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 [1]:
from __future__ import division

import numpy as np
import scipy as scp
import pylab as pyl
import matplotlib.pyplot as plt

from nt_toolbox.general import *
from nt_toolbox.signal import *

import warnings
warnings.filterwarnings('ignore')

%matplotlib inline
%load_ext autoreload
%autoreload 2

Sound Mixing

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

Sound loading.

In [2]:
from nt_toolbox.load_sound import *

n = 1024*16
s = 3 #number of sounds
p = 2 #number of micros

x = np.zeros([n,3])
x[:,0] = load_sound("nt_toolbox/data/bird.wav",n)
x[:,1] = load_sound("nt_toolbox/data/female.wav",n)
x[:,2] = load_sound("nt_toolbox/data/male.wav",n)

Normalize the energy of the signals.

In [3]:
x = x/np.tile(np.std(x,0),(n,1))

We mix the sound using a $2\mathrm{x}3$ transformation matrix. Here the direction are well-spaced, but you can try with more complicated mixing matrices.

Compute the mixing matrix

In [4]:
theta = np.linspace(0, np.pi, s + 1)[:-1]
theta[0] = .2
M = np.vstack((np.cos(theta), np.sin(theta)))

Compute the mixed sources.

In [5]:
y = np.dot(x,np.transpose(M))

Display of the sounds and their mix.

In [6]:
plt.figure(figsize = (10,10))

for i in range(s):
    plt.subplot(s, 1, i+1)
    plt.plot(x[:, i])
    plt.xlim(0,n)
    plt.title("Source #%i" %(i+1))

Display of the micro output.

In [7]:
plt.figure(figsize = (10,7))
    
for i in range(p):
    plt.subplot(p, 1, i+1)
    plt.plot(y[:, i])
    plt.xlim(0,n)
    plt.title("Micro #%i" %(i+1))

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 [8]:
w = 128   #size of the window
q = w//4  #overlap of the window

Compute the STFT of the sources.

In [9]:
from nt_toolbox.perform_stft import *
from nt_toolbox.plot_spectrogram import *

X = np.zeros([w,4*w+1,s])
Y = np.zeros([w,4*w+1,p])

for i in range(s):
    X[:,:,i] = perform_stft(x[:,i],w,q,n)
    plt.figure(figsize = (15,10))
    plot_spectrogram(X[:,:,i],"Source #%i" %(i+1))

Exercise 1

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

In [10]:
run -i nt_solutions/audio_2_separation/exo1
In [11]:
## 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 [12]:
mf = np.shape(Y)[0]
mt = np.shape(Y)[1]
P = np.reshape(Y, (mt*mf,p))
P = np.vstack((np.real(P), np.imag(P)))

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

Display some points in the original (spacial) domain.

Number of displayed points.

In [13]:
npts = 6000

Display the original points.

In [14]:
from numpy import random

sel = random.permutation(n)

sel = sel[:npts]

plt.figure(figsize = (7,5))
plt.plot(y[sel,0], y[sel,1], ".", ms = 3)
plt.xlim(-5,5)
plt.ylim(-5,5)
plt.title('Time domain')
plt.show()

Exercise 2

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

In [15]:
run -i nt_solutions/audio_2_separation/exo2
In [16]:
## Insert your code here.

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

In [17]:
import math 

nrow = np.shape(P)[0]
Theta = np.zeros(nrow)
for i in range(nrow):
    Theta[i] = math.atan2(P[i,1],P[i,0])%np.pi

Display histogram.

In [18]:
nbins = 100
t = np.linspace(np.pi/200,np.pi,nbins)
hist = np.histogram(Theta,t)
h = hist[0]/np.sum(hist[0])
t = t[:-1]

plt.figure(figsize = (7,5))
plt.bar(t, h, width = np.pi/nbins, color = "darkblue", edgecolor = "darkblue")
plt.xlim(0,np.pi)
plt.ylim(0,np.max(h))
plt.show()

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. Compute the energy of each point. Extract only a small sub-set.

In [19]:
run -i nt_solutions/audio_2_separation/exo3
In [20]:
## Insert your code here.

Exercise 4

Detect the direction $M_1$ 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. Sort in descending order.

In [21]:
run -i nt_solutions/audio_2_separation/exo4
--- M ---
[[ 0.98006658  0.5        -0.5       ]
 [ 0.19866933  0.8660254   0.8660254 ]]
--- M1 ---
[[ 0.98190316  0.50797406 -0.49567393]
 [ 0.18938372  0.86137237  0.8685087 ]]
In [22]:
## 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 [23]:
A = np.reshape(Y, (mt*mf,p))

Compute the projection of the coefficients on the directions.

In [24]:
C = abs(np.dot(np.transpose(M1),np.transpose(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 [25]:
tmp = np.max(C,0)
I = np.argmax(C,0)
I = np.reshape(I, (mf,mt)) + 1

An additional denoising is achieved by removing small coefficients.

In [26]:
T = .05
D = np.sqrt(np.sum(abs(Y)**2, 2))
I = I*(D > T)

We can display the segmentation of the time frequency plane.

In [27]:
plt.figure(figsize = (15,10))
imageplot(I[:mf//2,:])
plt.imshow(I[:mf//2,:], cmap = plt.get_cmap("jet"), interpolation = "nearest")
plt.show()

The recovered coefficients are obtained by projection.

In [28]:
Proj = np.dot(np.transpose(M1),np.transpose(A))
Xr = np.zeros([w,4*w+1,s])
for i in range(s):
    Xr[:,:,i] = np.reshape(Proj[i,:], (mf,mt))*(I == i)

The estimated signals are obtained by inverting the STFT.

In [29]:
xr = np.zeros([n,s])
for i in range(s):
    xr[:,i] = perform_stft(Xr[:,:,i], w, q, n)

One can display the recovered signals.

In [30]:
plt.figure(figsize = (10,10))

for i in range(s):
    plt.subplot(s,1,i+1)
    plt.plot(xr[:,i])
    plt.xlim(0,n)
    plt.title("Estimated source #%i" %(i+1))

One can listen to the recovered sources.

In [31]:
i = 1
from IPython.display import Audio
Audio(x[:,i], rate = 15000)
Out[31]: