User manual THE MATHWORKS WAVELET TOOLBOX 4

[. . . ] Wavelet ToolboxTM 4 User's Guide Michel Misiti Yves Misiti Georges Oppenheim Jean-Michel Poggi How to Contact The 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. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc. [. . . ] opt = 'lvd' and thr is a vector for level dependent threshold. keepapp = 1 to keep approximation coefficients, as previously and keepapp = 0 to allow approximation coefficients thresholding. x is the signal to be de-noised and wav, n, sorh are the same as above. De-Noising in Action We begin with examples of one-dimensional de-noising methods with the first example credited to Donoho and Johnstone. You can use the following file to get the first test function using wnoise. % Set signal to noise ratio and set rand seed. sqrt_snr = 4; init = 2055615866; % Generate original signal xref and a noisy version x adding % a standard Gaussian white noise. [xref, x] = wnoise(1, 11, sqrt_snr, init); % De-noise noisy signal using soft heuristic SURE thresholding % and scaled noise option, on detail coefficients obtained % from the decomposition of x, at level 3 by sym8 wavelet. xd = wden(x, 'heursure', 's', 'one', 3, 'sym8'); 6-103 6 Advanced Concepts Original signal 20 10 0 -10 -20 0 20 10 0 -10 -20 0 20 10 0 -10 -20 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 De-noised signal - Signal to noise ratio = 4 0. 8 0. 9 1 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 Noisy signal 0. 7 0. 8 0. 9 1 Figure 6-29: Blocks Signal De-Noising Since only a small number of large coefficients characterize the original signal, the method performs very well (see Figure 6-29). If you want to see more about how the thresholding works, use the GUI (see "De-Noising Signals" on page 3-18). As a second example, let us try the method on the highly perturbed part of the electrical signal studied above. According to this previous analysis, let us use db3 wavelet and decompose at level 3. To deal with the composite noise nature, let us try a level-dependent noise size estimation. 6-104 Wavelet Applications: More Detail % Load electrical signal and select part of it. load leleccum; indx = 2000:3450; x = leleccum(indx); % Find first value in order to avoid edge effects. deb = x(1); % De-noise signal using soft fixed form thresholding % and unknown noise option. xd = wden(x-deb, 'sqtwolog', 's', 'mln', 3, 'db3')+deb; Original electrical Signal 600 500 400 300 200 100 2000 2500 De-noised Signal 600 500 400 300 200 100 2000 2500 3000 3500 3000 3500 Figure 6-30: Electrical Signal De-Noising The result is quite good in spite of the time heterogeneity of the nature of the noise after and before the beginning of the sensor failure around time 2450. 6-105 6 Advanced Concepts Extension to Image De-Noising The de-noising method described for the one-dimensional case applies also to images and applies well to geometrical images. A direct translation of the one-dimensional model is s ( i, j ) = f ( i, j ) + e ( i, j ) where e is a white Gaussian noise with unit variance. The two-dimensional de-noising procedure has the same three steps and uses two-dimensional wavelet tools instead of one-dimensional ones. For the threshold selection, prod(size(s)) is used instead of length(s) if the fixed form threshold is used. Note that except for the "automatic" one-dimensional de-noising case, de-noising and compression are performed using wdencmp. As an example, you can use the following file illustrating the de-noising of a real image. % Load original image. init = 2055615866; randn('seed', init); x = X + 15*randn(size(X)); % Find default values. In this case fixed form threshold % is used with estimation of level noise, thresholding % mode is soft and the approximation coefficients are % kept. [thr, sorh, keepapp] = ddencmp('den', 'wv', x); % thr is equal to estimated_sigma*sqrt(log(prod(size(X)))) thr thr = 107. 6428 % De-noise image using global thresholding option. [. . . ] You can have a look at a one-dimensional example in the ex1_rwvt file and at a two-dimensional example in the ex2_rwvt file located in the toolbox/wavelet/wavedemo folder. These programs can be used directly, but they are also useful to learn how to build new object-oriented programming functions. B-24 Advanced Use of Objects The definition of the new class is described below. Class RWVTREE (parent class: WTREE) Fields dummy wtree Not used Parent object Methods rwvtree merge plot split Constructor for the class RWVTREE. Split (decompose) the data of a terminal node. Running This Example The following figure is obtained using the example ex1_rwvt and clicking the node 14. [. . . ]