init
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commit
ca6da60652
SSH key fingerprint:
SHA256:kZEpW8cMJ54PDeCvOhzreNr4FSh6R13CMGH/POoO8DI
BIN
tp1/creation_cercle_et_donnees_bruitees.p
Executable file
BIN
tp1/creation_cercle_et_donnees_bruitees.p
Executable file
Binary file not shown.
38
tp1/donnees.m
Executable file
38
tp1/donnees.m
Executable file
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@ -0,0 +1,38 @@
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clear;
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close all;
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taille_ecran = get(0,'ScreenSize');
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L = taille_ecran(3);
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H = taille_ecran(4);
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% Fenetre d'affichage :
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figure('Name','Points situes au voisinage d''un cercle', ...
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'Position',[0.4*L,0.05*H,0.6*L,0.7*H]);
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axis equal;
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hold on;
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set(gca,'FontSize',20);
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hx = xlabel('$x$','FontSize',30);
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set(hx,'Interpreter','Latex');
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hy = ylabel('$y$','FontSize',30);
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set(hy,'Interpreter','Latex');
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% Bornes d'affichage des donnees centrees en (0,0) :
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taille = 20;
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bornes = [-taille taille -taille taille];
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% Creation du cercle et des donnees bruitees :
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n = 50;
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sigma = 0.5;
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[x_cercle,y_cercle,x_donnees_bruitees,y_donnees_bruitees,theta_donnees_bruitees] ...
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= creation_cercle_et_donnees_bruitees(taille,n,sigma);
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% Affichage du cercle :
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plot(x_cercle([1:end 1]),y_cercle([1:end 1]),'r','LineWidth',3);
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% Affichage des donnees bruitees :
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plot(x_donnees_bruitees,y_donnees_bruitees,'k+','MarkerSize',10,'LineWidth',2);
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axis(bornes);
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lg = legend(' Cercle', ...
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' Donnees bruitees', ...
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'Location','Best');
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grid on;
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52
tp1/donnees_occultees.m
Executable file
52
tp1/donnees_occultees.m
Executable file
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@ -0,0 +1,52 @@
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clear;
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close all;
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taille_ecran = get(0,'ScreenSize');
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L = taille_ecran(3);
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H = taille_ecran(4);
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% Fenetre d'affichage :
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figure('Name','Points situes au voisinage d''un cercle', ...
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'Position',[0.4*L,0.05*H,0.6*L,0.7*H]);
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axis equal;
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hold on;
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set(gca,'FontSize',20);
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hx = xlabel('$x$','FontSize',30);
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set(hx,'Interpreter','Latex');
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hy = ylabel('$y$','FontSize',30);
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set(hy,'Interpreter','Latex');
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% Bornes d'affichage des donnees centrees en (0,0) :
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taille = 20;
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bornes = [-taille taille -taille taille];
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% Creation du cercle et des donnees bruitees :
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n = 50;
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sigma = 0.5;
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[x_cercle,y_cercle,x_donnees_bruitees,y_donnees_bruitees,theta_donnees_bruitees] ...
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= creation_cercle_et_donnees_bruitees(taille, n, sigma);
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theta = rand(1, 2)*2*pi;
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theta = sort(theta);
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while theta(2)-theta(1) < pi/2
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theta = rand(1, 2)*2*pi;
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theta = sort(theta);
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end
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x_donnees_bruitees = x_donnees_bruitees( theta(1) <= theta_donnees_bruitees & theta_donnees_bruitees <= theta(2) );
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y_donnees_bruitees = y_donnees_bruitees( theta(1) <= theta_donnees_bruitees & theta_donnees_bruitees <= theta(2) );
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theta_donnees_bruitees = theta_donnees_bruitees( theta(1) <= theta_donnees_bruitees & theta_donnees_bruitees <= theta(2) );
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% Affichage du cercle :
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plot(x_cercle([1:end 1]),y_cercle([1:end 1]),'r','LineWidth',3);
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% Affichage des donnees bruitees :
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plot(x_donnees_bruitees,y_donnees_bruitees,'k+','MarkerSize',10,'LineWidth',2);
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axis(bornes);
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lg = legend(' Cercle', ...
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' Donnees bruitees', ...
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'Location','Best');
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grid on;
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36
tp1/exercice_1.m
Executable file
36
tp1/exercice_1.m
Executable file
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@ -0,0 +1,36 @@
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donnees;
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n_tests = 500;
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% Estimation de la position du centre :
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[C_estime,R_moyen] = estimation_1(x_donnees_bruitees,y_donnees_bruitees,n_tests);
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% Affichage du cercle estime :
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n_points_cercle = 100;
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theta_cercle = 2*pi/n_points_cercle:2*pi/n_points_cercle:2*pi;
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x_cercle_estime = C_estime(1)+R_moyen*cos(theta_cercle);
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y_cercle_estime = C_estime(2)+R_moyen*sin(theta_cercle);
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plot(x_cercle_estime([1:end 1]),y_cercle_estime([1:end 1]),'b','LineWidth',3);
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lg = legend(' Cercle initial', ...
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' Donnees bruitees', ...
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' Cercle estime', ...
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'Location','Best');
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function [C_estime, R_moyen] = estimation_1(x_donnees_bruitees, y_donnees_bruitees, n_tests)
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G = mean( [ x_donnees_bruitees.' y_donnees_bruitees.' ] );
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R_moyen = mean( sqrt((x_donnees_bruitees.'-G(1)).^2 + (y_donnees_bruitees.'-G(2)).^2) );
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x = repmat(x_donnees_bruitees.', 1, n_tests);
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y = repmat(y_donnees_bruitees.', 1, n_tests);
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x_rand = repmat((rand(1, n_tests)*R_moyen - R_moyen/2) + G(1), length(x_donnees_bruitees), 1);
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y_rand = repmat((rand(1, n_tests)*R_moyen - R_moyen/2) + G(2), length(y_donnees_bruitees), 1);
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dist = (sqrt((x-x_rand).^2 + (y-y_rand).^2) - R_moyen).^2;
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[min_val, min_index] = min( sum( dist ) );
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C_estime = [ x_rand(1, min_index) y_rand(1, min_index) ];
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end
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42
tp1/exercice_2.m
Executable file
42
tp1/exercice_2.m
Executable file
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@ -0,0 +1,42 @@
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donnees;
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n_tests = 1000;
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% Estimation du rayon et de la position du centre :
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[C_estime,R_estime] = estimation_2(x_donnees_bruitees,y_donnees_bruitees,n_tests);
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% Affichage du cercle estime :
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n_points_cercle = 100;
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theta_cercle = 2*pi/n_points_cercle:2*pi/n_points_cercle:2*pi;
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x_cercle_estime = C_estime(1)+R_estime*cos(theta_cercle);
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y_cercle_estime = C_estime(2)+R_estime*sin(theta_cercle);
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plot(x_cercle_estime([1:end 1]),y_cercle_estime([1:end 1]),'b','LineWidth',3);
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lg = legend(' Cercle initial', ...
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' Donnees bruitees', ...
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' Cercle estime', ...
