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.

Details D3 and D4 contain the medium sine wave. The slow sine is clearly isolated in approximation A5, from which the higher-frequency information has been filtered.

Discussion

The deterministic part of the signal may undergo abrupt changes such as a jump, or a sharp change in the first or second derivative. In image processing, one of the major problems is edge detection, which also involves detecting abrupt changes. Also in this category, we find signals with very rapid evolutions such as transient signals in dynamic systems. The main characteristic of these phenomena is that the change is localized in time or in space. The purpose of the analysis is to determine The site of the change (e.g., time or position) The type of change (a rupture of the signal, or an abrupt change in its first or second derivative) The amplitude of the change The local aspects of wavelet analysis are well adapted for processing this type of event, as the processing scales are linked to the speed of the change.
Guidelines for Detecting Discontinuities
Short wavelets are often more effective than long ones in detecting a signal rupture. In the initial analysis scales, the support is small enough to allow fine analysis. The shapes of discontinuities that can be identified by the smallest wavelets are simpler than those that can be identified by the longest wavelets. Therefore, to identify A signal discontinuity, use the haar wavelet A rupture in the j-th derivative, select a sufficiently regular wavelet with at least j vanishing moments. (See Detecting Discontinuities and Breakdown Points II on page 1-6.)
The presence of noise, which is after all a fairly common situation in signal processing, makes identification of discontinuities more complicated. If the first levels of the decomposition can be used to eliminate a large part of the noise, the rupture is sometimes visible at deeper levels in the decomposition. Check, for example, the sample analysis File > Example Analysis > Basic Signals > ramp + white noise (MAT-file wnoislop). The rupture is visible in the level-six approximation (A6) of this signal.

t 1 + sin(0.3t) t s10 (t) = + sin(0.3t) 500
Example Example 13: A Triangle + A Sine + Noise on page 2-31
Equation A triangle + a sine + a noise:

s11(t)

wntrsin
501 t 1000, 1000 t + sin(0.3t) + b1 (t) 500 t t 500, s11 (t) = + sin(0.3t) + b1 (t) 500 s11 (t) =
Example 14: A Real Electricity Consumption Signal on page 2-33
A real electricity consumption signal

leleccum

Please note that All the decompositions use Daubechies wavelets. The examples show the signal, the approximations, and the details. The examples include specific comments and feature distinct domains for instance, if the level of decomposition is 5, The left column contains the signal and the approximations A5 to A1. The right column contains the signal and the details D5 to D1. The approximation A1 is located under A2, A2 under A3, and so on; the same is true for the details. The abscissa axis represents the time; the unit for the ordinate axis for approximations and details is the same as that of the signal. When the approximations do not provide enough information, they are replaced by details obtained by changing wavelets. The examples include questions for you to think about:
What can be seen on the figure? What additional questions can be studied?

Advice to the Reader

You should follow along and process these examples on your own, using either the graphical interface or the command line functions. Use the graphical interface for immediate signal processing. To execute the analyses included in the figures,
1 To open the Wavelet Toolbox Main Menu, type
2 Select the Wavelet 1-D menu option to open the Wavelet 1-D tool. 3 From the Wavelet 1-D tool, choose the File > Example Analysis menu

option.

4 From the dialog box, select the sample analysis in question.
This triggers the execution of the examples. When using the command line, follow the process illustrated in this file to conduct calculations:
% Load original 1-D signal. load sumsin; s = sumsin; % Perform the decomposition of s at level 5, using coif3. w = 'coif3' [c,l] = wavedec(s,5,w); % Reconstruct the approximation signals and detail signals at % using the wavelet decomposition structure [c,l]. for i = 1:5 A(i,:) = wrcoef('a',c,l,w,i); D(i,:) = wrcoef('d',c,l,w,i); end
Note This loop replaces 10 separate wrcoef statements defining approximations and details. The variable A contains the five approximations and the variable D contains the five details.

