A simple geometrical construction with 2D homogeneous coordinates

Our goal is to complete the drawing of a cuboid. We use need six vertices on the contour of the cuboid.

Data input
Finding and plotting vanishing points
Finding unknown points
Drawing
Rectify a face

Data input

im=imread('bluecube.jpg');
imshow(im);
hold on;
[x y]=getpts;
plot(x,y,'or','MarkerSize',12);
a=[x(1) y(1) 1]';
b=[x(2) y(2) 1]';
f=[x(3) y(3) 1]';
h=[x(4) y(4) 1]';
g=[x(5) y(5) 1]';
c=[x(6) y(6) 1]';

Warning: Image is too big to fit on screen;
displaying at 50%

Finding and plotting vanishing points

vab=cross(cross(a,b),cross(h,g));
vac=cross(cross(a,c),cross(f,h));
vae=cross(cross(b,f),cross(c,g));

vab=vab/vab(3);
vac=vac/vac(3);
vae=vae/vae(3);
plot(vab(1),vab(2),'x');
plot(vac(1),vac(2),'x');
plot(vae(1),vae(2),'x');

Finding unknown points

d=cross(cross(b,vac),cross(c,vab));
d=d/d(3);
e=cross(cross(g,vac),cross(f,vab));
e=e/e(3);

Drawing

we can now finally draw the cube.

myline=[a';b';d';c';a'];
line(myline(:,1),myline(:,2),'LineWidth',5);
myline=[e';f';h';g';e'];
line(myline(:,1),myline(:,2),'LineWidth',5);
myline=[a';e'];
line(myline(:,1),myline(:,2),'LineWidth',5);
myline=[b';f'];
line(myline(:,1),myline(:,2),'LineWidth',5);
myline=[c';g'];
line(myline(:,1),myline(:,2),'LineWidth',5);
myline=[d';h'];
line(myline(:,1),myline(:,2),'LineWidth',5);

Rectify a face

We can also transform the image so that a face becomes a square.

T = maketform('projective',[a(1:2)';b(1:2)';f(1:2)';e(1:2)'],[0 0;300,0;300,300;0,300]);
figure, imshow(imtransform(im,T));

Warning: Image is too big to fit on screen;
displaying at 67%

Such transformation is called homography, and will be presented later during the Image Analysis and Synthesis course. Still, it is a linear transformation, so it has the same form of the simpler 2D transformations (translations and rotations) which have already been introduced, a 3x3 matrix:

T.tdata.T

ans =

    0.4007    0.1674   -0.0001
   -0.2914    0.4613    0.0001
  -32.2264 -189.8333    1.0000