Contents

Build a simple camera matrix

(assuming a 640x480 sensor)

K=[240 0 320; 0 240 240; 0 0 1];
P=[K,[0;0;0]];

Consider several points in homogeneous coordinates

A=[+1,+1,1,1]';
B=[+1,+1,2,1]';
C=[+1,+1,3,1]';
%D=[+1,+1,10000,1]'; % A point at the infinity! (try it)
D=[+1,+1,4,1]';

... and project them

a=P*A;
b=P*B;
c=P*C;
d=P*D;

a=a/a(3);
b=b/b(3);
c=c/c(3);
d=d/d(3);

Plot results

plot(a(1),a(2),'o', 'MarkerEdgeColor','k',...
                'MarkerFaceColor','g',...
                'MarkerSize',10);
hold on;
plot(b(1),b(2),'o', 'MarkerEdgeColor','k',...
                'MarkerFaceColor','g',...
                'MarkerSize',10);
plot(c(1),c(2),'o', 'MarkerEdgeColor','k',...
                'MarkerFaceColor','g',...
                'MarkerSize',10);
plot(d(1),d(2),'o', 'MarkerEdgeColor','k',...
                'MarkerFaceColor','g',...
                'MarkerSize',10);
axis([0 640 0 480]); drawnow;

Verify cross ratio properties:

first compute distances in real world ...

AB=norm(A-B);
AC=norm(A-C);
AD=norm(A-D);
BC=norm(B-C);
BD=norm(B-D);
CD=norm(C-D);

... then distances in the image.

ab=norm(a-b);
ac=norm(a-c);
ad=norm(a-d);
bc=norm(b-c);
bd=norm(b-d);
cd=norm(c-d);

now check that the cross ratio is invariant to projection

(bc/ac)/(bd/ad)
(BC/AC)/(BD/AD)

ans =

    0.7500


ans =

    0.7500

Animate the moving point!

(run the code below)

% clf; hold on;
% axis([0 640 0 480]);
% for t=linspace(0,10,300)
%     A=[1-t,1,t,1]';
%     a=P*A;
%     a=a/a(3);
%     B=[1-t,-1,t,1]';
%     b=P*B;
%     b=b/b(3);
%
%     plot(a(1),a(2),'o', 'MarkerEdgeColor','k',...
%                     'MarkerFaceColor','g',...
%                     'MarkerSize',10);
%     plot(b(1),b(2),'o', 'MarkerEdgeColor','k',...
%                     'MarkerFaceColor','g',...
%                     'MarkerSize',10);
%     drawnow;
% end
%
% % what happens when t=linspace(20,-20,1200) ?
% % what happens when A=[+1+t,+1,t,1]';
%
%

Published with MATLAB® 7.8