All the figures in this paragraph are generated using the graphical user interface tools, but the user can also process the analysis using the command line mode. The following example corresponds to a command line equivalent for producing the figure below.
% Load the original 1-D signal, decompose, reconstruct details % and plot. % Load the signal. load leleccum; s = leleccum; % Decompose the signal s at level 5 using the wavelet db3. w = 'db3'; [c,l] = wavedec(s,5,w); % Reconstruct the details using the decomposition structure. for i = 1:5 D(i,:) = wrcoef('d',c,l,w,i); end
Note This loop replaces five separate wrcoef statements defining the details. The variable D contains the five details.
% Avoid edge effects by suppressing edge values and plot. tt = 1+100:length(s)-100; subplot(6,1,1); plot(tt,s(tt),'r'); title('Electrical Signal and Details'); for i = 1:5, subplot(6,1,i+1); plot(tt,D(5-i+1,tt),'g'); end
Suggestions for Further Analysis
Let us now make some suggestions for possible further analysis starting from the details of the decomposition at level 5 of 3 days.
Identify the Sensor Failure
Focus on the wavelet decomposition and try to identify the sensor failure directly on the details D1, D2, and D3, and not the other ones. Try to identify the other part of the noise. Indication: see figure below.

Suppress the Noise

Suppress measurement noise. Try by yourself and afterwards use the de-noising tools. Indication: study the approximations and compare two successive days, the first without sensor failure and the second corrupted by failure (see figure below).
Identify Patterns in the Details
The idea here is to identify a pattern in the details typical of relay-switched water heaters. Indication: the figure below gives an example of such a period. Focus on details D2, D3, and D4 around abscissa 1350, 1383, and 1415 to detect abrupt changes of the signal induced by automatic switches.
Locate and Suppress Outlying Values
Suppress the outliers by setting the corresponding values of the details to 0. Indication: The figure below gives two examples of outliers around t = 1193 and t = 1215. The effect produced on the details is clear when focusing on the low levels. As far as outliers are concerned, D1 and D2 are synchronized with s, while D3 shows a delayed effect.

Study Missing Data

Missing data have been crudely substituted (around observation 2870) by the estimation of 30 minutes of sampled data and spline smoothing for the intermediate time points. You can improve the interpolation by using an approximation and portions of the details taken elsewhere, thus implementing a sort of graft. Indication: see the figure below focusing around time 2870, and use the small variations part of D1 to detect the missing data.
About Wavelet Packet Analysis on page 3-2 One-Dimensional Wavelet Packet Analysis on page 3-7 Two-Dimensional Wavelet Packet Analysis on page 3-22 Importing and Exporting from Graphical Tools on page 3-31

Before concluding this analysis, it is worth turning our attention to the colored coefficients for terminal nodes plot and considering the best tree decomposition for this image.
This plot is shown in the lower right side of the Wavelet Packet 2-D tool. The plot shows us which details have been decomposed and which have not. Larger squares represent details that have not been broken down to as many levels as smaller squares. Consider, for example, this level 2 decomposition pattern:
Looking at the pattern of small and large squares in the fingerprint analysis shows that the best tree algorithm has apparently singled out the diagonal details, often sparing these from further decomposition. Why is this? If we consider the original image, we realize that much of its information is concentrated in the sharp edges that constitute the fingerprints pattern. Looking at these edges, we see that they are predominantly oriented horizontally and vertically. This explains why the best tree algorithm has
chosen not to decompose the diagonal details they do not provide very much information.
Importing and Exporting from Graphical Tools
The Wavelet Packet 1-D and Wavelet Packet 2-D tools let you import information from and export information to your disk. If you adhere to the proper file formats, you can Save decompositions as well as synthesized signals and images from the wavelet packet graphical tools to disk Load signals, images, and one- and two-dimensional decompositions from disk into the Wavelet Packet 1-D and Wavelet Packet 2-D graphical tools
Saving Information to Disk
Using specific file formats, the graphical tools let you save synthesized signals or images, as well as one- or two-dimensional wavelet packet decomposition structures. This feature provides flexibility and allows you to combine command line and graphical interface operations.
Saving Synthesized Signals
You can process a signal in the Wavelet Packet 1-D tool, and then save the processed signal to a MAT-file. For example, load the example analysis: File > Example Analysis > db1 depth: 2 ent: shannon > sumsin and perform a compression or de-noising operation on the original signal. When you close the Wavelet Packet 1-D De-noising or Wavelet Packet 1-D Compression window, update the synthesized signal by clicking Yes in the dialog box. Then, from the Wavelet Packet 1-D tool, select the File > Save > Synthesized Signal menu option. A dialog box appears allowing you to select a folder and filename for the MAT-file. For this example, choose the name synthsig.
To load the signal into your workspace, simply type

load synthsig whos

synthsig valTHR wname

1x1000 1x1 1x3

double array double array char array
The synthesized signal is given by synthsig. In addition, the parameters of the de-noising or compression process are given by the wavelet name (wname) and the global threshold (valTHR).

