfs(1:end,end-NBcols+1:end) = tmp; newt = write(newt,'data',node,cfs); end plot(newt) % Change Node Label from Depth_position to Index and % click on the node (0). You get the following figure.
You can use this method for a more useful purpose. Lets see a de-noising example.
Example 4: Thresholding Wavelet Packets
load noisbloc x = noisbloc; t = wpdec(x,3,'sym4'); plot(t); % Change Node Label from Depth_position to Index and % click the node (0). You get the following plot.
% Global thresholding. t1 = t; sorh = 'h'; thr = wthrmngr('wp1ddenoGBL','penalhi',t); cfs = read(t,'data'); cfs = wthresh(cfs,sorh,thr); t1 = write(t1,'data',cfs); plot(t1) % Change Node Label from Depth_position to Index and % click the node (0). You get the following plot.
% Node by node thresholding. t2 = t; sorh = 's'; thr(1) = wthrmngr('wp1ddenoGBL','penalhi',t); thr(2) = wthrmngr('wp1ddenoGBL','sqtwologswn',t); tn = leaves(t); for k=1:length(tn) node = tn(k); cfs = read(t,'data',node); numthr = rem(node,2)+1; cfs = wthresh(cfs,sorh,thr(numthr)); t2 = write(t2,'data',node,cfs); end plot(t2) % Change Node Label from Depth_position to Index and % click the node (0). You get the following plot.
Detailed Description of Objects in the Wavelet Toolbox Software
The following sections describe the objects in the Wavelet Toolbox software: WTBO Object on page B-16 NTREE Object on page B-17 DTREE Object on page B-18 WPTREE Object on page B-20

WTBO Object

Class WTBO (Wavelet Toolbox Object) -- Parent class: none

Fields

wtboInfo ud
Object information (Not used) Userdata field

Methods

wtbo get set
Constructor for the class WTBO. Get WTBO object field contents. Set WTBO object field contents.

Comments

Since any object in the toolbox is parented by a WTBO object, you can associate your own data to an object using the 'ud' field, and then access it. If Obj is an object (parented by a WTBO object), use
Obj = set(Obj,'ud',MyData)

to define the data.

To retrieve the data, use

MyData = get(O,'ud')

NTREE Object
Class NTREE (New Tree) -- Parent class: WTBO
wtbo order depth spsch tn
Parent object Tree order Tree depth Split scheme for nodes Column vector with terminal nodes indices
Constructor for the class NTREE. Find active nodes. Get NTREE object field contents. Recompose node(s). Split (decompose) node(s). Plot NTREE object. Set NTREE object field contents. Labels for the nodes of a tree. Manager for NTREE object.
findactn get nodejoin nodesplt plot set tlabels wtreemgr

Private

locnumcn tabofasc
Local number for a child node Table of ascendants of nodes

DTREE Object

Class DTREE (Data Tree) -- Parent class: NTREE

ntree allNI terNI

Parent object All Nodes Information Terminal Nodes Information

Fields Description

allNI is a NBnodes-by-3 array such that allNI(N,:) = [ind,size(1,1),size(1,2)]
ind = index of the node N size = size of data associated with the node N
terNI is a 1-by-2 cell array such that
terNI{1} is an NB_TerminalNodes-by-2 array such that
terNI{1}(N,:) is the size of coefficients associated with the N-th terminal node. The nodes are numbered from left to right and from top to bottom. The root index is 0.
terNI{2} is a row vector containing the previous coefficients stored row-wise in the above specified order.
dtree expand fmdtree nodejoin
Constructor for the class DTREE. Expand data tree. Field manager for DTREE object. Recompose node.
nodesplt rnodcoef defaninf get plot read set write merge recons split
Split (decompose) node. Reconstruct node coefficients. Define node information (all nodes). Get DTREE object field contents. Plot DTREE object. Read values in DTREE object fields. Set DTREE object field contents. Write values in DTREE object fields. Merge (recompose) the data of a node. Reconstruct node coefficients. Split (decompose) the data of a terminal node.
After the constructor, the first set of methods (between line separators) might not be overloaded (or only with great care). The second set of methods can be overloaded. The third set of methods must be overloaded to recompose, reconstruct, or decompose nodes data. The method nodejoin calls the method merge, the method nodesplt calls the method split, and the method rnodcoef calls the method recons. To define nodes information, you must overload the method defaninf. For each node N, the basic information is given by