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'Location','Best');
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function [C_estime,R_estime] = estimation_2(x_donnees_bruitees,y_donnees_bruitees,n_tests);
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G = mean( [ x_donnees_bruitees.' y_donnees_bruitees.' ] );
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R_moyen = mean( sqrt((x_donnees_bruitees.'-G(1)).^2 + (y_donnees_bruitees.'-G(2)).^2) );
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x = repmat(x_donnees_bruitees.', 1, n_tests, n_tests);
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y = repmat(y_donnees_bruitees.', 1, n_tests, n_tests);
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x_rand = repmat((rand(1, n_tests)*2 - 1) + G(1), length(x_donnees_bruitees), 1, n_tests);
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y_rand = repmat((rand(1, n_tests)*2 - 1) + G(2), length(y_donnees_bruitees), 1, n_tests);
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r_rand = rand(1, 1, n_tests) - 1/2 + R_moyen;
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R = repmat(r_rand, length(x_donnees_bruitees), n_tests, 1);
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dist = (sqrt((x-x_rand).^2 + (y-y_rand).^2) - R).^2;
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somme = reshape(sum(dist), [n_tests, n_tests]);
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[min_val, idx] = min(somme(:));
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[row, col] = ind2sub(size(somme), idx);
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C_estime = [ x_rand(1, row ) y_rand(1, row ) ];
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R_estime = r_rand(col);
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end
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42
tp1/exercice_3.m
Executable file
42
tp1/exercice_3.m
Executable file
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@ -0,0 +1,42 @@
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donnees_occultees;
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n_tests = 1000;
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% Estimation du rayon et de la position du centre :
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[C_estime,R_estime] = estimation_2(x_donnees_bruitees,y_donnees_bruitees,n_tests);
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% Affichage du cercle estime :
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n_points_cercle = 100;
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theta_cercle = 2*pi/n_points_cercle:2*pi/n_points_cercle:2*pi;
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x_cercle_estime = C_estime(1)+R_estime*cos(theta_cercle);
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y_cercle_estime = C_estime(2)+R_estime*sin(theta_cercle);
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plot(x_cercle_estime([1:end 1]),y_cercle_estime([1:end 1]),'b','LineWidth',3);
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lg = legend(' Cercle initial', ...
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' Donnees bruitees', ...
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' Cercle estime', ...
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'Location','Best');
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function [C_estime,R_estime] = estimation_2(x_donnees_bruitees,y_donnees_bruitees,n_tests);
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G = mean( [ x_donnees_bruitees.' y_donnees_bruitees.' ] );
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R_moyen = mean( sqrt((x_donnees_bruitees.'-G(1)).^2 + (y_donnees_bruitees.'-G(2)).^2) );
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x = repmat(x_donnees_bruitees.', 1, n_tests, n_tests);
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y = repmat(y_donnees_bruitees.', 1, n_tests, n_tests);
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x_rand = repmat((rand(1, n_tests)*3*R_moyen - 1.5*R_moyen) + G(1), length(x_donnees_bruitees), 1, n_tests);
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y_rand = repmat((rand(1, n_tests)*3*R_moyen - 1.5*R_moyen) + G(2), length(y_donnees_bruitees), 1, n_tests);
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r_rand = rand(1, 1, n_tests)*3*R_moyen - 1.5*R_moyen;
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R = repmat(r_rand, length(x_donnees_bruitees), n_tests, 1);
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dist = (sqrt((x-x_rand).^2 + (y-y_rand).^2) - R).^2;
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somme = reshape(sum(dist), [n_tests, n_tests]);
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[min_val, idx] = min(somme(:));
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[row, col] = ind2sub(size(somme), idx);
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C_estime = [ x_rand(1, row ) y_rand(1, row ) ];
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R_estime = r_rand(col);
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end
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42
tp1/exercice_4.m
Executable file
42
tp1/exercice_4.m
Executable file
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@ -0,0 +1,42 @@
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donnees_occultees;
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n_tests = 1000;
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% Estimation du rayon et de la position du centre :
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[C_estime,R_estime] = estimation_2(x_donnees_bruitees,y_donnees_bruitees,n_tests);
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% Affichage du cercle estime :
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n_points_cercle = 100;
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theta_cercle = 2*pi/n_points_cercle:2*pi/n_points_cercle:2*pi;
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x_cercle_estime = C_estime(1)+R_estime*cos(theta_cercle);
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y_cercle_estime = C_estime(2)+R_estime*sin(theta_cercle);
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plot(x_cercle_estime([1:end 1]),y_cercle_estime([1:end 1]),'b','LineWidth',3);
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lg = legend(' Cercle initial', ...
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' Donnees bruitees', ...
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' Cercle estime', ...
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'Location','Best');
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function [C_estime,R_estime] = estimation_2(x_donnees_bruitees,y_donnees_bruitees,n_tests);
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G = mean( [ x_donnees_bruitees.' y_donnees_bruitees.' ] );
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R_moyen = mean( sqrt((x_donnees_bruitees.'-G(1)).^2 + (y_donnees_bruitees.'-G(2)).^2) );
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x = repmat(x_donnees_bruitees.', 1, n_tests, n_tests);
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y = repmat(y_donnees_bruitees.', 1, n_tests, n_tests);
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x_rand = repmat((randn(1, n_tests)*R_moyen - R_moyen/2) + G(1), length(x_donnees_bruitees), 1, n_tests);
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y_rand = repmat((randn(1, n_tests)*R_moyen - R_moyen/2) + G(2), length(y_donnees_bruitees), 1, n_tests);
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r_rand = randn(1, 1, n_tests)*R_moyen - R_moyen/2;
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R = repmat(r_rand, length(x_donnees_bruitees), n_tests, 1);
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dist = (sqrt((x-x_rand).^2 + (y-y_rand).^2) - R).^2;
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somme = reshape(sum(dist), [n_tests, n_tests]);
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|
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[min_val, idx] = min(somme(:));
|
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[row, col] = ind2sub(size(somme), idx);
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|
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C_estime = [ x_rand(1, row ) y_rand(1, row ) ];
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R_estime = r_rand(col);
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end
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70
tp2/donnees.m
Executable file
70
tp2/donnees.m
Executable file
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@ -0,0 +1,70 @@
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clear;
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close all;
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|
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taille_ecran = get(0,'ScreenSize');
|
||||
L = taille_ecran(3);
|
||||
H = taille_ecran(4);
|
||||
|
||||
% Fenetre d'affichage :
|
||||
figure('Name','Points situes au voisinage d''une droite', ...
