小波去噪MATLAB_小波去噪csdn资源-CSDN文库

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小波去噪 MATLAB （242个子文件）

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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"> <html xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8">  <title>Image Denoising with Wavelets</title> <meta name="generator" content="MATLAB 7.4"> <meta name="date" content="2008-10-15"> <meta name="m-file" content="index"> <LINK REL="stylesheet" HREF="style.css" TYPE="text/css"> </head> <body> <div class="content"> <h1>Image Denoising with Wavelets</h1> <introduction> This numerical tour uses wavelets to perform both linear and non-linear image denoising. </introduction> <h2>Contents</h2> <div> <ul> <li><a href="#1">Installing toolboxes and setting up the path.</a></li> <li><a href="#8">Thresholding Estimator and Sparsity</a></li> <li><a href="#14">Image loading and adding Gaussian noise.</a></li> <li><a href="#17">Hard Thresholding vs. Soft Thresholding</a></li> <li><a href="#19">Orthogonal Wavelet Denoising</a></li> <li><a href="#26">Estimating the noise level</a></li> <li><a href="#29">Translation Invariant Wavelet Transform</a></li> <li><a href="#33">Translation Invariant Wavelet Denoising</a></li> <li><a href="#39">Wavelet Block Thresholding</a></li> </ul> </div> <h2>Installing toolboxes and setting up the path.<a name="1"></a></h2> You need to download the <a href="../toolbox_general.zip">general purpose toolbox</a> and the <a href="../toolbox_signal.zip">signal toolbox</a>. You need to unzip these toolboxes in your working directory, so that you have <tt>toolbox_general/</tt> and <tt>toolbox_signal/</tt> in your directory. For Scilab user: you must replace the Matlab comment '%' by its Scilab counterpart '//'. Recommandation: You should create a text file named for instance <tt>numericaltour.sce</tt> (in Scilabe) or <tt>numericaltour.m</tt> to write all the Scilab/Matlab command you want to execute. Then, simply run <tt>exec('numericaltour.sce');</tt> (in Scilab) or <tt>numericaltour;</tt> (in Matlab) to run the commands. Execute this line only if you are using Matlab.<pre class="codeinput">getd = @(p)path(path,p); % scilab users must *not* execute this </pre>Then you can add these toolboxes to the path.<pre class="codeinput">% Add some directories to the path getd('toolbox_signal/'); getd('toolbox_general/'); </pre><h2>Thresholding Estimator and Sparsity<a name="8"></a></h2> The idea of non-linear denoising is to use an orthogonal basis in which the coefficients <tt>x</tt> of the signal or image <tt>M0</tt> is sparse (a few large coefficients). In this case, the noisy coefficients <tt>x</tt> of the noisy data <tt>M</tt> (perturbated with Gaussian noise) are <tt>x0+noise</tt> where <tt>noise</tt> is Gaussian. A thresholding set to 0 the noise coefficients that are below <tt>T</tt>. The threshold level <tt>T</tt> should be chosen judiciously to be just above the noise level. First we generate a spiky signal.<pre class="codeinput">% dimension n = 4096; % probability of spiking rho = .05; % location of the spike x0 = rand(n,1)<rho; % random amplitude in [-1 1] x0 = 2 * x0 .* ( rand(n,1)-.5 ); </pre>We add some gaussian noise<pre class="codeinput">sigma = .1; x = x0 + randn(size(x0))*sigma; % display clf; subplot(2,1,1); plot(x0); axis([1 n -1 1]); set_graphic_sizes([], 20); title('Original signal'); subplot(2,1,2); plot(x); axis([1 n -1 1]); set_graphic_sizes([], 20); title('Noisy signal'); </pre><img vspace="5" hspace="5" src="index_01.png"> Exercice 1: (the solution is <a href="../private/denoising_wavelet/exo1.m">exo1.m</a>) What is the optimal threshold <tt>T</tt> to remove as much as possible of noise ? Try several values of <tt>T</tt>. <pre class="codeinput">exo1; </pre><img vspace="5" hspace="5" src="index_02.png"> In order to be optimal without knowing in advance the amplitude of the coefficients of <tt>x0</tt>, one needs to set <tt>T</tt> just above the noise level. This means that <tt>T</tt> should be roughly equal to the maximum value of a Gaussian white noise of size <tt>n</tt>. Exercice 2: (the solution is <a href="../private/denoising_wavelet/exo2.m">exo2.m</a>) The theory predicts that the maximum of <tt>n</tt> Gaussian variable of variance <tt>sigma^2</tt> is smaller than <tt>sqrt(2*log(n))</tt> with large probability (that tends to 1 when <tt>n</tt> increases). This is also a sharp result. Check this numerically by computing with Monte Carlo sampling the maximum with <tt>n</tt> increasing (in power of 2). Check also the deviation of the maximum when you perform several trial with <tt>n</tt> fixed. <pre class="codeinput">exo2; </pre><img vspace="5" hspace="5" src="index_03.png"> <h2>Image loading and adding Gaussian noise.<a name="14"></a></h2> A simple noise model is additive Gaussian noise. First we load an image.<pre class="codeinput">name = 'boat'; n = 256; M0 = load_image(name,n); M0 = rescale( M0, .05, .95 ); </pre>Then we add some gaussian noise to it.<pre class="codeinput">sigma = .08; % noise level M = M0 + sigma*randn(size(M0)); clf; imageplot(M0, 'Original', 1,2,1); imageplot(clamp(M), 'Noisy', 1,2,2); </pre><img vspace="5" hspace="5" src="index_04.png"> <h2>Hard Thresholding vs. Soft Thresholding<a name="17"></a></h2> A thresholding is a 1D non-linear function applied to each wavelet coefficients. The most important thresholding are the hard thresholding (related to L0 minimization) and the soft thresholding (related to L1 minimization). <pre class="codeinput">% threshold value T = 1; v = -linspace(-3,3,2000); % hard thresholding of the t values v_hard = v.*(abs(v)>T); % soft thresholding of the t values v_soft = max(1-T./abs(v), 0).*v; % display clf; hold('on'); plot(v, v_hard); plot(v, v_soft, 'r--'); axis('equal'); axis('tight'); legend('Hard thresholding', 'Soft thresholding'); hold('off'); </pre><img vspace="5" hspace="5" src="index_05.png"> <h2>Orthogonal Wavelet Denoising<a name="19"></a></h2> It is possible to perform non linear denoising by thresholding the wavelet coefficients. This allows to better respect the sharp features of the image. First we compute the wavelet coefficients of the noisy image.<pre class="codeinput">options.ti = 0; Jmin = 4; MW = perform_wavelet_transf(M,Jmin,+1,options); </pre>Then we hard threshold the coefficients below the noise level. In practice a threshold of <tt>3*sigma</tt> is close to optimal for natural images. <pre class="codeinput">T = 3*sigma; MWT = perform_thres

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