Unlike conventional techniques, wavelet decomposition produces a family of hierarchically organized decompositions. The selection of a suitable level for the hierarchy will depend on the signal and experience. Often the level is chosen based on a desired low-pass cutoff frequency. At each level j, we build the j-level approximation Aj, or approximation at level j, and a deviation signal called the j-level detail Dj, or detail at level j. We can consider the original signal as the approximation at level 0, denoted
by A0. The words approximation and detail are justified by the fact that A1 is an approximation of A0 taking into account the low frequencies of A0, whereas the detail D1 corresponds to the high frequency correction. Among the figures presented in Reconstructing Approximations and Details in the Wavelet Toolbox Getting Started Guide, one of them graphically represents this hierarchical decomposition. One way of understanding this decomposition consists of using an optical comparison. Successive images A1, A2, A3 of a given object are built. We use the same type of photographic devices, but with increasingly poor resolution. The images are successive approximations; one detail is the discrepancy between two successive images. Image A2 is, therefore, the sum of image A4 and intermediate details D4, D3: A 2 = A 3 + D3 = A 4 + D 4 + D 3
Finer and Coarser Resolutions
The organizing parameter, the scale a, is related to level j by a = 2j. If we define resolution as 1/a, then the resolution increases as the scale decreases. The greater the resolution, the smaller and finer are the details that can be accessed. j Scale Resolution 1/2

1 1/2 2

2 1/4 4
From a technical point of view, the size of the revealed details for any j is proportional to the size of the domain in which the wavelet or analyzing
x function of the variable x, is not too close to 0. a

Wavelet Shapes

One-dimensional analysis is based on one scaling function and one wavelet. Two-dimensional analysis (on a square or rectangular grid) is based on one scaling function (x1, x2) and three wavelets.
The following figure shows and for each wavelet, except the Morlet wavelet and the Mexican hat, for which does not exist. All the functions decay quickly to zero. The Haar wavelet is the only noncontinuous function with three points of discontinuity (0, 0.5, 1). The functions oscillate more than associated functions. coif2 exhibits some angular points; db6 and sym6 are quite smooth. The Morlet and Mexican hat wavelets are symmetrical.
Various One-Dimensional Wavelets
Wavelets and Associated Families
In the one-dimensional context, we distinguish the wavelet from the associated function , called the scaling function. Some properties of and are

Lifting Functions

The lifting functions of the toolbox are organized into five groups: Lifting Schemes on page 4-56 Biorthogonal Quadruplets of Filters and Lifting Schemes on page 4-56 Usual Biorthogonal Quadruplets on page 4-56 Lifting Wavelet Transform (LWT) on page 4-57 Laurent Polynomials and Matrices on page 4-57

Lifting Schemes

Function Name

lsinfo displs addlift

Description Information about lifting schemes Display a lifting scheme Add primal or dual elementary lifting steps to a lifting scheme
Biorthogonal Quadruplets of Filters and Lifting Schemes
These functions connect lifting schemes to biorthogonal quadruplets of filters and associated scaling and wavelet function pairs. Function Name
liftfilt filt2ls ls2filt bswfun
Description Apply elementary lifting steps on quadruplet of filters Transform a quadruplet of filters to a lifting scheme Transform a lifting scheme to a quadruplet of filters Compute and plot biorthogonal scaling and wavelet functions
Usual Biorthogonal Quadruplets
These functions provide some basic lifting schemes associated with some usual orthogonal or biorthogonal (true) wavelets and the lazy one. These schemes can be used to initialize a lifting procedure. Function Name
wavenames liftwave wave2lp
Description Provides usual wavelet names available for LWT Provides lifting scheme associated with a usual wavelet Provides Laurent polynomials associated with a usual wavelet
Lifting Wavelet Transform (LWT)
These functions contain the direct and inverse lifting wavelet transform (LWT) files for both 1-D and 2-D signals. LWT reduces to the polyphase version of the DWT algorithm with zero-padding extension mode and without extra-coefficients. Function Name
lwt ilwt lwtcoef lwt2 ilwt2 lwtcoef2
Description 1-D lifting wavelet transform Inverse 1-D lifting wavelet transform Extract or reconstruct 1-D LWT wavelet coefficients 2-D lifting wavelet transform Inverse 2-D lifting wavelet transform Extract or reconstruct 2-D LWT wavelet coefficients
Laurent Polynomials and Matrices

h l2 ( Z), g l2 ( Z)

1 x 1 = hn ( x n) 2 nZ 1 x 1 = gn ( x n) 2 nZ
By rewriting these formulas using Fourier transforms (expressed using a hat) we obtain

1 (2 ) = h( ) ( ) 2

(2 ) =

g( ) ( )

There are functions for which the h has a finite impulse response (FIR): there is only a finite number of nonzero hn coefficients. The associated wavelets were built by I. Daubechies (see [Dau92] in References on page 4-168) and are used extensively in the toolbox. The reader can refer to page 164 and Chapter 10 of the book Wavelets and Filter Banks, by Strang and Nguyen (see [StrN96] in References on page 4-168).
What Is the Link Between Wavelet and Fourier Analysis?
Wavelet analysis complements the Fourier analysis for which there are several functions: fft in MATLAB software and spectrum and sptool in Signal Processing Toolbox software. Fourier analysis uses the basic functions sin(t), cos(t), and exp(it). In the frequency domain, these functions are perfectly localized. The functions are suited to the analysis and synthesis of signals with a simple spectrum, which is very well localized in frequency; for example, sin(1 t) + 0.5 sin(2 t) cos(3 t). In the time domain, these functions are not localized. It is difficult for them to analyze or synthesize complex signals presenting fast local variations such as transients or abrupt changes: the Fourier coefficients for a frequency will depend on all values in the signal. To limit the difficulties involved, it is possible to window the signal using a regular function, which is zero or nearly zero outside a time segment [m, m]. We then build a well localized slice as I. Daubechies calls it (see page 2 of [Dau92] in References on page 4-168). The windowed-Fourier analysis coefficients are the doubly indexed coefficients:

Gs ( , t) =

s(u) g ( t u)e
The analogy of this formula with that of the wavelet coefficients is obvious:

C(a, t) =

s(u)

1 t u du a a

The large values of a correspond to small values of. The Fourier coefficient Gs(,t) depends on the values of the signal s on the segment [t m, t + m] with a constant width. If , like g, is zero outside of [m, m], the C(a,t) coefficients will depend on the values of the signal s on the segment [t am, t + am] of width 2am, which varies as a function of a. This slight difference solves several difficulties, allowing a kind of time-windowed analysis, different at the various scales a. The wavelets stay competitive, however, even in contexts considered favorable for the Fourier technique. I. Daubechies (see [Dau92] pages 3 to 6) gives an example of windowed-Fourier processing and complex Morlet wavelet processing, (t) = Cet / a (ei t e a /4 ) with a = 4, of a signal composed mainly of the sum of two sines. This wavelet analysis gives good results.

Because y is a standard Gaussian white noise, we expect that each method kills roughly all the coefficients and returns the result f(x) = 0. For Steins Unbiased Risk Estimate and minimax thresholds, roughly 3% of coefficients are saved. For other selection rules, all the coefficients are set to 0. We know that the detail coefficients vector is the superposition of the coefficients of f and the coefficients of e, and that the decomposition of e leads to detail coefficients, which are standard Gaussian white noises. So minimax and SURE threshold selection rules are more conservative and would be more convenient when small details of function f lie near the noise range. The two other rules remove the noise more efficiently. The option 'heursure' is a compromise. In this example, the fixed form threshold wins. Recalling step 2 of the de-noise procedure, the function thselect performs a threshold selection, and then each level is thresholded. This second step can be done using wthcoef, directly handling the wavelet decomposition structure of the original signal s.
Dealing with Unscaled Noise and Nonwhite Noise
Usually in practice the basic model cannot be used directly. We examine here the options available to deal with model deviations in the main de-noising function wden. The simplest use of wden is
sd = wden(s,tptr,sorh,scal,n,wav)
which returns the de-noised version sd of the original signal s obtained using the tptr threshold selection rule. Other parameters needed are sorh, scal, n, and wav. The parameter sorh specifies the thresholding of details coefficients of the decomposition at level n of s by the wavelet called wav. The remaining parameter scal is to be specified. It corresponds to thresholds rescaling methods. Option

'one' 'sln' 'mln'