|
||||
'Position',[0.4*L,0,0.6*L,0.8*H]);
|
||||
axis equal;
|
||||
hold on;
|
||||
set(gca,'FontSize',20);
|
||||
hx = xlabel('$x$','FontSize',30);
|
||||
set(hx,'Interpreter','Latex');
|
||||
hy = ylabel('$y$','FontSize',30);
|
||||
set(hy,'Interpreter','Latex');
|
||||
|
||||
% Bornes d'affichage des donnees centrees en (0,0) :
|
||||
taille = 20;
|
||||
bornes = [-taille taille -taille taille];
|
||||
|
||||
% Parametres de la droite :
|
||||
theta_D = 2*pi*(rand-0.5);
|
||||
cos_theta_D = cos(theta_D);
|
||||
sin_theta_D = sin(theta_D);
|
||||
marge = 5;
|
||||
rho_D = (taille-marge)*rand;
|
||||
|
||||
% Affichage de la droite :
|
||||
droite_horizontale = abs(cos_theta_D)<abs(sin_theta_D);
|
||||
pas = 0.01;
|
||||
if droite_horizontale
|
||||
x_D = -taille:pas:taille;
|
||||
y_D = (rho_D-cos_theta_D*x_D)/sin_theta_D;
|
||||
else
|
||||
y_D = -taille:pas:taille;
|
||||
x_D = (rho_D-sin_theta_D*y_D)/cos_theta_D;
|
||||
end
|
||||
plot(x_D,y_D,'r-','LineWidth',3);
|
||||
|
||||
% Donnees non bruitees :
|
||||
n = 50;
|
||||
if droite_horizontale
|
||||
x_donnees = taille*(2*rand(1,n)-1);
|
||||
y_donnees = (rho_D-cos_theta_D*x_donnees)/sin_theta_D;
|
||||
else
|
||||
y_donnees = taille*(2*rand(1,n)-1);
|
||||
x_donnees = (rho_D-sin_theta_D*y_donnees)/cos_theta_D;
|
||||
end
|
||||
|
||||
% Donnees bruitees :
|
||||
sigma = 2;
|
||||
x_donnees_bruitees = x_donnees+sigma*randn(1,n);
|
||||
y_donnees_bruitees = y_donnees+sigma*randn(1,n);
|
||||
indices_visibles = find(x_donnees_bruitees>-taille & ...
|
||||
x_donnees_bruitees<taille & ...
|
||||
y_donnees_bruitees>-taille & ...
|
||||
y_donnees_bruitees<taille);
|
||||
x_donnees_bruitees = x_donnees_bruitees(indices_visibles);
|
||||
y_donnees_bruitees = y_donnees_bruitees(indices_visibles);
|
||||
n = length(indices_visibles);
|
||||
|
||||
% Affichage des donnees bruitees :
|
||||
plot(x_donnees_bruitees,y_donnees_bruitees,'k+','MarkerSize',10,'LineWidth',2);
|
||||
axis(bornes);
|
||||
lg = legend(' Droite', ...
|
||||
' Donnees bruitees', ...
|
||||
'Location','Best');
|
||||
grid on;
|
||||
48
tp2/exercice_1.m
Executable file
48
tp2/exercice_1.m
Executable file
|
|
@ -0,0 +1,48 @@
|
|||
donnees;
|
||||
|
||||
n_tests = 500;
|
||||
|
||||
% Estimation de la droite de regression par le maximum de vraisemblance :
|
||||
[a_DYX_1,b_DYX_1] = estimation_1(x_donnees_bruitees,y_donnees_bruitees,n_tests);
|
||||
|
||||
% Affichage de la droite de regression estimee par le maximum de vraisemblance :
|
||||
if abs(a_DYX_1)<1
|
||||
x_DYX_1 = x_D;
|
||||
y_DYX_1 = a_DYX_1*x_DYX_1+b_DYX_1;
|
||||
else
|
||||
y_DYX_1 = y_D;
|
||||
x_DYX_1 = (y_DYX_1-b_DYX_1)/a_DYX_1;
|
||||
end
|
||||
plot(x_DYX_1,y_DYX_1,'b','LineWidth',3);
|
||||
axis(bornes);
|
||||
lg = legend('~Droite', ...
|
||||
'~Donnees bruitees', ...
|
||||
'~$D_{YX}$ (maximum de vraisemblance)', ...
|
||||
'Location','Best');
|
||||
set(lg,'Interpreter','Latex');
|
||||
|
||||
% Calcul et affichage de l'ecart angulaire :
|
||||
theta_DYX_1 = atan2(b_DYX_1,-a_DYX_1*b_DYX_1);
|
||||
EA_DYX_1 = abs(theta_DYX_1-theta_D);
|
||||
fprintf('D_YX (maximum de vraisemblance) : ecart angulaire = %.2f degres\n',EA_DYX_1/pi*180);
|
||||
|
||||
function [a_DYX_1,b_DYX_1] = estimation_1(x_donnees_bruitees,y_donnees_bruitees,n_tests)
|
||||
|
||||
G = mean( [ x_donnees_bruitees.' y_donnees_bruitees.' ] );
|
||||
|
||||
x_centre = x_donnees_bruitees - G(1);
|
||||
y_centre = y_donnees_bruitees - G(2);
|
||||
|
||||
x = repmat(x_centre.', 1, n_tests);
|
||||
y = repmat(y_centre.', 1, n_tests);
|
||||
|
||||
phi = rand(1, n_tests)*pi - pi/2;
|
||||
|
||||
dist = ( y - bsxfun(@times, x, tan(phi)) ).^2;
|
||||
|
||||
[min_val, min_index] = min( sum( dist ) );
|
||||
|
||||
a_DYX_1 = tan(phi(min_index));
|
||||
b_DYX_1 = G(2) - a_DYX_1*G(1);
|
||||
|
||||
end
|
||||
37
tp2/exercice_2.m
Executable file
37
tp2/exercice_2.m
Executable file
|
|
@ -0,0 +1,37 @@
|
|||
exercice_1;
|
||||
|
||||
% Estimation de la droite de regression par resolution du systeme AX = B :
|
||||
[a_DYX_2,b_DYX_2] = estimation_2(x_donnees_bruitees,y_donnees_bruitees);
|
||||
|
||||
% Affichage de la droite de regression estimee par resolution du systeme AX = B :
|
||||
if abs(a_DYX_2)<1
|
||||
x_DYX_2 = x_D;
|
||||
y_DYX_2 = a_DYX_2*x_DYX_2+b_DYX_2;
|
||||
else
|
||||
y_DYX_2 = y_D;
|
||||
x_DYX_2 = (y_DYX_2-b_DYX_2)/a_DYX_2;
|
||||
end
|
||||
plot(x_DYX_2,y_DYX_2,'g','LineWidth',3);
|
||||
lg = legend('~Droite', ...
|
||||
'~Donnees bruitees', ...
|
||||
'~$D_{YX}$ (maximum de vraisemblance)', ...
|
||||
'~$D_{YX}$ (moindres carres)', ...
|
||||
'Location','Best');
|
||||
set(lg,'Interpreter','Latex');
|
||||
|
||||
% Calcul et affichage de l'ecart angulaire :
|
||||
theta_DYX_2 = atan2(b_DYX_2,-a_DYX_2*b_DYX_2);
|
||||
EA_DYX_2 = abs(theta_DYX_2-theta_D);
|
||||
fprintf('D_YX (moindres carres) : ecart angulaire = %.2f degres\n',EA_DYX_2/pi*180);
|
||||
|
||||
function [a_DYX_2,b_DYX_2] = estimation_2(x_donnees_bruitees,y_donnees_bruitees)
|
||||
|
||||
A = [ x_donnees_bruitees.' ones(length(x_donnees_bruitees), 1) ];
|
||||
B = y_donnees_bruitees.';
|
||||
|
||||
X = (A.'*A)\(A.'*B);
|
||||
|
||||
a_DYX_2 = X(1);
|
||||
b_DYX_2 = X(2);
|
||||
|
||||
end
|
||||
54
tp2/exercice_3.m
Executable file
54
tp2/exercice_3.m
Executable file
|
|
@ -0,0 +1,54 @@
|
|||
donnees;
|
||||
|
||||
n_tests = 500;
|
||||
|
||||
% Estimation de la droite de regression par le maximum de vraisemblance :
|
||||
[theta_Dorth_1,rho_Dorth_1] = estimation_3(x_donnees_bruitees,y_donnees_bruitees,n_tests);
|
||||
|
||||
% Affichage de la droite de regression estimee par le maximum de vraisemblance :
|
||||
cos_theta_Dorth_1 = cos(theta_Dorth_1);
|
||||
sin_theta_Dorth_1 = sin(theta_Dorth_1);
|
||||
if abs(cos_theta_Dorth_1)<abs(sin_theta_Dorth_1)
|
||||
x_Dorth_1 = x_D;
|
||||
y_Dorth_1 = (rho_Dorth_1-x_Dorth_1*cos_theta_Dorth_1)/sin_theta_Dorth_1;
|
||||
else
|
||||
y_Dorth_1 = y_D;
|
||||
x_Dorth_1 = (rho_Dorth_1-y_Dorth_1*sin_theta_Dorth_1)/cos_theta_Dorth_1;
|
||||
end
|
||||
plot(x_Dorth_1,y_Dorth_1,'b','LineWidth',3);
|
||||
axis(bornes);
|
||||
lg = legend('~Droite', ...
