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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"> <!--This HTML is auto-generated from an M-file.To make changes, update the M-file and republish this document. --> <title>Source Separation with Sparsity</title> <meta name="generator" content="MATLAB 7.4"> <meta name="date" content="2008-10-19"> <meta name="m-file" content="index"> <LINK REL="stylesheet" HREF="style.css" TYPE="text/css"> </head> <body> <div class="content"> <h1>Source Separation with Sparsity</h1> <introduction> <p>This numerical tour explore local Fourier analysis of sounds, and its application to source separation from stereo measurements.</p> </introduction> <h2>Contents</h2> <div> <ul> <li><a href="#1">Installing toolboxes and setting up the path.</a></li> <li><a href="#8">Sound Mixing</a></li> <li><a href="#14">Local Fourier analysis of sound.</a></li> <li><a href="#18">Estimation of Mixing Direction by Clustering</a></li> <li><a href="#26">Separation of the Sources using Clustering</a></li> </ul> </div> <h2>Installing toolboxes and setting up the path.<a name="1"></a></h2> <p>You need to download the <a href="../toolbox_general.zip">general purpose toolbox</a> and the <a href="../toolbox_signal.zip">signal toolbox</a>. </p> <p>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. </p> <p><b>For Scilab user:</b> you must replace the Matlab comment '%' by its Scilab counterpart '//'. </p> <p><b>Recommandation:</b> 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. </p> <p>Execute this line only if you are using Matlab.</p><pre class="codeinput">getd = @(p)path(path,p); <span class="comment">% scilab users must *not* execute this</span></pre><p>Then you can add these toolboxes to the path.</p><pre class="codeinput"><span class="comment">% Add some directories to the path</span>getd(<span class="string">'toolbox_signal/'</span>);getd(<span class="string">'toolbox_general/'</span>);</pre><h2>Sound Mixing<a name="8"></a></h2> <p>We load 3 sounds and simulate a stero recording by performing a linear blending of the sounds.</p> <p>For Scilab users, it is safer to extend the stack size. For Matlab users this does nothing.</p><pre class="codeinput">extend_stack_size(4);</pre><p>Sound loading.</p><pre class="codeinput">n = 1024*16;s = 3; <span class="comment">% number of sound</span>p = 2; <span class="comment">% number of micros</span>options.subsampling = 1;x = zeros(n,3);[x(:,1),fs] = load_sound(<span class="string">'bird'</span>, n, options);[x(:,2),fs] = load_sound(<span class="string">'female'</span>, n, options);[x(:,3),fs] = load_sound(<span class="string">'male'</span>, n, options);<span class="comment">% normalize the energy of the signals</span>x = x./repmat(std(x,1), [n 1]);</pre><p>We mix the sound using a <tt>2x3</tt> transformation matrix. Here the direction are well-spaced, but you can try with more complicated mixing matrices. </p><pre class="codeinput"><span class="comment">% compute the mixing matrix</span>theta = linspace(0,pi(),s+1); theta(s+1) = [];theta(1) = .2;M = [cos(theta); sin(theta)];<span class="comment">% compute the mixed sources</span>y = x*M';</pre><p>Display of the sounds and their mix.</p><pre class="codeinput">clf;<span class="keyword">for</span> i=1:s subplot(s,1,i); plot(x(:,i)); axis(<span class="string">'tight'</span>); set_graphic_sizes([], 20); title(strcat(<span class="string">'Source #'</span>,num2str(i)));<span class="keyword">end</span></pre><img vspace="5" hspace="5" src="index_01.png"> <p>Display of the micro output.</p><pre class="codeinput">clf;<span class="keyword">for</span> i=1:p subplot(p,1,i); plot(y(:,i)); axis(<span class="string">'tight'</span>); set_graphic_sizes([], 20); title(strcat(<span class="string">'Micro #'</span>,num2str(i)));<span class="keyword">end</span></pre><img vspace="5" hspace="5" src="index_02.png"> <h2>Local Fourier analysis of sound.<a name="14"></a></h2> <p>In order to perform the separation, one performs a local Fourier analysis of the sound. The hope is that the sources will be well-separated over the Fourier domain because the sources are sparse after a STFT. </p> <p>First set up parameters for the STFT.</p><pre class="codeinput">options.n = n;w = 128; <span class="comment">% size of the window</span>q = w/4; <span class="comment">% overlap of the window</span></pre><p>Compute the STFT of the sources.</p><pre class="codeinput">clf; X = []; Y = [];<span class="keyword">for</span> i=1:s X(:,:,i) = perform_stft(x(:,i),w,q, options); subplot(s,1,i); plot_spectrogram(X(:,:,i)); set_graphic_sizes([], 20); title(strcat(<span class="string">'Source #'</span>,num2str(i)));<span class="keyword">end</span></pre><img vspace="5" hspace="5" src="index_03.png"> <p><i>Exercice 1:</i> (the solution is <a href="../private/audio_separation/exo1.m">exo1.m</a>) Compute the STFT of the micros, and store them into a matrix <tt>Y</tt>. </p><pre class="codeinput">exo1;</pre><img vspace="5" hspace="5" src="index_04.png"> <h2>Estimation of Mixing Direction by Clustering<a name="18"></a></h2> <p>Since the sources are quite sparse over the Fourier plane, the directions are well estimated by looking as the direction emerging from a point clouds of the transformed coefficients. </p> <p>First we compute the position of the point cloud.</p><pre class="codeinput">mf = size(Y,1);mt = size(Y,2);P = reshape(Y, [mt*mf p]);P = [real(P);imag(P)];</pre><p>Then we keep only the 5% of points with largest energy.