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This numerical tour explores the use of L1 optimization to find sparse representation in a redundant Gabor dictionary. It shows application to denoising and stereo separation.
addpath('toolbox_signal')
addpath('toolbox_general')
addpath('solutions/audio_3_gabor')
The Gabor transform is a collection of short time Fourier transforms (STFT) computed with several windows. The redundancy |K*L| of the transform depends on the number |L| of windows used and of the overlapping factor |K| of each STFT.
We decide to use a collection of windows with dyadic sizes.
Sizes of the windows.
wlist = 32*[4 8 16 32];
L = length(wlist);
Overlap of the window, so that |K=2|.
K = 2;
qlist = wlist/K;
Overall redundancy.
disp( strcat(['Approximate redundancy of the dictionary=' num2str(K*L) '.']) );
Approximate redundancy of the dictionary=8.
We load a sound.
n = 1024*32;
options.n = n;
[x0,fs] = load_sound('glockenspiel', n);
Compute its short time Fourier transform with a collection of windows.
options.multichannel = 0;
S = perform_stft(x0,wlist,qlist, options);
Exercise 1
Compute the true redundancy of the transform. Check that the transform is a tight frame (energy conservation).
exo1()
True redundancy of the dictionary=8.0586.
%% Insert your code here.
Display the coefficients.
plot_spectrogram(S, x0);
Reconstructs the signal using the inverse Gabor transform.
x1 = perform_stft(S,wlist,qlist, options);
Check for reconstruction error.
e = norm(x0-x1)/norm(x0);
disp(strcat(['Reconstruction error (should be 0) = ' num2str(e, 3)]));
Reconstruction error (should be 0) = 1.58e-16
We can perform denoising by thresholding the Gabor representation.
We add noise to the signal.
sigma = .05;
x = x0 + sigma*randn(size(x0));
Denoising with soft thresholding. Setting correctly the threshold is quite difficult because of the redundancy of the representation.
transform
S = perform_stft(x,wlist,qlist, options);
threshold
T = sigma;
ST = perform_thresholding(S, T, 'soft');
reconstruct
xT = perform_stft(ST,wlist,qlist, options);
Display the result.
err = snr(x0,xT);
clf
plot_spectrogram(ST, xT);
subplot(length(ST)+1,1,1);
title(strcat(['Denoised, SNR=' num2str(err,3), 'dB']));
Exercise 2
Find the best threshold, that gives the smallest error.
exo2()
%% Insert your code here.
Since the representation is highly redundant, it is possible to improve the quality of the representation using a basis pursuit denoising that optimize the L1 norm of the coefficients.
The basis pursuit finds a set of coefficients |S1| by minimizing
|min_{S1} 1/2norm(x-x1)^2 + lambdanorm(S1,1) (*)|
Where |x1| is the signal reconstructed from the Gabor coefficients |S1|.
The parameter |lambda| should be optimized to match the noise level. Increasing |lambda| increases the sparsity of the solution, but then the approximation |x1| deviates from the noisy observations |x1|.
Basis pursuit denoising |(*)| is solved by iterative thresholding, which iterates between a step of gradient descent, and a step of thresholding.
Initialization of |x1| and |S1|.
lambda = .1;
x1 = x;
S1 = perform_stft(x1,wlist,qlist, options);
Step 1: gradient descent of |norm(x-x1)^2|.
residual
r = x - x1;
Sr = perform_stft(r, wlist, qlist, options);
S1 = cell_add(S1, Sr);
Step 2: thresholding and update of |x1|.
threshold
S1 = perform_thresholding(S1, lambda, 'soft');
update the denoised signal
x1 = perform_stft(S1,wlist,qlist, options);
The difficulty is to set the value of |lambda|. If the basis were orthogonal, it should be set to approximately 3/2*sigma (soft thresholding). Because of the redundancy of the representation in Gabor frame, it should be set to a slightly larger value.