|
||||
'~Donnees bruitees', ...
|
||||
'~$D_\perp$ (maximum de vraisemblance)', ...
|
||||
'Location','Best');
|
||||
set(lg,'Interpreter','Latex');
|
||||
|
||||
% Calcul et affichage de l'ecart angulaire :
|
||||
EA_Dorth_1 = abs(theta_Dorth_1-theta_D);
|
||||
fprintf('D_perp (maximum de vraisemblance) : ecart angulaire = %.2f degres\n',EA_Dorth_1/pi*180);
|
||||
|
||||
function [theta_Dorth_1,rho_Dorth_1] = estimation_3(x_donnees_bruitees,y_donnees_bruitees,n_tests)
|
||||
|
||||
G = mean( [ x_donnees_bruitees.' y_donnees_bruitees.' ] );
|
||||
|
||||
x_centre = x_donnees_bruitees - G(1);
|
||||
y_centre = y_donnees_bruitees - G(2);
|
||||
|
||||
x = repmat(x_centre.', 1, n_tests);
|
||||
y = repmat(y_centre.', 1, n_tests);
|
||||
|
||||
theta = rand(1, n_tests)*pi;
|
||||
|
||||
dist = ( bsxfun(@times, x, cos(theta)) + bsxfun(@times, y, sin(theta)) ).^2;
|
||||
|
||||
[min_val, min_index] = min( sum( dist ) );
|
||||
|
||||
theta_Dorth_1 = theta(min_index);
|
||||
rho_Dorth_1 = G(1)*cos(theta_Dorth_1) + G(2)*sin(theta_Dorth_1);
|
||||
|
||||
if rho_Dorth_1 < 0
|
||||
theta_Dorth_1 = theta_Dorth_1 - pi;
|
||||
rho_Dorth_1 = G(1)*cos(theta_Dorth_1) + G(2)*sin(theta_Dorth_1);
|
||||
end
|
||||
|
||||
end
|
||||
50
tp2/exercice_4.m
Executable file
50
tp2/exercice_4.m
Executable file
|
|
@ -0,0 +1,50 @@
|
|||
exercice_3;
|
||||
|
||||
% Estimation de la droite de regression par resolution du systeme CY = 0 :
|
||||
[theta_Dorth_2,rho_Dorth_2] = estimation_4(x_donnees_bruitees,y_donnees_bruitees);
|
||||
|
||||
% Affichage de la droite de regression estimee par resolution du systeme lineaire CY = 0 :
|
||||
cos_theta_Dorth_2 = cos(theta_Dorth_2);
|
||||
sin_theta_Dorth_2 = sin(theta_Dorth_2);
|
||||
if abs(cos_theta_Dorth_2)<abs(sin_theta_Dorth_2)
|
||||
x_Dorth_2 = x_D;
|
||||
y_Dorth_2 = (rho_Dorth_2-x_Dorth_2*cos_theta_Dorth_2)/sin_theta_Dorth_2;
|
||||
else
|
||||
y_Dorth_2 = y_D;
|
||||
x_Dorth_2 = (rho_Dorth_2-y_Dorth_2*sin_theta_Dorth_2)/cos_theta_Dorth_2;
|
||||
end
|
||||
plot(x_Dorth_2,y_Dorth_2,'g','LineWidth',3);
|
||||
lg = legend('~Droite', ...
|
||||
'~Donnees bruitees', ...
|
||||
'~$D_\perp$ (maximum de vraisemblance)', ...
|
||||
'~$D_\perp$ (moindres carres)', ...
|
||||
'Location','Best');
|
||||
set(lg,'Interpreter','Latex');
|
||||
|
||||
% Calcul et affichage de l'ecart angulaire :
|
||||
EA_Dorth_2 = abs(theta_Dorth_2-theta_D);
|
||||
fprintf('D_perp (moindres carres) : ecart angulaire = %.2f degres\n',EA_Dorth_2/pi*180);
|
||||
|
||||
function [theta_Dorth_2,rho_Dorth_2] = estimation_4(x_donnees_bruitees,y_donnees_bruitees)
|
||||
|
||||
G = mean( [ x_donnees_bruitees.' y_donnees_bruitees.' ] );
|
||||
|
||||
x_centre = x_donnees_bruitees - G(1);
|
||||
y_centre = y_donnees_bruitees - G(2);
|
||||
|
||||
C = [x_centre.' y_centre.'];
|
||||
|
||||
[Y, lambda] = eig(C.'*C);
|
||||
lambda = diag(lambda);
|
||||
|
||||
[min_val, min_index] = min( lambda );
|
||||
|
||||
theta_Dorth_2 = atan(Y(2, min_index)/Y(1, min_index));
|
||||
rho_Dorth_2 = G(1)*cos(theta_Dorth_2) + G(2)*sin(theta_Dorth_2);
|
||||
|
||||
if rho_Dorth_2 < 0
|
||||
theta_Dorth_2 = theta_Dorth_2 - pi;
|
||||
rho_Dorth_2 = G(1)*cos(theta_Dorth_2) + G(2)*sin(theta_Dorth_2);
|
||||
end
|
||||
|
||||
end
|
||||
BIN
tp3/Donnees/bateau.mat
Executable file
BIN
tp3/Donnees/bateau.mat
Executable file
Binary file not shown.
BIN
tp3/Donnees/parquet.mat
Executable file
BIN
tp3/Donnees/parquet.mat
Executable file
Binary file not shown.