Corresponding Model Basic model Basic model with unscaled noise Basic model with nonwhite noise
Option scal = 'one' corresponds to the basic model. In general, you can ignore the noise level and it must be estimated. The detail coefficients cD1 (the finest scale) are essentially noise coefficients with standard deviation equal to. The median absolute deviation of the coefficients is a robust estimate of. The use of a robust estimate is crucial for two reasons. The first one is that if level 1 coefficients contain f details, then these details are concentrated in a few coefficients if the function f is sufficiently regular. The second reason is to avoid signal end effects, which are pure artifacts due to computations on the edges. Option scal = 'sln' handles threshold rescaling using a single estimation of level noise based on the first-level coefficients. When you suspect a nonwhite noise e, thresholds must be rescaled by a level-dependent estimation of the level noise. The same kind of strategy as in the previous option is used by estimating lev level by level.

Various examples illustrating either the command-line mode or GUI tools for true compression using wavelets are in Two-Dimensional True Compression in the Wavelet Toolbox Getting Started Guide. More details on how to use the main command-line function are in the Reference document (see wcompress). More information on the true compression for images and more precisely on the compression methods is in [Wal99], [Sha93], [Sai96], [StrN96], and [Chr06]. See References on page 4-168.

Wavelet Packets

The wavelet packet method is a generalization of wavelet decomposition that offers a richer signal analysis. Wavelet packet atoms are waveforms indexed by three naturally interpreted parameters: position, scale (as in wavelet decomposition), and frequency. For a given orthogonal wavelet function, we generate a library of bases called wavelet packet bases. Each of these bases offers a particular way of coding signals, preserving global energy, and reconstructing exact features. The wavelet packets can be used for numerous expansions of a given signal. We then select the most suitable decomposition of a given signal with respect to an entropy-based criterion. There exist simple and efficient algorithms for both wavelet packet decomposition and optimal decomposition selection. We can then produce adaptive filtering algorithms with direct applications in optimal signal coding and data compression.
From Wavelets to Wavelet Packets: Decomposing the Details
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 as shown in the following figure.
Wavelet Packet Decomposition Tree at Level 3
The idea of this decomposition is to start from a scale-oriented decomposition, and then to analyze the obtained signals on frequency subbands.
Wavelet Packets in Action: An Introduction
The following simple examples illustrate certain differences between wavelet analysis and wavelet packet analysis.
Example 1: Analyzing a Sine Function
The signal to be analyzed, called sinper8, is a 256-length sampled sine function of period 8. The Haar wavelet is used to decompose the signal at level 7. The following figure contains the time-frequency plot (x-axis is time and y-axis is frequency, high to low from the top to the bottom) for the wavelet decomposition (on the left) and for the wavelet packet decomposition (on the right). Wavelet decomposition localizes the period of the sine within the interval [8,16]. Wavelet packets provide a more precise estimation of the actual period. How to Obtain and Explain These Graphs. You can reproduce these graphs by typing at the MATLAB prompt

=================================== Mexican_hat mexh =================================== Morlet morl =================================== Complex Gaussian cgau -----------------------------cgau1 cgau2 cgau3 cgau4 cgau5 cgau** =================================== Shannon shan -----------------------------shan1-1.5 shan1-1 shan1-0.5 shan1-0.1 shan2-3 shan** =================================== Frequency B-Spline fbsp -----------------------------fbsp1-1-1.5 fbsp1-1-1 fbsp1-1-0.5 fbsp2-1-1 fbsp2-1-0.5 fbsp2-1-0.1 fbsp** =================================== Complex Morlet cmor -----------------------------cmor1-1.5 cmor1-1 cmor1-0.5 cmor1-1 cmor1-0.5 cmor1-0.1 cmor** =================================== % Add new family of orthogonal wavelets. % You must define: % % Family Name: Lemarie % Family Short Name: lem % Type of wavelet: 1 (orth) % Wavelets numbers: 5 % File driver: lemwavf % % The function lemwavf.m must be as follow: % function w = lemwavf(wname) % where the input argument wname is a string: % wname = 'lem1' or 'lem2'. i.e., % wname = sh.name + number % and w the corresponding scaling filter.
% The addition is obtained using: wavemngr('add','Lemarie','lem',1,'5','lemwavf'); % The ascii file 'wavelets.asc' is saved as % 'wavelets.prv', then it is modified and % the MAT file 'wavelets.inf' is generated. % List wavelets families. wavemngr('read') ans = =================================== Haar haar Daubechies db Symlets sym Coiflets coif BiorSplines bior ReverseBior rbio Meyer meyr DMeyer dmey Gaussian gaus Mexican_hat mexh Morlet morl Complex Gaussian cgau Shannon shan Frequency B-Spline fbsp Complex Morlet cmor Lemarie lem ===================================
After Adding a New Wavelet Family
When you use the wavemngr command to add a new wavelet, the toolbox creates three wavelet extension files in the current folder: the two ASCII files wavelets.asc and wavelets.prv, and the MAT-file wavelets.inf. If you want to use your own extended wavelet families with the Wavelet Toolbox software, you should
1 Create a new folder specifically to hold the wavelet extension files. 2 Move the previously mentioned files into this new folder. 3 Prepend this folder to the MATLAB folder search path (see the reference
entry for the path command).
4 Use this same folder for subsequent modifications. Allowing many wavelet
extension files to proliferate in different folders may lead to unpredictable results.
5 Define a file called <fsn>info.m (for example, see dbinfo.m or morlinfo.m).