Exercise 3
Perform the iterative thresholding by progressively decaying the value of |lambda| during the iterations, starting from |lambda=1.5sigma| until |lambda=.5sigma|. Retain the solution |xbp| together with the coefficients |Sbp| that provides the smallest error.
exo3()
%% Insert your code here.
Display the solution computed by basis pursuit.
e = snr(x0,xbp);
clf
plot_spectrogram(Sbp, xbp);
subplot(length(Sbp)+1,1,1);
title(strcat(['Denoised, SNR=' num2str(e,3), 'dB']));
The increase of sparsity produced by L1 minimization is helpful to improve audio stereo separation.
Load 3 sounds.
n = 1024*32;
options.n = n;
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('male', n, options);
[x(:,3),fs] = load_sound('glockenspiel', n, options);
normalize the energy of the signals
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
theta = linspace(0,pi(),s+1); theta(s+1) = [];
theta(1) = .2;
M = [cos(theta); sin(theta)];
compute the mixed sources
y = x*M';
We transform the stero pair using the multi-channel STFT (each channel is transformed independantly.
options.multichannel = 1;
S = perform_stft(y, wlist, qlist, options);
check for reconstruction
y1 = perform_stft(S, wlist, qlist, options);
disp(strcat(['Reconstruction error (should be 0)=' num2str(norm(y-y1,'fro')/norm(y, 'fro')) '.' ]));
Reconstruction error (should be 0)=1.5966e-16.
Now we perform a multi-channel basis pursuit to find a sparse approximation of the coefficients.
regularization parameter
lambdaV = .2;
initialization
y1 = y;
S1 = S;
niter = 100;
err = [];
iterations
for i=1:niter
% gradient
r = y - y1;
Sr = perform_stft(r, wlist, qlist, options);
S1 = cell_add(S1, Sr);
% multi-channel thresholding
%%% BUG HERE %%%%
% S1 = perform_thresholding(S1, lambdaV, 'soft-multichannel');
% update the value of lambdaV to match noise
y1 = perform_stft(S1,wlist,qlist, options);
end
Create the point cloud of both the tight frame and the sparse BP coefficients.
P1 = []; P = [];
for i=1:length(S)
Si = reshape( S1{i}, [size(S1{i},1)*size(S1{i},2) 2] );
P1 = cat(1, P1, Si);
Si = reshape( S{i}, [size(S{i},1)*size(S{i},2) 2] );
P = cat(1, P, Si);
end
P = [real(P);imag(P)];
P1 = [real(P1);imag(P1)];
Display the two point clouds.
p = size(P,1);
m = 10000;
sel = randperm(p); sel = sel(1:m);
clf;
subplot(1,2,1);
plot( P(sel,1),P(sel,2), '.' );
title('Tight frame coefficients');
axis([-10 10 -10 10]);
subplot(1,2,2);
plot( P1(sel,1),P1(sel,2), '.' );
title('Basis Pursuit coefficients');
axis([-10 10 -10 10]);
Compute the angles of the points with largest energy.
d = sqrt(sum(P.^2,2));
d1 = sqrt(sum(P1.^2,2));
I = find( d>.2 );
I1 = find( d1>.2 );
compute angles
Theta = mod(atan2(P(I,2),P(I,1)), pi());
Theta1 = mod(atan2(P1(I1,2),P1(I1,1)), pi());
Compute and display the histogram of angles. We reaint only a small sub-set of most active coefficients.
compute histograms
nbins = 150;
[h,t] = hist(Theta, nbins);
h = h/sum(h);
[h1,t1] = hist(Theta1, nbins);
h1 = h1/sum(h1);
display histograms
clf;
subplot(2,1,1);
bar(t,h); axis('tight');
set_graphic_sizes([], 20);
title('Tight frame coefficients');
subplot(2,1,2);
bar(t1,h1); axis('tight');
set_graphic_sizes([], 20);
title('Sparse coefficients');
Exercise 4
Compare the source separation obtained by masking with a tight frame Gabor transform and with the coefficients computed by a basis pursuit sparsification process.
exo4()
%% Insert your code here.