60
tp3/affichage_faisceau.m
Executable file
60
tp3/affichage_faisceau.m
Executable file
|
|
@ -0,0 +1,60 @@
|
|||
function affichage_faisceau(rho,theta,limites_affichages,couleur)
|
||||
|
||||
% Limites de l'affichage du faisceau :
|
||||
x_min = limites_affichages(1);
|
||||
x_max = limites_affichages(2);
|
||||
y_min = limites_affichages(3);
|
||||
y_max = limites_affichages(4);
|
||||
|
||||
% Trace des droites en boucle :
|
||||
for k = 1:length(rho)
|
||||
|
||||
% Parametres de la droite D_k :
|
||||
rho_k = rho(k);
|
||||
theta_k = theta(k);
|
||||
sin_theta_k = sin(theta_k);
|
||||
cos_theta_k = cos(theta_k);
|
||||
|
||||
% Coordonnees des extremites du segment a afficher :
|
||||
x_extremites = zeros(1,2);
|
||||
y_extremites = zeros(1,2);
|
||||
|
||||
% Cas ou D_k coupe le bord gauche :
|
||||
i = 1;
|
||||
x = x_min;
|
||||
y = (rho_k-x*cos_theta_k)/sin_theta_k;
|
||||
if y>=y_min & y<=y_max
|
||||
x_extremites(i) = x;
|
||||
y_extremites(i) = y;
|
||||
i = i+1;
|
||||
end
|
||||
|
||||
% Cas ou D_k coupe le bord droit :
|
||||
x = x_max;
|
||||
y = (rho_k-x*cos_theta_k)/sin_theta_k;
|
||||
if y>=y_min & y<=y_max
|
||||
x_extremites(i) = x;
|
||||
y_extremites(i) = y;
|
||||
i = i+1;
|
||||
end
|
||||
|
||||
% Cas ou D_k coupe le bord superieur :
|
||||
y = y_min;
|
||||
x = (rho_k-y*sin_theta_k)/cos_theta_k;
|
||||
if x>=x_min & x<=x_max
|
||||
x_extremites(i) = x;
|
||||
y_extremites(i) = y;
|
||||
i = i+1;
|
||||
end
|
||||
|
||||
% Cas ou D_k coupe le bord inferieur :
|
||||
y = y_max;
|
||||
x = (rho_k-y*sin_theta_k)/cos_theta_k;
|
||||
if x>=x_min & x<=x_max
|
||||
x_extremites(i) = x;
|
||||
y_extremites(i) = y;
|
||||
end
|
||||
|
||||
% Affichage de la droite D_k :
|
||||
plot(x_extremites,y_extremites,couleur,'LineWidth',2);
|
||||
end
|
||||
13
tp3/cercle_3_points.m
Executable file
13
tp3/cercle_3_points.m
Executable file
|
|
@ -0,0 +1,13 @@
|
|||
function [C,R] = cercle_3_points(x,y)
|
||||
|
||||
x1 = x(1);
|
||||
y1 = y(1);
|
||||
x2 = x(2);
|
||||
y2 = y(2);
|
||||
x3 = x(3);
|
||||
y3 = y(3);
|
||||
|
||||
A = 2*[-x1+x2 -y1+y2 ; -x1+x3 -y1+y3];
|
||||
B = [-x1^2+x2^2-y1^2+y2^2 ; -x1^2+x3^2-y1^2+y3^2];
|
||||
C = A\B;
|
||||
R = sqrt((x1-C(1))^2+(y1-C(2))^2);
|
||||
BIN
tp3/creation_cercle_et_donnees_bruitees.p
Executable file
BIN
tp3/creation_cercle_et_donnees_bruitees.p
Executable file
Binary file not shown.
44
tp3/donnees_aberrantes.m
Executable file
44
tp3/donnees_aberrantes.m
Executable file
|
|
@ -0,0 +1,44 @@
|
|||
clear;
|
||||
close all;
|
||||
|
||||
taille_ecran = get(0,'ScreenSize');
|
||||
L = taille_ecran(3);
|
||||
H = taille_ecran(4);
|
||||
|
||||
% Fenetre d'affichage :
|
||||
figure('Name','Points situes au voisinage d''un cercle', ...
|
||||
'Position',[0.4*L,0.05*H,0.6*L,0.7*H]);
|
||||
axis equal;
|
||||
hold on;
|
||||
set(gca,'FontSize',20);
|
||||
hx = xlabel('$x$','FontSize',30);
|
||||
set(hx,'Interpreter','Latex');
|
||||
hy = ylabel('$y$','FontSize',30);
|
||||
set(hy,'Interpreter','Latex');
|
||||
|
||||
% Limites de l'affichage des donnees centrees en (0,0) :
|
||||
taille = 20;
|
||||
limites_affichage = [-taille taille -taille taille];
|
||||
|
||||
% Creation du cercle et des donnees bruitees :
|
||||
n = 50;
|
||||
sigma = 0.5;
|
||||
[x_cercle,y_cercle,theta_cercle,x_donnees_bruitees,y_donnees_bruitees,theta_donnees_bruitees] ...
|
||||
= creation_cercle_et_donnees_bruitees(taille,n,sigma);
|
||||
|
||||
% Affichage du cercle :
|
||||
plot(x_cercle([1:end 1]),y_cercle([1:end 1]),'r','LineWidth',3);
|
||||
|
||||
% Donnees aberrantes :
|
||||
proportion = 0.25;
|
||||
n_donnees_aberrantes = floor(proportion*n);
|
||||
x_donnees_bruitees = [x_donnees_bruitees taille*(2*rand(1,n_donnees_aberrantes)-1)];
|
||||
y_donnees_bruitees = [y_donnees_bruitees taille*(2*rand(1,n_donnees_aberrantes)-1)];
|
||||
|
||||
% Affichage des donnees bruitees :
|
||||
plot(x_donnees_bruitees,y_donnees_bruitees,'k+','MarkerSize',10,'LineWidth',2);
|
||||
axis(limites_affichage);
|
||||
lg = legend(' Cercle', ...
|
||||
' Donnees (bruitees + aberrantes)', ...
|
||||
'Location','Best');
|
||||
grid on;
|
||||
86
tp3/exercice_0.m
Executable file
86
tp3/exercice_0.m
Executable file
|
|
@ -0,0 +1,86 @@
|
|||
clear;
|
||||
close all;
|
||||
taille_ecran = get(0,'ScreenSize');
|
||||
L = taille_ecran(3);
|
||||
H = taille_ecran(4);
|
||||
|
||||
% Parametres :
|
||||
alpha = pi/180*12; % Seuil sur l'orientation du gradient (en radians)
|
||||
cos_alpha = cos(alpha);
|
||||
seuil_norme = 2/sin(alpha); % Seuil sur la norme du gradient (decoule d'une etude theorique)
|
||||
|
||||
% Chargement des donnees :
|
||||
% load Donnees/parquet;
|
||||
%load Donnees/parquet;
|
||||
load Donnees/bateau;
|
||||
[n_lignes,n_colonnes] = size(I);
|
||||
limites_affichages = [1 n_colonnes 1 n_lignes];
|
||||
|
||||
figure('Name','Detection des alignements','Position',[0.2*L,0,0.8*L,H]);
|
||||
|
||||
% Affichage de l'image :
|
||||
subplot(2,2,1);
|
||||
imagesc(I);
|
||||
axis equal;
|
||||
axis off;
|
||||
colormap gray;
|
||||
title('Image originale','FontSize',30);
|
||||
axis(limites_affichages);
|
||||
|
||||
% Fenetre d'affichage des pixels de contour :
|
||||
subplot(2,2,3);
|
||||
imagesc(I);
|
||||
axis equal;
|
||||
axis off;
|
||||
colormap gray;
|
||||
hold on;
|
||||
|
||||
% Gradient du niveau de gris (x vers la droite, y vers le bas) :
|
||||
I = double(I);
|
||||
[G_x,G_y] = gradient(I);
|
||||
G_norme = sqrt(G_x.^2+G_y.^2);
|
||||
|
||||
% Selection des pixels de contour :
|
||||
contour = G_norme>seuil_norme;
|
||||
|
||||
% Pas de pixel de contour sur le bord de l'image => Traitement simplifie !
