<span style="font-size:18px;">function exercise
clear all;
clf;
close all;
%%定义一个相机1
au1=100;av1=120;uo1=128;vo1=128;
%%定义第二个相机
au2=90;av2=110;uo2=128;vo2=128;
ax = 0.1; by = pi/4; cz = 0.2;
tx = -1000; ty = 190; tz = 230;
%%两个相机的本征矩阵和转移矩阵
A1=[au1 0 uo1
0 av1 vo1
0 0 1];
A2=[au2 0 uo2
0 av2 uo2
0 0 1];
%相机到图像平面的 转移矩阵
%z轴的旋转矩阵 相机间的旋转矩阵
rz=[cos(cz) -sin(cz) 0
sin(cz) cos(cz) 0
0 0 1];
ry=[cos(by) 0 sin(by)
0 1 0
-sin(by) 0 cos(by)];
rx=[1 0 0
0 cos(ax) -sin(ax)
0 sin(ax) cos(ax)];
%旋转矩阵
R=rx*ry*rz;
%平移矩阵
t=[tx;ty;tz];
K_2respto1=[R t]; %相机2到1的变换
K_1respto2=[R' -R'*t];%相机1到的变换 就是公式推导出来的
%% Step 4 Get the Fundamental matrix analytically as the product of matrices defined in step 3
t_antisym=[0 -tz ty
tz 0 -tx
-ty tx 0]; %反对称矩阵
F1=inv(A2')*R'*t_antisym*inv(A1);%求得是基础矩阵F
F2=inv(A1')*t_antisym*R*inv(A2);
%% Step 5 Define the following set of object points with respect to the
% world coordinate system (orcamera 1 coordinate system)
V(:,1) = [100;-400;2000;1];
V(:,2) = [300;-400;3000;1];
V(:,3) = [500;-400;4000;1];
V(:,4) = [700;-400;2000;1];
V(:,5) = [900;-400;3000;1];
V(:,6) = [100;-50;4000;1];
V(:,7) = [300;-50;2000;1];
V(:,8) = [500;-50;3000;1];
V(:,9) = [700;-50;4000;1];
V(:,10) = [900;-50;2000;1];
V(:,11) = [100;50;3000;1];
V(:,12) = [300;50;4000;1];
V(:,13) = [500;50;2000;1];
V(:,14) = [700;50;3000;1];
V(:,15) = [900;50;4000;1];
V(:,16) = [100;400;2000;1];
V(:,17) = [300;400;3000;1];
V(:,18) = [500;400;4000;1];
V(:,19) = [700;400;2000;1];
V(:,20) = [900;400;3000;1];
%% Step 6 Compute the couples of image points in both image planes by using the matrices of step3
%世界坐标的点转移到图像坐标
intr1=[A1 zeros(3,1)];
intr2=[A2 zeros(3,1)];%相机坐标系到图像坐标系的变化
K_wrespto1=[eye(3) zeros(3,1)];%世界坐标系到相机坐标系的变换
extr1=[K_wrespto1;zeros(1,3) 1];
K_wrespto2=K_1respto2;
extr2=[K_wrespto2;zeros(1,3) 1];
V_I1=point_projection(intr1,extr1,V);
V_I2=point_projection(intr2,extr2,V);
%% Step 7 Open two windows in matlab, which will be used as both image
%planes, and draw the 2D points obtained in step 6
figure(15)
subplot(1,2,1);hold on;
x=[0;256;256;0];y=[0;0;256;256];
fill(x,y,[0.6 0.6 1]);
text(256,256,' image plane I1');
stem(V_I1(1,:),V_I1(2,:),'LineStyle','none','Marker','o','Color','m');
title('Camera 1');
subplot(1,2,2);hold on;
fill(x,y,[1 0.6 1]);
text(256,256,' image plane I2');
stem(V_I2(1,:),V_I2(2,:),'LineStyle','none','Marker','o','Color','r');
title('Camera 2');
figure(1)
draw_object_points(V_I1,V_I2);
%% Step 8. Compute the fundamental matrix by using the 8-point method and least-squares by means of the 2D points obtained in step 6
F1_8pm=eightpointsmethod1(V_I1,V_I2); %在极线配准那里的基础矩阵x2T * F *x1=0;
F1_normal=F1/F1(3,3);
F2_8pm=eightpointsmethod2(V_I1,V_I2);
F2_normal=F2/F2(3,3);
%% Step 9. Compare the step 8 matrix with the one obtained in step 4
different1=F1_8pm-F1_normal;
different2=F2_8pm-F2_normal;
%% Step 10. Draw in the windows of step 7 all the epipolar geometry
figure(1)
subplot(1,2,2);
ylabel('epipolar geometry using LS 8 points method without noise');
%%
%boundary of x-axis
x1_max=600;
x1_min=-600;
x2_max=600;
x2_min=-600;
draw_epipolar_geometry(F1_normal,F2_normal,V_I1,V_I2,x1_min,x1_max,x2_min,x2_max);
draw_focal_point(t,intr1,extr1,intr2,extr2);
%% Step 11 Add some Gaussian noise to the 2D points producing discrepancies
% between the range [-1,+1] pixels for the 95% of points.
