angle o, the roll angle y and the height Z of the camera 
(cf. fig. 1). Since the azimuth 3 of the viewing direction 
is at our disposal, the rotation matrix from the object to 
the camera system reads R = Rz (y) - Rx (1/2 + à) and 
with the normal to the plane in the object coordinate sys- 
tem E = (0,0,1)" the normal in the camera coordinate 
systems becomes 
n=RE = (nx,ny,nz)'. (1) 
The relationships between the normal and the angles is 
nz nx 
«a = arctan , 7 = —arctan | — (2) 
ny ny 
and n! = N (tan(—y), 1, tan(a)). 
  
Without loss of generality, the projection center Z = 
(0,0. Z)" is chosen. The origin of the object coordinate 
system lies in the reference plane, the Z-axis runs through 
the projection center of the camera. The Y-axis is defined 
by the projection of the optical axis onto the plane, the X- 
axis is perpendicular to both (cf. fig. 1). 
  
Figure 1: shows the definition of the involved coordinate 
systems and the projection of a height into the image. 
Camera Model. For the camera a straight line preserv- 
ing pinhole model is introduced with the principal distance 
c, the scale factor m, the shear s and the principal point 
(2, yo) as the intrinsic camera parameters. With the ho- 
mogeneous calibration matrix 
¢ sc a 
K= | 0 me vw 
0 ^0 1 
the homogeneous 3 x 4-projection matrix P = KR(h|-Z) 
projects an object point X ; into the image point x; via the 
linear transformation x; = PX. 
With the presented approach and a camera in general posi- 
tion, two of the five intrinsic parameters can be determined 
— preferably the principal distance and the scale factor. 
Therefore, initially the used calibration matrix has diago- 
nal shape: 
K = Diag(c, me, 1). 
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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
Observations. For each object the four coordinates x}, 
Yi» yp, and y; (bottom, top) of the foot and head points are 
available as observations. 
2.2 Concept of the Virtual Camera 
The mapping of an object foot pointay «(5 ur into 
the corresponding head point g/ - (x y) can be ex- 
pressed by the projective transformation 
x” = Hx], (3) 
called a homography with eight independent parameters 
due to the homogenity. The 3 x 3-transformation matrix 
H is constant for objects of equal height and can be deter- 
mined by four point correspondences. In the following we 
show how H can be expressed as a function of the unknown 
parameters and how a given transformation matrix can be 
decomposed accordingly. 
2.2.1 Virtual Homography. With the notion of corre- 
sponding head and foot points being identical in space, the 
situation can also be described with the help of a second 
virtual camera (cf. fig. 2). A plane induced homography 
results from the images of two cameras observing the same 
object on a plane. With the calibration matrices K' and K" 
of these two cameras, the distance Z of the first camera to 
the plane and the baseline vector £ the homography reads 
HzK" (e T z^) Ki. (4) 
with the rotation matrix R” of the second camera in re- 
spect to the first camera coordinate system; cf. (Faugeras 
and Lustman, 1988) or (Hartley and Zisserman, 2000) for 
an alternative derivation. The term in brackets is called the 
motion matrix M. In our case we have one camera observ- 
ing the scene from two altitudes with unchanged viewing 
direction, thus K° = K” = K, R" — Hl, and t — Hn 
(cf. fig. 2). The homography (4) becomes 
HK C = 7m) Kt (5) 
Observe that the baseline length ||¢|| is identical to the ob- 
ject height H. 
The transformation (5) is a so-called planar homology 
(Hartley and Zisserman, 2000, p. 585) since with the hori- 
zon line I' = Kn and the vanishing point v = Kn 
— normally the nadir — equation (5) reads 
al 
Hz Hh + (pn — By 
with (y — 1) 2 H/(Zv''Y). The planar homology has 
five degrees of freedom — the vertex v (2 dof), the axis 
l' (2 dof) and the characteristic ratio jj (Semple and Knee- 
bone, 1952) and can therefore be determined by 2.5 point 
correspondences. 
As H contains 5 dof, we can determine two intrinsic pa- 
rameters in addition to the three parameters a, y and Z of 
   
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