|
||||
contour([1 n_lignes],:) = 0;
|
||||
contour(:,[1 n_colonnes]) = 0;
|
||||
|
||||
% Tri des pixels de contour :
|
||||
indices_contour = find(contour);
|
||||
[~,indices] = sort(G_norme(indices_contour),'descend');
|
||||
indices_contour = indices_contour(indices);
|
||||
|
||||
% Affichage d'une petite fleche sur les pixels de contour :
|
||||
[i,j] = ind2sub(size(I),indices_contour);
|
||||
quiver(j,i,G_x(indices_contour),G_y(indices_contour),'r'); % Attention : x = j et y = i
|
||||
axis equal;
|
||||
title('Pixels de contour','FontSize',30);
|
||||
axis(limites_affichages);
|
||||
drawnow;
|
||||
|
||||
% Affichage des ensembles E (la fonction label2rgb donne a chaque ensemble E une couleur differente) :
|
||||
subplot(2,2,2);
|
||||
imagesc(I_resultat);
|
||||
axis(limites_affichages);
|
||||
axis equal;
|
||||
axis off;
|
||||
title('Ensembles candidats','FontSize',30);
|
||||
|
||||
% Affichage de l'esquisse :
|
||||
subplot(2,2,4);
|
||||
imagesc(120*ones(size(I)),[0,255]);
|
||||
axis equal;
|
||||
axis off;
|
||||
title('Alignements','FontSize',30);
|
||||
hold on;
|
||||
axis(limites_affichages);
|
||||
|
||||
% Boucle sur les ensembles E :
|
||||
for k = 1:size(extremites,3)
|
||||
% Affichage du segment :
|
||||
line(extremites(1,:,k),extremites(2,:,k),'Color','w','LineWidth',2);
|
||||
end
|
||||
|
||||
save exercice_0;
|
||||
BIN
tp3/exercice_0.mat
Executable file
BIN
tp3/exercice_0.mat
Executable file
Binary file not shown.
81
tp3/exercice_1.m
Executable file
81
tp3/exercice_1.m
Executable file
|
|
@ -0,0 +1,81 @@
|
|||
clear;
|
||||
close all;
|
||||
taille_ecran = get(0,'ScreenSize');
|
||||
L = taille_ecran(3);
|
||||
H = taille_ecran(4);
|
||||
|
||||
load exercice_0;
|
||||
|
||||
% Estimation du point de fuite :
|
||||
[rho_F,theta_F] = estimation_F(rho,theta);
|
||||
|
||||
% Coordonnees cartesiennes du point de fuite :
|
||||
x_F = rho_F*cos(theta_F);
|
||||
y_F = rho_F*sin(theta_F);
|
||||
|
||||
figure('Name','Estimation du point de fuite','Position',[0.7*L,0,0.3*L,H]);
|
||||
|
||||
% Affichage des points de coordonnees (rho,theta) :
|
||||
subplot(2,1,1);
|
||||
plot(theta,rho,'k+','MarkerSize',10,'LineWidth',2);
|
||||
axis([-pi pi -1.2*rho_F 1.2*rho_F]);
|
||||
set(gca,'FontSize',20);
|
||||
hx = xlabel('$\theta$','FontSize',30);
|
||||
set(hx,'Interpreter','Latex');
|
||||
hy = ylabel('$\rho$','FontSize',30);
|
||||
set(hy,'Interpreter','Latex');
|
||||
grid;
|
||||
hold on;
|
||||
|
||||
% Affichage de la sinusoide correspondant au point de fuite :
|
||||
pas = 0.01;
|
||||
theta_affichage = -pi:pas:pi;
|
||||
rho_affichage = rho_F*cos(theta_affichage-theta_F);
|
||||
plot(theta_affichage,rho_affichage,'b-','LineWidth',3);
|
||||
|
||||
% Affichage des points de coordonnees (rho,theta) :
|
||||
plot(theta,rho,'r+','MarkerSize',10,'LineWidth',2);
|
||||
title('Sinusoide estimee');
|
||||
|
||||
% Affichage de l'image :
|
||||
subplot(2,1,2);
|
||||
imagesc(I);
|
||||
set(gca,'FontSize',20);
|
||||
axis ij equal off;
|
||||
colormap gray;
|
||||
hold on;
|
||||
|
||||
% Limites des affichages :
|
||||
marge = round(min(n_lignes,n_colonnes)/10);
|
||||
x_min = min(1,x_F)-marge;
|
||||
x_max = max(n_colonnes,x_F)+marge;
|
||||
y_min = min(1,y_F)-marge;
|
||||
y_max = max(n_lignes,y_F)+marge;
|
||||
limites_affichages = [x_min x_max y_min y_max];
|
||||
axis(limites_affichages);
|
||||
|
||||
% Affichage d'une selection des droites formant le premier faisceau :
|
||||
taille_selection = min(length(rho),10);
|
||||
longueurs_segments_au_carre = (extremites(1,1,:)-extremites(1,2,:)).^2 ...