mu=0;
sigma=0.5102;
V_I1_noise=V_I1+[normrnd(mu,sigma,2,20);zeros(1,20)];% add Gaussian noise
% to the point
V_I2_noise=V_I2+[normrnd(mu,sigma,2,20);zeros(1,20)];
%% Step 12. Again repeat step 8 up to 10 with the noisy 2D points.
%Compute the fundamental matrix by using the 8-point method and
%least-squares by means of the 2D points with noise obtained in step 11
F1_8pm_noise=eightpointsmethod1(V_I1_noise,V_I2_noise);
F2_8pm_noise=eightpointsmethod2(V_I1_noise,V_I2_noise);
%Draw all the epipolar geometry
figure(2);
subplot(1,2,2);
ylabel('epipolar geometry using LS 8 points method with noise (no uniqueeipiplar)');
%boundary of x-axis
x1_max=700;
x1_min=-700;
x2_max=700;
x2_min=-700;
draw_object_points(V_I1_noise,V_I2_noise);
draw_epipolar_geometry(F1_8pm_noise,F2_8pm_noise,V_I1_noise,V_I2_noise,x1_min,x1_max,x2_min,x2_max);
draw_focal_point(t,intr1,extr1,intr2,extr2);
%compute the closest rank-2 matrix
[Q1,S1,V1] = svd(F1_8pm_noise);
S1(3,3)=0;
F1_8pm_noise_svd=Q1*S1*V1';
[Q2,S2,V2] = svd(F2_8pm_noise);
S2(3,3)=0;
F2_8pm_noise_svd=Q2*S2*V2';
figure(3)
subplot(1,2,2);
ylabel('epipolar geometry using LS 8 points method with noise (uniqueeipiplar)');
x1_max=700;
x1_min=-700;
x2_max=700;
x2_min=-700;
draw_object_points(V_I1_noise,V_I2_noise);
draw_epipolar_geometry(F1_8pm_noise_svd,F2_8pm_noise_svd,V_I1_noise,V_I2_noise,x1_min,x1_max,x2_min,x2_max);
draw_focal_point(t,intr1,extr1,intr2,extr2);
%% Step 13. Increase the Gaussian noise of step 11 (now in the range [-2,+2]
% for the 95% of points) and repeat step 8-12.
mu=0;
sigma=0.5102;
sigma2=2*sigma;
V_I1_noise2=V_I1+[normrnd(mu,sigma2,2,20);zeros(1,20)];% add Gaussian noiseto the point
V_I2_noise2=V_I2+[normrnd(mu,sigma2,2,20);zeros(1,20)];
%Compute the fundamental matrix by using the 8-point method and
%least-squares by means of the 2D points with noise obtained above
F1_8pm_noise2=eightpointsmethod1(V_I1_noise2,V_I2_noise2);
F2_8pm_noise2=eightpointsmethod2(V_I1_noise2,V_I2_noise2);
%Draw all the epipolar geometry
figure(4)
subplot(1,2,2);
ylabel('epipolar geometry using LS 8 points method with noise2 (no uniqueeipiplar)');
%boundary of x-axis
x1_max=700;
x1_min=-700;
x2_max=700;
x2_min=-700;
draw_object_points(V_I1_noise2,V_I2_noise2);
draw_epipolar_geometry(F1_8pm_noise2,F2_8pm_noise2,V_I1_noise2,V_I2_noise2,x1_min,x1_max,x2_min,x2_max);
draw_focal_point(t,intr1,extr1,intr2,extr2);
%compute the closest rank-2 matrix
[Q1,S1,V1] = svd(F1_8pm_noise2);
S1(3,3)=0;
F1_8pm_noise2_svd=Q1*S1*V1';
[Q2,S2,V2] = svd(F2_8pm_noise2);
S2(3,3)=0;
F2_8pm_noise2_svd=Q2*S2*V2';
figure(5)
subplot(1,2,2);
ylabel('epipolar geometry using LS 8 points method with noise2 (uniqueeipiplar)');
x1_max=700;
x1_min=-700;
x2_max=700;
x2_min=-700;
draw_object_points(V_I1_noise2,V_I2_noise2);
draw_epipolar_geometry(F1_8pm_noise2_svd,F2_8pm_noise2_svd,V_I1_noise2,V_I2_noise2,x1_min,x1_max,x2_min,x2_max);
draw_focal_point(t,intr1,extr1,intr2,extr2);
%% Part 2.
