Feature Extraction & Image Processing, 2nd Edition

10.6 Example of camera models

This section illustrates the mapping of points into an image frame for the per-

spective and affine camera models. Code 10.1 contains the functions used to create

%***********************************************

%Draw a set of points

%-----------------------------------------------

function DrawPoints(P,colour)

%-----------------------------------------------

[r,c]=size(P);

plot3(P(1,1:c),P(2,1:c),P(3,1:c),'+','color',colour);

%***********************************************

%Draw a set of Image points

%----------------------------------------------

function DrawImagePoints(P,C,colour)

%----------------------------------------------

[r,c]=size(P);

for column=1:c

P(1:r-1,column)=P(1:r-1,column)/P(3,column);

end

plot(P(1,1:c),P(2,1:c),'+','color',colour);

axis([0 C(10) 0 C(11)]);

%***********************************************

%Draw a coordinate frame

%-----------------------------------------------

function DrawWorldFrame(x0,x1,y0,y1,z0,z1);

%-----------------------------------------------

axis equal; %same aspect ratio

axis([x0,x1,y0,y1,z0,z1]);

xlabel('X','FontSize',14);

ylabel('Y','FontSize',14);

zlabel('Z','FontSize',14);

grid on;

hold on;

%***********************************************

%Draw an image plane

%-----------------------------------------------

function DrawImagePlane(C,dx,dy);

%-----------------------------------------------

Appendix 2: Camera geometry fundamentals 371

%CW: Camera to world transformation

CW=CameraToWorld(C);

%transform coordinates to world coordinates

P(:,1)=CW*p(:,1);

P(:,2)=CW*p(:,2);

P(:,3)=CW*p(:,3);

P(:,4)=CW*p(:,4);

%Draw image plane

patch(P(1,:),P(2,:),P(3,:),[.9,.9,1]);

%*************************************************

%Draw a line between optical centre and 3D points

%

%-----------------------------------------------------

function DrawPerspectiveProjectionLines (o,P,colour);

%-----------------------------------------------------

[r,c]=size(P);

for i=1:c

plot3([o(1) P(1,i)],[o(2) P(2,i)],[o(3) P(3,i)],'color',colour);

%optic centre

end;

%***********************************************

%Draw a line between image plane and 3D points

%

%-----------------------------------------------

function DrawAffineProjectionLines(z,P,colour);

%-----------------------------------------------

[r,c]=size(P);

for column=1:c

plot3([P(1,column) P(1,column)],[P(2,column) P(2,column)],[z(3)

P(3,column)],'color',colour); %optic centre

end;

%coordinates of 4 points on the

%image plane (to draw a rectangle)

p(1,1)=-dx/2; p(2,1)=-dy/2; p(3,1)=C(7); p(4,1)=1;

p(1,2)=-dx/2; p(2,2)=+dy/2; p(3,2)=C(7); p(4,2)=1;

p(1,3)=+dx/2; p(2,3)=+dy/2; p(3,3)=C(7); p(4,3)=1;

p(1,4)=+dx/2; p(2,4)=-dy/2; p(3,4)=C(7); p(4,4)=1;

%Draw camera origin

plot3(C(1),C(2),C(3),'o'); %optic centre

plot3(C(1),C(2),C(3),'+'); %optic centre

Code 10.1 Drawing functions

372 Feature Extraction and Image Processing

figures: DrawPoints, DrawImagePoints, DrawWorldFrame, DrawImagePlane,

DrawPerspectiveProjectionLines and DrawAffineProjectionLines.

The function DrawPoints draws a set of points given its three world coordinates. In our

example, it is used to draw points in the world. The function DrawImagePoints draws points

on the image plane. It uses homogeneous coordinates, so it implements Equation 10.3. The

function DrawWorldFrame draws the three axes given the location of the origin. This is used to

illustrate the world frame and the origin is always set to zero. The function DrawImagePlane

draws a rectangle that represents the image plane. It also draws a point to illustrate the location

of the centre of the camera. In our examples, we assume that the centre of the camera is at the

position of the focal point, for both the perspective and the affine model.

The functions DrawPerspectiveProjectionLines and DrawAffine

ProjectionLines are used to illustrate the projection of points into the image plane for

the perspective and affine projections, respectively. These functions take a vector of points and

trace a line to the image plane that represent the projection. For perspective, the lines intersects

the focal point and for affine they are projected parallel to the image plane.

In the example implementation, we group the camera parameters in the vector

C =x0y0z0abgfu0v0 kx ky. The first six elements define the location and rotation

parameters. The value of f defines the focal length. The remaining parameters define the loca-

tion of the optical centre and the pixel size. For simplicity, we assume that there is no skew.

Code 10.2 contains the functions that compute camera transformations from camera parameters.

The function CameraToWorld computes a matrix that defines the position of the camera.

%***********************************************

%Compute matrix that transforms coordinates

%in the camera frame to the world frame%

%-----------------------------------------------

function CW=CameraToWorld(C);

%-----------------------------------------------

%rotation

Rx=[cos(C(6)) sin(C(6)) 0

-sin(C(6)) cos(C(6)) 0

0 0 1];

Ry=[cos(C(5)) 0 sin(C(5))

0 1 0

-sin(C(5)) 0 cos(C(5))];

Rz=[1 0 0

0 cos(C(4)) sin(C(4))

0 -sin(C(4)) cos(C(4))];

%translation

T=[C(1) C(2) C(3)]';

%transformation

CW=(Rz*Ry*Rx);

CW(:,4)=T;

Appendix 2: Camera geometry fundamentals 373

%***********************************************

%Convert from homogeneous coordinates in pixels

%to distance coordinates in the camera frame

%--------------------------------------------

function p=ImageToCamera(P,C);

%--------------------------------------------

%inverse of K

Ki=[1/C(10) 0 -C(8)/C(10)

0 1/C(11) -C(9)/C(11)

0 0 1 ];

%coordinate in distance units

p=Ki*P;

%coordinates in the image plane

p(1,:)=p(1,:)./p(3,:);

p(2,:)=p(2,:)./p(3,:);

p(3,:)=p(3,:)./p(3,:);

%the third coordinate gives the depth

%the focal length C(7) defines depth

p(3,:)=p(3,:).*C(7);

%include homogeneous coordinates

p(4,:)=p(1,:)/p(1,:);

Rz=[1 0 0

0 cos(C(4)) -sin(C(4))

0 sin(C(4)) cos(C(4))];

T=[-c(1) -c(2) -c(3)]';

%transformation

WC=(Rz*Ry*Rx); %rotation inverse

Tp=WC*T; %translation

WC(:,4)=Tp; %compose homogeneous form

%***********************************************

%Compute matrix that transforms coordinates

%in the world frame to the camera frame

%----------------------------------------------

function WC=WorldToCamera(c);

%----------------------------------------------

%translation T'=-R'T

%for R'=inverse rotation

Rx=[cos(C(6)) -sin(C(6)) 0

sin(C(6)) cos(C(6)) 0

0 0 1];

Ry=[cos(C(5)) 0 –sin(C(5))

0 1 0

sin(C(5)) 0 cos(C(5))];

Code 10.2 Transformation function

374 Feature Extraction and Image Processing