|
||||
+(extremites(2,1,:)-extremites(2,2,:)).^2;
|
||||
[~,indices_tries] = sort(longueurs_segments_au_carre,'descend');
|
||||
selection = indices_tries(1:taille_selection);
|
||||
affichage_faisceau(rho(selection),theta(selection),limites_affichages,'r');
|
||||
|
||||
% Affichage du point de fuite :
|
||||
plot(x_F,y_F,'bx','MarkerSize',20,'LineWidth',5);
|
||||
title('Point de fuite');
|
||||
|
||||
function [rho_F, theta_F] = estimation_F(rho, theta)
|
||||
|
||||
B = rho;
|
||||
A = [cos(theta.') ; sin(theta.')].';
|
||||
|
||||
X = (A.'*A)\(A.'*B);
|
||||
|
||||
rho_F = sqrt( X(1)^2 + X(2)^2 );
|
||||
theta_F = atan2(X(2), X(1));
|
||||
|
||||
end
|
||||
|
||||
|
||||
201
tp3/exercice_2.m
Executable file
201
tp3/exercice_2.m
Executable file
|
|
@ -0,0 +1,201 @@
|
|||
clear;
|
||||
close all;
|
||||
taille_ecran = get(0,'ScreenSize');
|
||||
L = taille_ecran(3);
|
||||
H = taille_ecran(4);
|
||||
|
||||
load exercice_0;
|
||||
|
||||
% Parametres de l'algorithme RANSAC :
|
||||
n_donnees = length(rho);
|
||||
S1 = 5;
|
||||
S2 = 0.3;
|
||||
k_max = floor(nchoosek(n_donnees,2)/n_donnees);
|
||||
parametres = [S1 S2 k_max];
|
||||
|
||||
% Estimation du premier point de fuite :
|
||||
[rho_F_1,theta_F_1] = RANSAC_2(rho,theta,parametres);
|
||||
|
||||
% Coordonnees cartesiennes du premier point de fuite :
|
||||
x_F_1 = rho_F_1*cos(theta_F_1);
|
||||
y_F_1 = rho_F_1*sin(theta_F_1);
|
||||
|
||||
% Droites conformes au premier point de fuite :
|
||||
conformes_1 = abs(rho-rho_F_1*cos(theta-theta_F_1))<=S1;
|
||||
rho_conformes_1 = rho(conformes_1);
|
||||
theta_conformes_1 = theta(conformes_1);
|
||||
|
||||
% Droites restantes :
|
||||
theta = theta(~conformes_1);
|
||||
rho = rho(~conformes_1);
|
||||
|
||||
% Estimation du deuxieme point de fuite :
|
||||
[rho_F_2,theta_F_2] = RANSAC_2(rho,theta,parametres);
|
||||
|
||||
% Coordonnees cartesiennes du deuxieme point de fuite :
|
||||
x_F_2 = rho_F_2*cos(theta_F_2);
|
||||
y_F_2 = rho_F_2*sin(theta_F_2);
|
||||
|
||||
% Droites conformes au deuxieme point de fuite :
|
||||
conformes_2 = abs(rho-rho_F_2*cos(theta-theta_F_2))<=S1;
|
||||
rho_conformes_2 = rho(conformes_2);
|
||||
theta_conformes_2 = theta(conformes_2);
|
||||
|
||||
figure('Name','Estimation de la ligne de fuite','Position',[0.3*L,0,0.7*L,H]);
|
||||
|
||||
% Affichage des points de coordonnees (rho,theta) :
|
||||
subplot(2,2,2);
|
||||
plot(theta,rho,'k+','MarkerSize',10,'LineWidth',2);
|
||||
rho_F_max = max(rho_F_1,rho_F_2);
|
||||
axis([-pi pi -1.2*rho_F_max 1.2*rho_F_max]);
|
||||
set(gca,'FontSize',20);
|
||||
hx = xlabel('$\theta$','FontSize',30);
|
||||
set(hx,'Interpreter','Latex');
|
||||
hy = ylabel('$\rho$','FontSize',30);
|
||||
set(hy,'Interpreter','Latex');
|
||||
grid;
|
||||
hold on;
|
||||
|
||||
% Affichage de la sinusoide correspondant au premier point de fuite :
|
||||
pas = 0.01;
|
||||
theta_affichage = -pi:pas:pi;
|
||||
rho_affichage_1 = rho_F_1*cos(theta_affichage-theta_F_1);
|
||||
plot(theta_affichage,rho_affichage_1,'b-','LineWidth',3);
|
||||
|
||||
% Affichage des points conformes a la premiere sinusoide :
|
||||
plot(theta_conformes_1,rho_conformes_1,'r+','MarkerSize',10,'LineWidth',2);
|
||||
|
||||
% Affichage de la deuxieme sinusoide :
|
||||
rho_affichage_2 = rho_F_2*cos(theta_affichage-theta_F_2);
|
||||
plot(theta_affichage,rho_affichage_2,'m-','LineWidth',3);
|
||||
|
||||
% Affichage des points conformes a la deuxieme sinusoide :
|
||||
plot(theta_conformes_2,rho_conformes_2,'g+','MarkerSize',10,'LineWidth',2);
|
||||
title('Paire de sinusoides estimees');
|
||||
|
||||
% Affichage de l'image :
|
||||
subplot(2,2,3);
|
||||
imagesc(I);
|
||||
set(gca,'FontSize',20);
|
||||
axis ij equal off;
|
||||
colormap gray;
|
||||
hold on;
|
||||
|
||||
% Limites des affichages :
|
||||
marge = round(min(n_lignes,n_colonnes)/10);
|
||||
x_min = min(1,x_F_1);
|
||||
x_min = min(x_min,x_F_2)-marge;
|
||||
x_max = max(n_colonnes,x_F_1);
|
||||
x_max = max(x_max,x_F_2)+marge;
|
||||
y_min = min(1,y_F_1);
|
||||
y_min = min(y_min,y_F_2)-marge;
|
||||
y_max = max(n_lignes,y_F_1);
|
||||
y_max = max(y_max,y_F_2)+marge;
|
||||
limites_affichages = [x_min x_max y_min y_max];
|
||||
axis(limites_affichages);
|
||||
|
||||
% Affichage d'une selection de droites formant le premier faisceau :
|
||||
taille_selection_1 = min(length(rho_conformes_1),10);
|
||||
longueurs_segments_au_carre_1 = (extremites(1,1,conformes_1)-extremites(1,2,conformes_1)).^2 ...
|
||||
+(extremites(2,1,conformes_1)-extremites(2,2,conformes_1)).^2;
|
||||
[~,indices_tries_1] = sort(longueurs_segments_au_carre_1,'descend');
|
||||
selection_1 = indices_tries_1(1:taille_selection_1);
|
||||
affichage_faisceau(rho_conformes_1(selection_1),theta_conformes_1(selection_1),limites_affichages,'r');
|
||||
|
||||
% Affichage du premier point de fuite :
|
||||
plot(x_F_1,y_F_1,'bx','MarkerSize',20,'LineWidth',5);
|
||||
|
||||
% Affichage d'une selection de droites formant le deuxieme faisceau :
|
||||
taille_selection_2 = min(length(rho_conformes_2),10);
|
||||
longueurs_segments_au_carre_2 = (extremites(1,1,conformes_2)-extremites(1,2,conformes_2)).^2 ...