%% Step 14. Compute the fundamental matrix by using the 8-point method and SVD from the points 2D obtained in step 6 and without noise.
F1_svd=svdmethod1(V_I1,V_I2);
F2_svd=svdmethod2(V_I1,V_I2);
different3=F1_svd-F1_normal;
different4=F2_svd-F2_normal;
%% Step 15. Repeat step 10 up to 13 (with the matrix of step 14 instead ofstep 8) for points with Gaussian noise
%first Gaussian noise in the rang [-1, 1] for the 95% of points
F1_svd_noise=svdmethod1(V_I1_noise,V_I2_noise);
F2_svd_noise=svdmethod2(V_I1_noise,V_I2_noise);
%Draw all the epipolar geometry
figure(6)
subplot(1,2,2);
ylabel('epipolar geometry using SVD 8 points method with noise (no uniqueeipiplar)');
%boundary of x-axis
x1_max=700;
x1_min=-700;
x2_max=700;
x2_min=-700;
draw_object_points(V_I1_noise,V_I2_noise);
draw_epipolar_geometry(F1_svd_noise,F2_svd_noise,V_I1_noise,V_I2_noise,x1_min,x1_max,x2_min,x2_max);
draw_focal_point(t,intr1,extr1,intr2,extr2);
%compute the closest rank-2 matrix
[Q1,S1,V1] = svd(F1_svd_noise);
S1(3,3)=0;
F1_svd_noise_svd=Q1*S1*V1';
[Q2,S2,V2] = svd(F2_svd_noise);
S2(3,3)=0;
F2_svd_noise_svd=Q2*S2*V2';
figure(7)
subplot(1,2,2);
ylabel('epipolar geometry using SVD 8 points method with noise (uniqueeipiplar)');
x1_max=700;
x1_min=-700;
x2_max=700;
x2_min=-700;
draw_object_points(V_I1_noise,V_I2_noise);
draw_epipolar_geometry(F1_svd_noise_svd,F2_svd_noise_svd,V_I1_noise,V_I2_noise,x1_min,x1_max,x2_min,x2_max);
draw_focal_point(t,intr1,extr1,intr2,extr2);
%second Gaussian noise in the rang [-2, 2] for the 95% of points
F1_svd_noise2=svdmethod1(V_I1_noise2,V_I2_noise2);
F2_svd_noise2=svdmethod2(V_I1_noise2,V_I2_noise2);
%Draw all the epipolar geometry
figure(8)
subplot(1,2,2);
ylabel('epipolar geometry using SVD 8 points method with noise2 (no uniqueeipiplar)');
%boundary of x-axis
x1_max=700;
x1_min=-700;
x2_max=700;
x2_min=-700;
draw_object_points(V_I1_noise2,V_I2_noise2);
draw_epipolar_geometry(F1_svd_noise2,F2_svd_noise2,V_I1_noise2,V_I2_noise2,x1_min,x1_max,x2_min,x2_max);
draw_focal_point(t,intr1,extr1,intr2,extr2);
%compute the closest rank-2 matrix
[Q1,S1,V1] = svd(F1_svd_noise2);
S1(3,3)=0;
F1_svd_noise2_svd=Q1*S1*V1';
[Q2,S2,V2] = svd(F2_svd_noise2);
S2(3,3)=0;
F2_svd_noise2_svd=Q2*S2*V2';
figure(9)
subplot(1,2,2);
ylabel('epipolar geometry using SVD 8 points method with noise2 (uniqueeipiplar)');
x1_max=700;
x1_min=-700;
x2_max=700;
x2_min=-700;
draw_object_points(V_I1_noise2,V_I2_noise2);
draw_epipolar_geometry(F1_svd_noise2_svd,F2_svd_noise2_svd,V_I1_noise2,V_I2_noise2,x1_min,x1_max,x2_min,x2_max);
draw_focal_point(t,intr1,extr1,intr2,extr2);
%% function
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function V_I1=point_projection(intr1,extr1,V)
V_I1=intr1*extr1*V;
V_I1(1,:)=V_I1(1,:)./V_I1(3,:);
V_I1(2,:)=V_I1(2,:)./V_I1(3,:);
V_I1(3,:)=1; %归一化后的处理 让图像上的z坐标为1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function draw_object_points(V_I1,V_I2)
subplot(1,2,1);hold on;
