International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
homogeneous) coordinates of the vanishing points. On the basis
of this equation, simultaneous adjustment of observed points on
lines imaged in each view allows to estimate the interior orien-
tation elements of a ‘normal’ camera (camera constant, princi-
pal point) along with the radial symmetric lens distortion coeffi-
cients and, optionally, the image aspect ratio. Our approach pre-
sents certain similarities with that of Wilczkowiak et al. (2002),
also a multi-view camera calibration approach not depending on
image-to-image correspondences. A notable difference rests in
the implicit involvement of vanishing points through the projec-
tion of parallelepipeds on the images, which apparently restricts
the applications to scenes containing such primitives.
The presented algorithm has been successfully applied to both
simulated and real data, and the outcome has been compared to
a rigorous photogrammetric bundle adjustment and to plane-ba-
sed calibration.
2. GEOMETRIC FORMULATION
2.1 The normal camera
As mentioned above, Gracie (1968) has given the six necessary
equations relating the three interior orientation elements (came-
ra constant, principal point) and image rotations (c, q, K) to the
vanishing points of three orthogonal directions. This is illustra-
ted in Fig. 1, where O is the projection centre, c the camera con-
stant and P(x,, y,) the principal point, with Vx, Vy, V7 being the
respective vanishing points of the three orthogonal space direc-
tions X, Y, Z. The principal point P, namely the projection of O
onto the image plane, is actually the orthocentre of the triangle
VXV«,Vz. Of course, the directions of the lines OV,, OVy, OV; are
respectively parallel to the X, Y, Z space axes (and consequently
they are mutually orthogonal).
Figure 1. Image geometry with three vanishing points.
As mentioned already, in case two orthogonal vanishing points
V;, V; are known on the image plane while the third remains un-
known, estimation of the interior orientation is feasible only if a
fixed principal point can be assumed (usually the image centre).
Hence, with only two vanishing points, the possible locations of
the projection centre are obviously infinite. Yet, a constraint is
always present: the image rays OV, and OV; form a right angle
for all possible locations of O. This bounds the projection centre
onto a geometrical locus, encompassing all points which see the
two vanishing points under a right angle. Therefore, all possible
locations of O in the 3D image space form a sphere (named here
a ‘calibration sphere’) of radius R, with the middle M of line seg-
ment V, V, as centre and the distance V,V, as its diameter. Every
point on this sphere represents a possible projection centre; the
camera constant c equals then its distance from the image plane,
while the principal point P is its projection onto it (see Fig. 2).
calibration
sphere
image
plane
Figure 2. Calibration sphere as locus of'the projection centre,
principal point locus (ppl) and isocentre circle (icc)
The analytical equation of the sphere can be written as:
2 2 5
(Xo 7 Xn) *(yo 7 Ym) +¢2=R (1)
4 X1 +X d yo
with Xm ET Ym a
Qu -x2Y +(yı-y2)
2
and p.d:
2
whereby (x, yi), (X», yz) are the two vanishing points, (Xn, Ym)
is the centre of the sphere, R its radius and d its diameter. Every
pair of orthogonal vanishing points gives one such Eq. (1). Two
pairs of such vanishing points, from the same or from different
images sharing identical internal parameters, define a circle (the
intersection of tite two spheres) as the locus for the projection
centre, whereas a third pair — i.e. a third sphere — can fully cali-
brate the camera. In Fig. 3 the definition of the projection centre
as intersection of three calibration spheres is illustrated. In actual
fact, there exist two intersection points: ‘above’ and ‘below’ the
image frame. This ambiguity is, of course, removed by kecping
the point with c > 0 (point ‘above’ the image). In this sense one
should speak of ‘calibration semi-spheres” rather than spheres.
Figure 3. Projection centre O as intersection of three calibration
spheres and principal point P as its projection on the image.
An equation equivalent to Eq. (1) is:
(So = An) (yo yd. =R?-¢? (2)
which describes a circle on the image plane with centre (xa, yn)
and the radius (R? — cy. This circle is the locus of the principal
point P for a fixed camera constant c. In fact, as seen in Fig. 2, a
fixed c constrains the projection centre on a circle of the sphere,
which, being an intersection of the sphere with a plane parallel
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