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ISPRS Commission III, Vol.34, Part 3A , Photogrammetric Computer Vision“, Graz, 2002 
  
  
  
Figure 1: Image of a high voltage switch gear and the as- 
sociated drawing. Many horizontal and vertical structures 
occur in the image, which correspond to lines and points in 
the drawing. 
the measured image and map quantities and to validate the 
achieved accuracy of the estimated projection matrix P. 
The result of such an image-drawing orientation may be 
used in augmented reality systems or just as approximate 
values for a subsequent rigorous bundle adjustment using 
all available images. 
The remainder of this paper is organized as follows: After 
introducing notation, geometric relations and statistical es- 
sentials in section 2, we describe the scene constraints in 
terms of projective geometry in section 3. In section 4 we 
provide both an algebraic solution for P and a statistically 
optimal one, along with statistical tools for self-diagnosis. 
Experiments on synthetic and real data are presented in 
section 5. We finally conclude in section 6. 
2 MATHEMATICAL FOUNDATIONS 
First, we briefly describe the represention of geometric en- 
tities, the assumed representation for uncertain geomet- 
ric entities, basic error propagation, estimation and testing 
techniques used later in the paper. 
Notation: We denote coordinate vectors of planar geo- 
metric objects with small bold face letters, e. g. x, in 3D 
space with capital bold face letters, e. g. .X. Vectors and 
matrices are denoted with slanted letters, thus x or R. Ho- 
mogeneous vectors and matrices are denoted with upright 
letters, e. g. x or A. We use the skew matrix S(x) = [x]x 
of a 3-vector x in order to represent the cross product by 
a x b = S(a)b = —b x a = —S(b)a. We will use the 
vec-operator collecting the columns of a matrix into a vec- 
tor, thus vecA — vec(a,, a», ...) — (al, al, ...)', and use 
the relation 
vec(ABC) = (A ® C')vec(B') 2 (C' & A)vecB. (1) 
with the Kronecker product A & B — {a;;B}. 
2.1 Geometry 
We mainly follow the representation of projective geome- 
try as introduced in (Hartley and Zisserman, 2000). Hence 
we represent points and lines both in 2D and 3D by homo- 
geneous vectors. We write 2D points as x = (u,v,w)" 
and 2D lines as 1 = (a,b,c). Lines in 2D can easily be 
constructed from two points by 1 — x x y and the incidence 
constraint of a point and a line is given by x'1 = 0. We 
only need points in 3D, they are represented analogously 
by X = (U,V, W, T). 
We can write the projection of a 3D point X into an image 
point x image plane as 
P1 
x=PX=| P! |X=0:3X")p (2) 
pr 
3 
with p — vec(P! ) 2 (P,, P5, P3), 
where P is the 3 x 4 projection matrix, PI its rows, and 
p = vec(P") its vector version. Note that due to its homo- 
geneity, only 11 of its elements are independent. 
Analogically, we can write the parallel projection yielding 
the drawing (cf. fig. 1) 
x PUE. with. (P^- 
SO 
© — © 
Ceo c. c 
m CQ 
inducing xz? — (x, y)! .The task is to derive P from ob- 
served points x and x^ in the image and the drawing and 
further constraints from the interpretaion of the drawing. 
Straight lines 1 in the image always are assumed to be de- 
rived by two measured points x and y, the same holds for 
lines in the drawing. 
2.2 Statistics 
Estimation with linear constraints: All constraints will 
have the form 
A(y)p=e=0 
where À is a matrix depending on measurements collected 
in the vector of observations y, and p is the vector of the 
unknown elements of the projection matrix, appearing lin- 
ear in the constraints.