x=[0;256;256;0];y=[0;0;256;256];
fill(x,y,[0.6 0.6 1]);
text(256,256,' image plane I1');
stem(V_I1(1,:),V_I1(2,:),'LineStyle','none','Marker','.','Color','m');
title('Camera 1');
subplot(1,2,2);hold on;
fill(x,y,[1 0.6 1]);
text(256,256,' image plane I2');
stem(V_I2(1,:),V_I2(2,:),'LineStyle','none','Marker','.','Color','r');
title('Camera 2');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function F1_8pm=eightpointsmethod1(V_I1,V_I2)
U1=[V_I1(1,:).*V_I2(1,:)
V_I1(2,:).*V_I2(1,:)
V_I2(1,:)
V_I1(1,:).*V_I2(2,:)
V_I1(2,:).*V_I2(2,:)
V_I2(2,:)
V_I1(1,:)
V_I1(2,:)]';
F1_8pm=-inv(U1'*U1)*U1'*ones(20,1);
F1_8pm=[F1_8pm(1) F1_8pm(2) F1_8pm(3)
F1_8pm(4) F1_8pm(5) F1_8pm(6)
F1_8pm(7) F1_8pm(8) 1];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function F2_8pm=eightpointsmethod2(V_I1,V_I2)
U2=[V_I2(1,:).*V_I1(1,:)
V_I2(2,:).*V_I1(1,:)
V_I1(1,:)
V_I2(1,:).*V_I1(2,:)
V_I2(2,:).*V_I1(2,:)
V_I1(2,:)
V_I2(1,:)
V_I2(2,:)]';
F2_8pm=-inv(U2'*U2)*U2'*ones(20,1);
F2_8pm=[F2_8pm(1) F2_8pm(2) F2_8pm(3)
F2_8pm(4) F2_8pm(5) F2_8pm(6)
F2_8pm(7) F2_8pm(8) 1];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function draw_epipolar_geometry(F1,F2,V_I1,V_I2,x1_min,x1_max,x2_min,x2_max)
l2=F1*V_I1;
%极线方程
l1=F2*V_I2; %由基础矩阵得到的就是极线方程 就是极线在另一个平面上的极线
x1=x1_min:x1_max;
x2=x2_min:x2_max;
m1=-l1(1,:)./l1(2,:);
d1=-l1(3,:)./l1(2,:);
m2=-l2(1,:)./l2(2,:);
d2=-l2(3,:)./l2(2,:);
for i=1:20
y1(i,:)=m1(i)*x1+d1(i);
y2(i,:)=m2(i)*x2+d2(i);
subplot(1,2,1);hold on;
plot(x1,y1(i,:),'Color','b');
xlim([x1_min x1_max]);
ylim([x1_min x1_max]);
axis square;
subplot(1,2,2);hold on;
plot(x2,y2(i,:),'Color','m');
axis square;
xlim([x2_min x2_max]);
ylim([x2_min x2_max]);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function draw_focal_point(t,intr1,extr1,intr2,extr2)
oc2_resptow=[t;1];
oc2_I1=intr1*extr1*oc2_resptow;
oc2_I1=oc2_I1/oc2_I1(3);
oc1_resptow=[0;0;0;1];
oc1_I2=intr2*extr2*oc1_resptow;
oc1_I2=oc1_I2/oc1_I2(3);
subplot(1,2,1);hold on;
stem(oc2_I1(1),oc2_I1(2),'LineStyle','none','Color','r');
subplot(1,2,2);hold on;
stem(oc1_I2(1),oc1_I2(2),'LineStyle','none','Color','b');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function F1_svd=svdmethod1(V_I1,V_I2)
U1_svd=[V_I1(1,:).*V_I2(1,:)
V_I1(2,:).*V_I2(1,:)
V_I2(1,:)
V_I1(1,:).*V_I2(2,:)
V_I1(2,:).*V_I2(2,:)
V_I2(2,:)
V_I1(1,:)
V_I1(2,:)
ones(1,20)]';
[Q3,S3,V3] = svd(U1_svd);
F1_svd=V3(:,9);
F1_svd=F1_svd/F1_svd(9);
F1_svd=[F1_svd(1) F1_svd(2) F1_svd(3)
F1_svd(4) F1_svd(5) F1_svd(6)
F1_svd(7) F1_svd(8) F1_svd(9)];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function F2_svd=svdmethod2(V_I1,V_I2)
U2_svd=[V_I2(1,:).*V_I1(1,:)
V_I2(2,:).*V_I1(1,:)
V_I1(1,:)
V_I2(1,:).*V_I1(2,:)
V_I2(2,:).*V_I1(2,:)
V_I1(2,:)
V_I2(1,:)
V_I2(2,:)
ones(1,20)]';
[Q4,S4,V4] = svd(U2_svd);
F2_svd=V4(:,9);
F2_svd=F2_svd/F2_svd(9);
F2_svd=[F2_svd(1) F2_svd(2) F2_svd(3)
F2_svd(4) F2_svd(5) F2_svd(6)
F2_svd(7) F2_svd(8) F2_svd(9)];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
</span>