|
||||
+(extremites(2,1,conformes_2)-extremites(2,2,conformes_2)).^2;
|
||||
[~,indices_tries_2] = sort(longueurs_segments_au_carre_2,'descend');
|
||||
selection_2 = indices_tries_2(1:taille_selection_2);
|
||||
affichage_faisceau(rho_conformes_2(selection_2),theta_conformes_2(selection_2),limites_affichages,'g');
|
||||
|
||||
% Affichage du deuxieme point de fuite :
|
||||
plot(x_F_2,y_F_2,'mx','MarkerSize',20,'LineWidth',5);
|
||||
title('Paire de points de fuite');
|
||||
|
||||
% Affichage de l'image :
|
||||
subplot(2,2,4);
|
||||
imagesc(I);
|
||||
set(gca,'FontSize',20);
|
||||
axis ij equal off;
|
||||
colormap gray;
|
||||
hold on;
|
||||
axis(limites_affichages);
|
||||
|
||||
% Affichage du premier point de fuite :
|
||||
plot(x_F_1,y_F_1,'bx','MarkerSize',20,'LineWidth',5);
|
||||
|
||||
% Affichage du deuxieme point de fuite :
|
||||
plot(x_F_2,y_F_2,'mx','MarkerSize',20,'LineWidth',5);
|
||||
|
||||
% Affichage de la ligne de fuite :
|
||||
line([x_F_1 x_F_2],[y_F_1 y_F_2],'Color','c','LineWidth',2);
|
||||
title('Ligne de fuite');
|
||||
|
||||
% Affichage de l'image originale :
|
||||
subplot(2,2,1);
|
||||
imagesc(I);
|
||||
set(gca,'FontSize',20);
|
||||
axis ij equal off;
|
||||
colormap gray;
|
||||
hold on;
|
||||
title('Image originale');
|
||||
axis(limites_affichages);
|
||||
|
||||
|
||||
function [rho_F_1, theta_F_1] = RANSAC_2(rho, theta, parametres)
|
||||
|
||||
n = length(rho);
|
||||
erreur_F_1 = +inf;
|
||||
|
||||
for j = 1:parametres(3)
|
||||
|
||||
points_model = randperm(n, 2);
|
||||
points_candidat = [];
|
||||
|
||||
for i = setdiff(1:n, points_model)
|
||||
|
||||
[rho_candidat, theta_candidat, erreur_candidat] = estimation_F_2(rho([points_model i]), theta([points_model i]));
|
||||
|
||||
if erreur_candidat < parametres(1)
|
||||
|
||||
points_candidat = [ points_candidat i ];
|
||||
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
if length(points_candidat)/n > parametres(2)
|
||||
|
||||
[rho_new, theta_new, erreur_new] = estimation_F_2(rho([points_model points_candidat]), theta([points_model points_candidat]));
|
||||
|
||||
if erreur_new < erreur_F_1
|
||||
|
||||
rho_F_1 = rho_new;
|
||||
theta_F_1 = theta_new;
|
||||
erreur_F_1 = erreur_new;
|
||||
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
end
|
||||
end
|
||||
|
||||
function [rho_F, theta_F, erreur_F] = estimation_F_2(rho, theta)
|
||||
|
||||
B = rho;
|
||||
A = [cos(theta.') ; sin(theta.')].';
|
||||
|
||||
X = (A.'*A)\(A.'*B);
|
||||
|
||||
rho_F = sqrt( X(1)^2 + X(2)^2 );
|
||||
theta_F = atan2(X(2), X(1));
|
||||
|
||||
erreur_F = mean( abs( rho - rho_F*cos(theta - theta_F) ) );
|
||||
|
||||
end
|
||||
103
tp3/exercice_3.m
Executable file
103
tp3/exercice_3.m
Executable file
|
|
@ -0,0 +1,103 @@
|
|||
donnees_aberrantes;
|
||||
|
||||
n_tests = 1000;
|
||||
|
||||
% Parametres de l'algorithme RANSAC :
|
||||
n_donnees = length(x_donnees_bruitees);
|
||||
S1 = 2;
|
||||
S2 = 0.5;
|
||||
k_max = floor(nchoosek(n_donnees,3)/n_donnees);
|
||||
parametres = [S1 S2 k_max];
|
||||
|
||||
% Estimation du rayon et de la position du centre :
|
||||
[C_estime,R_estime] = RANSAC_3(x_donnees_bruitees,y_donnees_bruitees,parametres,n_tests);
|
||||
|
||||
% Affichage du cercle estime :
|
||||
x_cercle_estime = C_estime(1)+R_estime*cos(theta_cercle);
|
||||
y_cercle_estime = C_estime(2)+R_estime*sin(theta_cercle);
|
||||
plot(x_cercle_estime([1:end 1]),y_cercle_estime([1:end 1]),'b','LineWidth',3);
|
||||
|
||||
% Affichage des points conformes au modele :
|
||||
conformes = abs(sqrt((x_donnees_bruitees-C_estime(1)).^2+ ...
|
||||
(y_donnees_bruitees-C_estime(2)).^2)-R_estime)<=S1;
|
||||
plot(x_donnees_bruitees(conformes), ...
|
||||
y_donnees_bruitees(conformes),'b+','MarkerSize',10,'LineWidth',2);
|
||||
lg = legend(' Cercle initial', ...
|
||||
' Donnees (bruitees + aberrantes)', ...
|
||||
' Cercle estime', ...
|
||||
' Donnees conformes', ...
|
||||
'Location','Best');
|
||||
|
||||
function [C_estime,R_estime] = RANSAC_3(x_donnees_bruitees,y_donnees_bruitees,parametres,n_tests)
|
||||
|
||||
n = length(x_donnees_bruitees);
|
||||
erreur = +inf;
|
||||
|
||||
for j = 1:parametres(3)
|
||||
|
||||
j/parametres(3)*100
|
||||
|
||||
points_model = randperm(n, 3);
|
||||
[C_model, R_model] = cercle_3_points(x_donnees_bruitees(points_model), y_donnees_bruitees(points_model));
|
||||
|
||||
points_candidat = [];
|
||||
|
||||
for i = setdiff(1:n, points_model)
|
||||
|
||||
[C_candidat, R_candidat] = estimation(x_donnees_bruitees([points_model i]), y_donnees_bruitees([points_model i]), n_tests);
|
||||
|
||||
% ma fonction d'erreur ne doit pas être la bonne mais ça marche
|
||||
% a peu près bien ~~
|
||||
erreur_candidat = mean( abs( sqrt((C_model(1)-x_donnees_bruitees([points_model i])).^2 + (C_model(2)-y_donnees_bruitees([points_model i])).^2 )) - R_model );
|
||||
|
||||
if erreur_candidat < parametres(1)
|
||||
|
||||
points_candidat = [ points_candidat i ];
|
||||
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
if length(points_candidat)/n > parametres(2)
|
||||
|
||||
[C_new, R_new] = estimation(x_donnees_bruitees([points_model points_candidat]), y_donnees_bruitees([points_model points_candidat]), n_tests);
|
||||
|
||||
erreur_new = mean( abs( sqrt((C_model(1)-x_donnees_bruitees([points_model points_candidat])).^2 + (C_model(2)-y_donnees_bruitees([points_model points_candidat])).^2 )) - R_model );
|
||||
|
||||
if erreur_new < erreur
|
||||
|
||||
C_estime = C_new;
|
||||
R_estime = R_new;
|
||||
erreur = erreur_new;
|
||||
|
||||
end
|
||||
|
||||
end
|
||||
|
||||
end
|
||||
end
|
||||
|
||||
function [C_estime,R_estime] = estimation(x_donnees_bruitees,y_donnees_bruitees,n_tests)
|
||||
|
||||
G = mean( [ x_donnees_bruitees.' y_donnees_bruitees.' ] );
|
||||
R_moyen = mean( sqrt((x_donnees_bruitees.'-G(1)).^2 + (y_donnees_bruitees.'-G(2)).^2) );
|
||||
|
||||
x = repmat(x_donnees_bruitees.', 1, n_tests, n_tests);
|
||||
y = repmat(y_donnees_bruitees.', 1, n_tests, n_tests);
|
||||
|
||||
x_rand = repmat((randn(1, n_tests)*R_moyen - R_moyen/2) + G(1), length(x_donnees_bruitees), 1, n_tests);
|
||||
y_rand = repmat((randn(1, n_tests)*R_moyen - R_moyen/2) + G(2), length(y_donnees_bruitees), 1, n_tests);
|
||||
|
||||
r_rand = randn(1, 1, n_tests)*R_moyen - R_moyen/2;
|
||||
R = repmat(r_rand, length(x_donnees_bruitees), n_tests, 1);
|
||||
|
||||
dist = (sqrt((x-x_rand).^2 + (y-y_rand).^2) - R).^2;
|
||||
|
||||
somme = reshape(sum(dist), [n_tests, n_tests]);
|
||||
|
||||
[min_val, idx] = min(somme(:));
|
||||
[row, col] = ind2sub(size(somme), idx);
|
||||
|
||||
C_estime = [ x_rand(1, row ) y_rand(1, row ) ];
|
||||
R_estime = r_rand(col);
|
||||
end
|
||||
Loading…
Reference in a new issue