Source Separation with Sparsity¶

Important: Please read the installation page for details about how to install the toolboxes. $\newcommand{\dotp}[2]{\langle #1, #2 \rangle}$ $\newcommand{\enscond}[2]{\lbrace #1, #2 \rbrace}$ $\newcommand{\pd}[2]{ \frac{ \partial #1}{\partial #2} }$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\umax}[1]{\underset{#1}{\max}\;}$ $\newcommand{\umin}[1]{\underset{#1}{\min}\;}$ $\newcommand{\uargmin}[1]{\underset{#1}{argmin}\;}$ $\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\abs}[1]{\left|#1\right|}$ $\newcommand{\choice}[1]{ \left\{ \begin{array}{l} #1 \end{array} \right. }$ $\newcommand{\pa}[1]{\left(#1\right)}$ $\newcommand{\diag}[1]{{diag}\left( #1 \right)}$ $\newcommand{\qandq}{\quad\text{and}\quad}$ $\newcommand{\qwhereq}{\quad\text{where}\quad}$ $\newcommand{\qifq}{ \quad \text{if} \quad }$ $\newcommand{\qarrq}{ \quad \Longrightarrow \quad }$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\EE}{\mathbb{E}}$ $\newcommand{\Zz}{\mathcal{Z}}$ $\newcommand{\Ww}{\mathcal{W}}$ $\newcommand{\Vv}{\mathcal{V}}$ $\newcommand{\Nn}{\mathcal{N}}$ $\newcommand{\NN}{\mathcal{N}}$ $\newcommand{\Hh}{\mathcal{H}}$ $\newcommand{\Bb}{\mathcal{B}}$ $\newcommand{\Ee}{\mathcal{E}}$ $\newcommand{\Cc}{\mathcal{C}}$ $\newcommand{\Gg}{\mathcal{G}}$ $\newcommand{\Ss}{\mathcal{S}}$ $\newcommand{\Pp}{\mathcal{P}}$ $\newcommand{\Ff}{\mathcal{F}}$ $\newcommand{\Xx}{\mathcal{X}}$ $\newcommand{\Mm}{\mathcal{M}}$ $\newcommand{\Ii}{\mathcal{I}}$ $\newcommand{\Dd}{\mathcal{D}}$ $\newcommand{\Ll}{\mathcal{L}}$ $\newcommand{\Tt}{\mathcal{T}}$ $\newcommand{\si}{\sigma}$ $\newcommand{\al}{\alpha}$ $\newcommand{\la}{\lambda}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\La}{\Lambda}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Si}{\Sigma}$ $\newcommand{\be}{\beta}$ $\newcommand{\de}{\delta}$ $\newcommand{\De}{\Delta}$ $\newcommand{\phi}{\varphi}$ $\newcommand{\th}{\theta}$ $\newcommand{\om}{\omega}$ $\newcommand{\Om}{\Omega}$

This numerical tour explore local Fourier analysis of sounds, and its application to source separation from stereo measurements.

In [2]:

addpath('toolbox_signal')
addpath('toolbox_general')
addpath('solutions/audio_2_separation')

Sound Mixing¶

We load 3 sounds and simulate a stero recording by performing a linear blending of the sounds.

For Scilab users, it is safer to extend the stack size. For Matlab users this does nothing.

In [3]:

extend_stack_size(4);

Sound loading.

In [4]:

n = 1024*16;
s = 3; % number of sound
p = 2; % number of micros
options.subsampling = 1;
x = zeros(n,3);
[x(:,1),fs] = load_sound('bird', n, options);
[x(:,2),fs] = load_sound('female', n, options);
[x(:,3),fs] = load_sound('male', n, options);

normalize the energy of the signals

In [5]:

x = x./repmat(std(x,1), [n 1]);

We mix the sound using a |2x3| transformation matrix. Here the direction are well-spaced, but you can try with more complicated mixing matrices.

compute the mixing matrix

In [6]:

theta = linspace(0,pi(),s+1); theta(s+1) = [];
theta(1) = .2;
M = [cos(theta); sin(theta)];

compute the mixed sources

In [7]:

y = x*M';

Display of the sounds and their mix.

