User manual MATHWORKS WAVELET TOOLBOX 4

[. . . ] Wavelet ToolboxTM 4 User's Guide Michel Misiti Yves Misiti Georges Oppenheim Jean-Michel Poggi How to Contact MathWorks Web Newsgroup www. mathworks. com/contact_TS. html Technical Support www. mathworks. com comp. soft-sys. matlab suggest@mathworks. com bugs@mathworks. com doc@mathworks. com service@mathworks. com info@mathworks. com Product enhancement suggestions Bug reports Documentation error reports Order status, license renewals, passcodes Sales, pricing, and general information 508-647-7000 (Phone) 508-647-7001 (Fax) The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 For contact information about worldwide offices, see the MathWorks Web site. Wavelet ToolboxTM User's Guide © COPYRIGHT 1997­2010 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. [. . . ] laurpoly([0. 125 -0. 125], 1)); [LoDN, HiDN, LoRN, HiRN] = liftfilt(LoD, HiD, LoR, HiR, twoels); % The biorthogonal wavelet bior1. 3 is obtained up to % an unsignificant sign. [LoDB, HiDB, LoRB, HiRB] = wfilters('bior1. 3'); samewavelet = . . . 4-60 Lifting Method for Constructing Wavelets isequal([LoDB, HiDB, LoRB, HiRB], [LoDN, -HiDN, LoRN, HiRN]) samewavelet = 1 % Visualize the two times two pairs of scaling and wavelet % functions. bswfun(LoDN, HiDN, LoRN, HiRN, 'plot'); 4-61 4 Advanced Concepts Frequently Asked Questions Continuous or Discrete Analysis? When is continuous analysis more appropriate than discrete analysis?To answer this, consider the related questions: Do you need to know all values of a continuous decomposition to reconstruct the signal s exactly?When the energy of the signal is finite, not all values of a decomposition are needed to exactly reconstruct the original signal, provided that you are using a wavelet that satisfies some admissibility condition (see [Dau92] pages 7, 24, and 27). In which case, a continuous-time signal s is characterized by the knowledge of the discrete transform . In such cases, discrete analysis is sufficient and continuous analysis is redundant. When the signal is recorded in continuous time or on a very fine time grid, both analyses are possible. It depends; each one has its own advantages: · Discrete analysis ensures space-saving coding and is sufficient for exact reconstruction. · Continuous analysis is often easier to interpret, since its redundancy tends to reinforce the traits and makes all information more visible. Thus, the analysis gains in "readability" and in ease of interpretation what it loses in terms of saving space. Why Are Wavelets Useful for Space-Saving Coding? The family of functions (0, k;j, l) j 0, used for the analysis is an orthogonal basis, therefore leading to nonredundancy. Let us remember that for a one-dimensional signal, stands for 4-62 Frequently Asked Questions For biorthogonal wavelets, the idea is similar. What Is the Advantage Having Zero Average and Sometimes Several Vanishing Moments? When the wavelet's k + 1 moments are equal to zero ( ) all the polynomial signals coefficients. This property ensures the suppression of signals that are polynomials of a degree lower or equal to k. What About the Regularity of a Wavelet ? In theoretical and practical studies, the notion of regularity has been increasing in importance. Wavelets are tools used to study regularity and to conduct local studies. Deterministic fractal signals or Brownian motion trajectories are locally very irregular; for example, the latter are continuous signals, but their first derivative exists almost nowhere. If the signal is s-time continuously differentiable at x0 and s is an integer ( ), then the regularity is s. If the derivative of f of order m resembles s = m + r with 0 < r < 1. locally around x0, then The regularity of f in a domain is that of its least regular point. The following table gives some indications for Daubechies wavelets. 4-63 4 Advanced Concepts Regularity db1 = Haar Discontinuous db2 0. 5 db3 0. 91 db4 1. 27 db5 1. 59 db7 2. 15 db10 2. 90 We have an asymptotic relation linking the size of the support of the Daubechies wavelets dbN and their regularity: when , length(support) = 2N, regularity . The functions are more regular at certain points than at others (see Zooming in on a db3 Wavelet on page 4-64). Zooming in on a db3 Wavelet Selecting a regularity and a wavelet for the regularity is useful in estimations of the local properties of functions or signals. This can be used, for example, to make sure that a signal has a constant regularity at all points. Work by Donoho, Johnstone, Kerkyacharian, and Picard on function estimation and nonlinear regression is currently under way to adapt the statistical estimators to unknown regularity. From a practical viewpoint, these questions arise in the world of finance in dealing with monetary and stock markets where detailed studies of very fast transactions are required. 4-64 Frequently Asked Questions Are Wavelets Useful in Fields Other Than Signal or Image Processing? · From a theoretical viewpoint, wavelets are used to characterize large sets of mathematical functions and are used in the study of operators linked to partial differential equations. · From a practical viewpoint, wavelets are used in several fields of numerical analysis, making certain complex calculations easier to handle or more precise. What Functions Are Candidates to Be a Wavelet? If a function f is continuous, has null moments, decreases quickly towards 0 when x tends towards infinity, or is null outside a segment of R, it is a likely candidate to become a wavelet. [. . . ] The plot method shows how to add Node Labels, Node Actions, and Tree Actions. You can have a look at the example in the ex1_edwt file located in the toolbox/wavelet/wavedemo folder. This program can be used directly, but it is also useful to learn how to build new object-oriented programming functions. B-27 B Object-Oriented Programming The definition of the new class is described below. Class EDWTTREE (parent class: DTREE) Fields dtree dwtMode wavInfo Parent object DWT extension mode Structure (wavelet information) Fields Description wavInfo wavName Lo_D Hi_D Lo_R Hi_R Wavelet Name Low Decomposition filter High Decomposition filter Low Reconstruction filter High Reconstruction filter Methods edwttree merge plot recons split Constructor for the class EDWTTREE. [. . . ]