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. [. . . ] 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. xd = wdencmp('gbl', x, 'sym4', 2, thr, sorh, keepapp); % Plots. colormap(pink(255)), sm = size(map, 1); 6-106 Wavelet Applications: More Detail subplot(221), image(wcodemat(X, sm)), title('Original Image') subplot(222), image(wcodemat(x, sm)), title('Noisy Image') subplot(223), image(wcodemat(xd, sm)), title('De-Noised Image') The result shown below is acceptable. Original Image 50 100 150 200 250 50 100 150 200 250 De-noised Image 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 Noisy Image Figure 6-31: Image De-Noising One-Dimensional Variance Adaptive Thresholding of Wavelet Coefficients Local thresholding of wavelet coefficients, for one- or two-dimensional data, is a capability available from a lot of graphical interface tools throughout Wavelet ToolboxTM software (see "Using Wavelets" on page 2-1). The idea is to define level by level time-dependent thresholds, and then increase the capability of the de-noising strategies to handle nonstationary variance noise models. More precisely, the model assumes (as previously) that the observation is equal to the interesting signal superimposed on a noise (see "De-Noising" on page 6-97). s ( n ) = f ( n ) + e ( n ) 6-107 6 Advanced Concepts But the noise variance can vary with time. There are several different variance values on several time intervals. [. . . ] You can have a look at a one-dimensional example in the ex1_wt file and at a two-dimensional example in the ex2_wt file located in the toolbox/wavelet/wavedemo folder. These examples can be used directly, but they are also useful to learn how to build new object-oriented programming functions. Class WTREE (parent class: DTREE) B-23 B Object-Oriented Programming Fields dtree dwtMode wavInfo Parent object DWT extension mode Structure (wavelet information) wavInfo Structure information wavName Lo_D Hi_D Lo_R Hi_R Wavelet Name Low Decomposition filter High Decomposition filter Low Reconstruction filter High Reconstruction filter Methods wtree merge split Constructor for the class WTREE. Split (decompose) the data of a terminal node. (RWVTREE) Example 2: Building a Right Wavelet Tree Object This example creates a new class of objects: RWVTREE. We define a right wavelet tree class starting from the class WTREE and overloading the methods split, merge, and plot (inherited from DTREE). [. . . ]