In [8]:

clf;
for i=1:s
    subplot(s,1,i);
    plot(x(:,i)); axis('tight');
    set_graphic_sizes([], 20);
    title(strcat('Source #',num2str(i)));
end

Display of the micro output.

In [9]:

clf;
for i=1:p
    subplot(p,1,i);
    plot(y(:,i)); axis('tight');
    set_graphic_sizes([], 20);
    title(strcat('Micro #',num2str(i)));
end

Local Fourier analysis of sound.¶

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.

First set up parameters for the STFT.

In [10]:

options.n = n;
w = 128;   % size of the window
q = w/4;    % overlap of the window

Compute the STFT of the sources.

In [11]:

clf; X = []; Y = [];
for 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('Source #',num2str(i)));
end

Exercise 1

Compute the STFT of the micros, and store them into a matrix |Y|.

In [12]:

exo1()

In [13]:

%% Insert your code here.

Estimation of Mixing Direction by Clustering¶

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.

First we compute the position of the point cloud.

In [14]:

mf = size(Y,1);
mt = size(Y,2);
P = reshape(Y, [mt*mf p]);
P = [real(P);imag(P)];

Then we keep only the 5% of points with largest energy.

Display some points in the original (spacial) domain.

number of displayed points

In [15]:

npts = 6000;

display original points

In [16]:

sel = randperm(n); sel = sel(1:npts);
clf;
plot(y(sel,1), y(sel,2), '.');
axis([-1 1 -1 1]*5);
set_graphic_sizes([], 20);
title('Time domain');

Exercise 2

Display some points of |P| in the transformed (time/frequency) domain.

In [17]:

exo2()

In [18]:

%% Insert your code here.

We compute the angle associated to each point over the transformed domain. The histograms shows the main direction of mixing.

In [19]:

Theta = mod(atan2(P(:,2),P(:,1)), pi());

display histograms

In [20]:

nbins = 100;
[h,t] = hist(Theta, nbins);
h = h/sum(h);
clf;
bar(t,h); axis('tight');

Exercise 3

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. ompute the energy of each point xtract only a small sub-set ompute histogram isplay histograms

In [21]:

exo3()

In [22]:

%% Insert your code here.

Exercise 4

Detect the direction |M1| approximating the true direction |M| by looking at the local maxima of the histogram. First detect the set of local maxima, and then keep only the three largest. ort in descending order

In [23]:

exo4()

M =

    0.9801    0.5000   -0.5000
    0.1987    0.8660    0.8660


M1 =

    0.9803    0.5010   -0.5028
    0.1973    0.8655    0.8644

In [24]:

%% Insert your code here.

Separation of the Sources using Clustering¶

Once the mixing direction are known, one can project the sources on the direction.

We compute the projection of the coefficients Y on each estimated direction.

In [25]:

A = reshape(Y, [mt*mf p]);

compute the projection of the coefficients on the directions

In [26]:

C = abs( M1'*A' );

At each point |x|, the index |I(x)| is the direction which creates the largest projection.

I is the index of the closest source

In [27]:

[tmp,I] = compute_max(C, 1);
I = reshape(I, [mf mt]);

An additional denoising is achieved by removing small coefficients.

do not take into account too small coefficients

In [28]:

T = .05;
D = sqrt(sum( abs(Y).^2, 3));
I = I .* (D>T);

We can display the segmentation of the time frequency plane.

In [29]:

clf;
imageplot(I(1:mf/2,:));
axis('normal');
set_colormap('jet');

The recovered coefficients are obtained by projection.

In [30]:

Proj = M1'*A';
Xr = [];
for 