</p> <p>Display some points in the original (spacial) domain.</p><pre class="codeinput"><span class="comment">% number of displayed points</span>npts = 6000;<span class="comment">% display original points</span>sel = randperm(n); sel = sel(1:npts);clf;plot(y(sel,1), y(sel,2), <span class="string">'.'</span>);axis([-1 1 -1 1]*5);set_graphic_sizes([], 20);title(<span class="string">'Time domain'</span>);</pre><img vspace="5" hspace="5" src="index_05.png"> <p><i>Exercice 2:</i> (the solution is <a href="../private/audio_separation/exo2.m">exo2.m</a>) Display some points of <tt>P</tt> in the transformed (time/frequency) domain. </p><pre class="codeinput">exo2;</pre><img vspace="5" hspace="5" src="index_06.png"> <p>We compute the angle associated to each point over the transformed domain. The histograms shows the main direction of mixing.</p><pre class="codeinput">Theta = mod(atan2(P(:,2),P(:,1)), pi());<span class="comment">% display histograms</span>nbins = 100;[h,t] = hist(Theta, nbins);h = h/sum(h);clf;bar(t,h); axis(<span class="string">'tight'</span>);</pre><img vspace="5" hspace="5" src="index_07.png"> <p><i>Exercice 3:</i> (the solution is <a href="../private/audio_separation/exo3.m">exo3.m</a>) The histogram computed from the whole set of points are not peacked enough. To stabilize the detection of mixing direction, compute an histogram from a reduced set of point that have the largest amplitude. </p><pre class="codeinput">exo3;</pre><img vspace="5" hspace="5" src="index_08.png"> <p><i>Exercice 4:</i> (the solution is <a href="../private/audio_separation/exo4.m">exo4.m</a>) Detect the direction <tt>M1</tt> approximating the true direction <tt>M</tt> by looking at the local maxima of the histogram. First detect the set of local maxima, and then keep only the three largest. </p><pre class="codeinput">exo4;</pre><pre class="codeoutput">M = 0.9801 0.5000 -0.5000 0.1987 0.8660 0.8660M1 = 0.9803 0.5010 -0.5028 0.1973 0.8655 0.8644</pre><h2>Separation of the Sources using Clustering<a name="26"></a></h2> <p>Once the mixing direction are known, one can project the sources on the direction.</p> <p>We compute the projection of the coefficients Y on each estimated direction.</p><pre class="codeinput">A = reshape(Y, [mt*mf p]);<span class="comment">% compute the projection of the coefficients on the directions</span>C = abs( M1'*A' );</pre><p>At each point <tt>x</tt>, the index <tt>I(x)</tt> is the direction which creates the largest projection. </p><pre class="codeinput"><span class="comment">% I is the index of the closest source</span>[tmp,I] = compute_max(C, 1);I = reshape(I, [mf mt]);</pre><p>An additional denoising is achieved by removing small coefficients.</p><pre class="codeinput"><span class="comment">% do not take into account too small coefficients</span>T = .05;D = sqrt(sum( abs(Y).^2, 3));I = I .* (D>T);</pre><p>We can display the segmentation of the time frequency plane.</p><pre class="codeinput">clf;imageplot(I(1:mf/2,:));axis(<span class="string">'normal'</span>);set_colormap(<span class="string">'jet'</span>);</pre><img vspace="5" hspace="5" src="index_09.png"> <p>The recovered coefficients are obtained by projection.</p><pre class="codeinput">Proj = M1'*A';Xr = [];<span class="keyword">for</span> i=1:s Xr(:,:,i) = reshape(Proj(i,:), [mf mt]) .* (I==i);<span class="keyword">end</span></pre><p>The estimated signals are obtained by inverting the STFT.</p><pre class="codeinput"><span class="keyword">for</span> i=1:s xr(:,i) = perform_stft(Xr(:,:,i),w,q, options);<span class="keyword">end</span></pre><p>One can display the recovered signals.</p><pre class="codeinput">clf;<span class="keyword">for</span> i=1:s subplot(s,1,i); plot(xr(:,i)); axis(<span class="string">'tight'</span>); set_graphic_sizes([], 20); title(strcat(<span class="string">'Estimated source #'</span>,num2str(i)));<span class="keyword">end</span></pre><img vspace="5" hspace="5" src="index_10.png"> <p>One can listen to the recovered sources.</p><pre class="codeinput">i = 1;sound(x(:,i),fs);sound(xr(:,i),fs);</pre><p class="footer"><br> Copyright ® 2008 Gabriel Peyre<br></p> </div> <!--##### SOURCE BEGIN #####%% Source Separation with Sparsity% This numerical tour explore local Fourier analysis of sounds, and its% application to source separation from stereo measurements.%% Installing toolboxes and setting up the path.%%% You need to download the % <../toolbox_general.zip general purpose toolbox>% and the <../toolbox_signal.zip signal toolbox>.%%% You need to unzip these toolboxes in your working directory, so% that you have |toolbox_general/| and |toolbox_signal/| 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% |numericaltour.sce| (in Scilabe) or |numericaltour.m| to write all the% Scilab/Matlab command you want to execute. Then, simply run% |exec('numericaltour.sce');| (in Scilab) or |numericaltour;| (in Matlab)% to run the commands. %%% Execute this line only if you are using Matlab.getd = @(p)path(path,p); % scilab users must *not* execute this%%% Then you can add these toolboxes to the path.% Add some directories to the pathgetd('toolbox_signal/');getd('toolbox_general/');%% Sound Mixing% We load 3 sounds and simulate a stero recording by performing a linear% blending of the sounds.%%?? 快捷鍵說(shuō)明
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