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Merge branch 'master' of https://github.com/epfl-lts2/unlocbox
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Nathanael Perraudin committed Sep 15, 2015
2 parents b34f9d1 + cf26542 commit 097db6b
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149 changes: 95 additions & 54 deletions demos/demo_dequantization.m
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% This demo shows how a quantized signal, sparse in the DCT domain, can be dequantized
% solving a convex problem using Douglas-Rachford algorithm
%DEMO_DEQUANTIZATION
% This demo shows how a quantized signal, sparse in the DCT domain, can be dequantized
% solving a convex problem using Douglas-Rachford algorithm
%
% Suppose signal y has been quantized. In this demo we use quantization levels
% that are uniformly spread between the min. and max. value of the
% signal. The resulting signal is y_Q.
%
% The problem can be expressed as
%
% .. argmin_x || x ||_1 s.t. ||Dx - y_Q||_infty <= alpha/2
%
% .. math:: arg \min_x \| x \|_{1} \text{ s.t. } \|Dx - y_Q\|_\infty \leq \frac{\alpha}{2}
%
% where D is the synthesis dictionary (DCT in our case) and $\alpha$ is the
% distance between quantization levels. The constraint basically
% represents the fact that the reconstructed signal samples must stay
% within the corresponding quantization stripes.
%
% After sparse coordinates are found, the dequantized signal
% is obtained simply by synthesis with the dictionary.
%
% The program is solved using Douglas-Rachford algorithm. We set
%
% * $f_1(x)=||x||_{1}$. Its respective prox is the soft thresholding operator.
%
% * $f_2(x)=i_C$ is the indicator function of the set C, defined as
% .. C = { x | ||Dx - y_Q||_infty <= alpha/2 }
%
% .. math:: C = \{ x | \|Dx - y_Q\|_\infty <= \frac{\alpha}{2} \}
% Its prox is the orthogonal projection onto that set, which is realized
% by entry-wise 1D projections onto the quantization stripes. This is
% realized for all the entries at once by function proj_box.
%
% As an alternative, setting algorithm = 'LP' switches to computing the
% result via linear programming (requires Matlab optimization toolbox).
%
% References: combettes2007douglas

% Authors: Pavel Rajmic, Pavel Záviška
% Date: August 2015


% 2015 Pavel Rajmic, Pavel Záviška

clc
clear
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y = A(x);

%or just load a fixed dataset
clear
load('dequantization_dataset_01')
% clear
% load('dequantization_dataset_01')

% show coefficients
fig_coef = figure;
Expand All @@ -61,7 +100,7 @@
max_y = max(max(y));
range = max_y - min_y;

quant_step = range / (d-1); %quantization step
quant_step = range / (d-1); %quantization step
dec_bounds = (min_y+quant_step/2) : quant_step : (max_y-quant_step/2); %decision boundaries lie in the middle
quant_levels = min_y : quant_step : max_y; %quantization levels

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info

sol = f2.prox(sol,[]); %final projection into the constraints
pause

%%%%%%%% manual version %%%%%%%%%%%%%
%starting point
DR_y = A(y_quant);
DR_x_old = DR_y;

relat_change_coefs = 1;
relat_change_obj = 1;
cnt = 1; %iteration counter
obj_eval = [];

% while relat_change_coefs > 0.00001
while relat_change_obj > paramsolver.tol
% DR: gamma = 1
DR_x = f2.prox(DR_y,[]);
obj_eval = [obj_eval, f1.eval(DR_x) + f2.eval(DR_x)]; %record values of objective function
DR_y = DR_y + paramsolver.lambda*(f1.prox(2*DR_x-DR_y, paramsolver.gamma)-DR_x);
if cnt > 1
relat_change_coefs = norm(DR_x-DR_x_old) / norm(DR_x_old);
relat_change_obj = norm(obj_eval(end) - obj_eval(end-1)) / norm(obj_eval(end-1));
if paramsolver.verbose > 1
fprintf(' relative change in coefficients: %e \n', relat_change_coefs);
fprintf(' relative change in objective fun: %e \n', relat_change_obj);
fprintf('\n');
end
end
DR_x_old = DR_x;
cnt = cnt + 1;

end

DR_x = f2.prox(DR_y); %final projection into the constraints
y_dequant = A(DR_x); %dequantized signal

disp(['Finished after ' num2str(cnt) ' iterations.'])


%compare UNLOCBOX with manual
figure
plot([y_dequant A(sol)])
norm(y_dequant - A(sol))
title('UNLOCBOX vs. manual solution')
y_dequant = A(sol);

%behaviour of objective through iterations
figure
plot(obj_eval)
title('Objective function value (after projection into constraints)')
% pause
%
% %%%%%%%% manual version %%%%%%%%%%%%%
% %starting point
% DR_y = A(y_quant);
% DR_x_old = DR_y;
%
% relat_change_coefs = 1;
% relat_change_obj = 1;
% cnt = 1; %iteration counter
% obj_eval = [];
%
% % while relat_change_coefs > 0.00001
% while relat_change_obj > paramsolver.tol
% % DR: gamma = 1
% DR_x = f2.prox(DR_y,[]);
% obj_eval = [obj_eval, f1.eval(DR_x) + f2.eval(DR_x)]; %record values of objective function
% DR_y = DR_y + paramsolver.lambda*(f1.prox(2*DR_x-DR_y, paramsolver.gamma)-DR_x);
% if cnt > 1
% relat_change_coefs = norm(DR_x-DR_x_old) / norm(DR_x_old);
% relat_change_obj = norm(obj_eval(end) - obj_eval(end-1)) / norm(obj_eval(end-1));
% if paramsolver.verbose > 1
% fprintf(' relative change in coefficients: %e \n', relat_change_coefs);
% fprintf(' relative change in objective fun: %e \n', relat_change_obj);
% fprintf('\n');
% end
% end
% DR_x_old = DR_x;
% cnt = cnt + 1;
%
% end
%
% DR_x = f2.prox(DR_y); %final projection into the constraints
% y_dequant = A(DR_x); %dequantized signal
%
% disp(['Finished after ' num2str(cnt) ' iterations.'])
%
%
% %compare UNLOCBOX with manual
% figure
% plot([y_dequant A(sol)])
% norm(y_dequant - A(sol))
% title('UNLOCBOX vs. manual solution')
%
% %behaviour of objective through iterations
% figure
% plot(obj_eval)
% title('Objective function value (after projection into constraints in each iteration)')

end

Expand All @@ -242,7 +283,7 @@
axis tight
legend([h_quant_noise h_dequant_error], 'Quantizat. error', 'Error of reconstr.')

%coefficients of recontructed signal
%coefficients of reconstructed signal
figure(fig_coef)
hold on
h_coefs_dequant = bar(sol,'FaceColor',green);
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