(n 
e 
Wolfgang Fórstner 
Figure 1: shows the geometric situation for one, two and three images 
X 
to 
    
C cs Ty 1 8 Ty c0. 0 
K]-0 slm) yk ll 00irm y 0 0.0 (2) 
0 0 1 0 0 1 0 0 1 
; It is an upper diagonal matrix and can be arbitrarily scaled, if no interpretation of its elements is required. Observe the part 
R(I| — X,) transforms the object coordinates into the camera system, the second factor Diag(c, c, 1) of the calibration 
matrix performs the projection and the first factor the calibration. The projection matrix in general has rank 3 and its null 
space is the projection centre as PX, = 0. Therefore the three row vectors 1, 2 and 3 of the projection matrix P can be 
interpreted as the parameters of planes. The vector 1 is a plane through the line ' 2 0asu! — 1-X —« 1, X »—0,VX 
and passes through the projection centre. Similarily 2 is a plane through v' — 0, and 3 is the focal plane parallel to the 
image plane, as then w' — 3-X = 0. The three planes intersect in the projection centre: X, — 12783. 
32 LINES 
A similar projection relation holds for 3D lines. The image line l 2 x' ^ y! of a3D line L — X ^ Y can be expressed as 
a function of the images x' — PX — (1-X,2-X,3-X)' and y' = PY = (1-Y,2-Y,3-Y)" of two object points X and 
Y, namely I' z x' x y' or 
Y= (dV, dV =(1-X 2X 3X) x(1Y,2Y 3Y) 
This expression can be simplified. E. g. the first element a’ is a’ = (2-X)(3-Y) — (2-Y)(3-X) — XxT(23T —32"1Y = 
(X AY)N(2N3) = (2N3)-L. Similarily we obtain expressions for b' and c'. We therefore obtain the direct linear 
transformation of 3D lines 
rzPL ^o Yz(d23$)!L-(üL2L3L! ^ wn |P-(2n3,n1,10)2) 
with a 3 x 6 projection matrix P: Its three rows are 6-vectors representing 3D lines, namely the intersections of the 
principle planes, thus the three coordinate axes of the camera system. 
3.3 INVERSION 
Inversion of the projection leads to projection rays L’ for image points x’ and projection planes A’ for image lines I' 
L'zPx-w2n34v3n01«wiln2 A' 2 P'l'—o14 024 c3 
The expression for L' results from the incidence relation x'! l' — 0 for all lines I" passing through x’, leading to (x'' P) L 
= < L',L >= 0. The expression for A' results from the incidence relation Y'x' — 0 for all points x' on the line ', 
leading to (i! P) X —« A', X »— 0. The projection ray and the projection plane can be expressed as a function of the 
Lon a 
3D point and the 3D line resp. showing the concatenated matrix PP only to depend on the projection centre 
AZ" T -T 
DL -PPXeX.AXAGQX. o A GGP'PLZXOALZA(XJL. with PP=AX) 
4 ONEIMAGE 
4.1 OBSERVATION EQUATIONS AND CONSTRAINTS 
We now easily can write down the observation equations for points in one image, i. e. the collinearity equations 
j uw 51-X QU. X 
I= = mm 
u! 7 3-X y uw "3X 
  
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 299