ticular we shall show that a necessary and sufficient condition 
for the system to be singular is for the configuration to belong 
to one of two classes: configurations in which the optical axes 
and baseline are coplanar; and configurations in which there 
is coplanarity of one optical axis, the baseline and the vector 
perpendicular to both the baseline and the other optical axis. 
The paper is organised as follows. The next section defines 
the notation, sets out the geometry of the problem and re- 
views some basic concepts. The third section shows that 
self-calibration can be reduced to solving a linear system from 
whose solution the focal lengths can be easily calculated. The 
fourth section derives conditions under which the linear sys- 
tem is singular (so no unique solution exists), and shows that 
these correspond to the geometric configurations described 
in the previous paragraph. Finally the last section presents 
a solution for the special case when both focal lengths are 
known a priori to be equal: here, the unknown focal length 
can be read off from the roots of a quadratic. 
2 PROBLEM FORMULATION 
The following notation will be used in the paper. World points 
are written in upper case, image points in lower case, vectors 
in bold lower case, and matrices in bold upper case. a; de- 
notes the j-th column of the corresponding matrix A, while 
Aj; denotes the 27-th element. In particular, I denotes the 
3 x 3 identity matrix and i; denotes the j-th column of I, 
i.e. the unit vector with a one in the j-th position and zeroes 
elsewhere. (Note that by definition a; — Ai;.) The sym- 
bol 7 denotes matrix and vector transpose, while 77 denotes 
the transpose of the inverse matrix. The notation u + v 
(U ~ V) indicates that the vectors u and v (matrices U 
and V) are the same up to an arbitrary scale factor. Finally 
entities related to the right image are marked with a ' 
We now define the geometry of the model problem. Figure 1 
shows a scene being stereoscopically imaged. The global ref- 
erence system O' — X'Y'Z' is taken to coincide with the 
coordinate system of the right image, with the origin at the 
optical centre. R = R(a,/,y) denotes the matrix associ- 
ated with a rotation of angles o, 3, y about the X"-, Y"-, 
and Z'-axes, respectively, that renders the left image parallel 
to the right image. b — (b,,b,, b;)7 denotes the baseline 
vector connecting the optical centres of the cameras. The 
parameters o, B, v, b,, by, b; are said to specify the relative 
orientation of a stereo pair of images. Since the scene can 
only reconstructed to within an overall scale factor, it is usual 
to remove this ambiguity by assuming that ||b|| — 1, leav- 
ing 5 independent parameters to be determined in relative 
orientation. 
We next assume that the location of the principal point (the 
intersection of the optical axis with the image plane) is known 
in each image, and that the image coordinate system is Eu- 
clidean (i.e. no skewness or differing scales on different axes). 
In this case a simple translation of all image points will en- 
sure that the principal points coincide with the origins in the 
image plane coordinates. Thus the only unknown intrinsic 
parameter associated with the formation of each image is the 
focal length. 
Now let M be a visible point in the scene, and m = (z,y)” 
and m' = (z',y')" be its projections onto the left and right 
image planes. Relative to the global reference system O' — 
X'Y'"Z' at C' and the coordinate system O — XY Z at C, 
the points m' and m can be expressed in vector form as 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
  
Figure 1: Stereo camera setup. 
q' — (z',y', —f")* and q « (z,y, —f)*. It is clear from 
Figure 1 that the three vectors q, q', and b are coplanar. This 
relationship is encapsulated in the epipolar equation which 
relates corresponding image points by 
q'-(b x Rq) - 0. (1) 
Let f > 0 and f' » 0 be the two focal lengths, A and A’ 
be the intrinsic parameter matrices of the two cameras, and 
B be the skew-symmetric matrix containing elements of b. 
Then under the above assumptions these matrices have the 
form 
1a 0 0 1240 0 
A= 150,1 0 Az i0 71 0 
00° 1/4 0550 V2 f^ 
0 — by 
B = b. 0 —bz , (2) 
—b, b 0 
and (1) may be expressed in matrix form as 
q' Eq - 0, (3) 
where E is the essential matrix given by 
E = BR. (4) 
Now let m' — (z', y', 1)? and m — (z, y, 1)? be alternative 
representations of the image points in homogeneous coordi- 
nates. Observe that 
q= A7'm (5) 
q' = A'”'m'. (6) 
We may now immediately infer that 
m' Fm = 0, (7) 
where F is the fundamental matrix [2] embodying both ex- 
trinsic and intrinsic imaging parameters, and is given by 
F=A""BRA™. (8) 
   
If suffi 
the tw 
Fest. * 
up to 
sectior 
so tha! 
of the 
we sha 
a suffi 
that (€ 
Thuss 
various 
In this 
efficien 
matrix 
values 
calling 
given i 
Propo: 
(i) 
(ii) | 
(iii) 
We ne: 
of F (s 
Here U 
matrix 
singula 
01,02: 
It also 
for whi 
that 
so it fo 
We ma 
tion 1 ! 
Multipl 
and the