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
allNI(N,1:3): [index,size(1,1),size(1,2)];
You can add other information by adding columns to allNI. See the WPTREE object method for an example. If the method get is not overloaded, using the DTREE get method you can get some object field contents (but not all).
For example, if T is parented by a DTREE object of order 2 and if 'Tfield' is a field of T, whose content is Tval, [a,b] = get(t,'order','Tfield') returns a = 2 and b = 'errorWTBX'. Nevertheless, using a nondocumented method you can get the right values. Namely: [a,b] = getwtbo(t,'order','Tfield') returns a = 2 and b=Tval.

WPTREE Object

Class WPTREE (Wavelet Packet Tree) -- Parent class: DTREE

dtree wavInfo entInfo

Parent object Structure (wavelet information) Structure (entropy information)

bases. See analysis,wavelet packets besttree function 4-167 binning density estimation 4-120 regression estimation 4-125 biorthogonal quadruplets 4-53 biorthogonal wavelets 4-78 definition 4-78 See also analysis border distortion boundary value replication 4-35 periodic extension 4-35 periodic padding 4-36 periodized wavelet transform 4-45 smooth padding 4-36 symmetric extension 4-35 symmetrization 4-35 zero-padding 4-35 breakdowns peak 2-33 proximal slopes 2-19 rupture 2-17 second derivative 2-21 variance 4-112

centfrq function 4-69

Index-1
chirp signal example analysis 4-146 coefficients approximation fast wavelet transform 4-23 coloration A-21 detail fast wavelet transform 4-23 coiflets definition 4-77 Coloration Mode color coding A-2 controlling A-7 controlling the colormap A-6 colored AR(3) noise example 2-13 complex frequency B-spline wavelets 4-88 complex Gaussian wavelets 4-87 complex Morlet wavelets 4-87 complex Shannon wavelets 4-89 compressing images fingerprint example 1-27 true compression 4-136 compression ddencmp function 3-4 difference with de-noising 4-116 energy ratio 4-118 methods 4-132 norm recovery 4-118 number of zeros 4-119 predefined strategies 4-128 procedure wavelet packets 3-5 wavelets 4-115 retained energy 4-118 thresholding strategies 4-132 true 4-136 using wavelet packets 3-26
Daubechies wavelets definition 4-74 de-noising basic model one-dimensional 4-102 two-dimensional 4-111 default values 3-4 fixed form threshold 4-105 methods 4-132 minimax performance 4-105 noise size estimate 4-107 nonwhite noise 4-107 predefined strategies 4-128 procedure wavelet packets 3-5 wavelets 4-103 SURE estimate 4-105 using SWT 2-D analysis example 1-24 variance adaptive 4-112 white noise 4-101 de-noising images 2-D wavelet analysis and 2-D stationary wavelet analysis 1-21 two-dimensional procedure 4-111 de-noising signals wavelet analysis 1-18 decimation. See downsampling decomposition best-level 4-164 choosing optimal 4-158 entropy-based criteria 4-158 hierarchical organization 4-10 optical comparison 4-6 density estimation definition 4-119 details decomposition 4-143 mathematical definition 4-17

Index-2

notation 4-3 orientation 4-25 wavelet decomposition 4-6 dilation equation twin-scale relation 4-19 discontinuities 2-19 detecting 1-3 See also breakdowns discrete Meyer wavelet 4-86 downsampling one-dimensional 4-24 two-dimensional 4-25