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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
9 
Q, 
  
  
  
  
X" e 
H 
X' m > 
H 
04 
Z 
X = X” ~ 
Figure 2: shows the true configuration (top) and the equiv- 
alent situation with a second virtual camera (bottom). In 
both cases the observations of the object foot and head 
points X' and X" are identical. 
the exterior orientation. With the special calibration matrix 
K — Diag(c, mc, 1) the planar homology explicitly reads 
HnX-Z Hnxny cHnxnz 
7 Z 
  
  
  
7 dm : 
Hz m Hnxny Hny —Z mcHnymnz . (6) 
y Z 7 
Hnxnz Hnyng Hnz-Z 
Zc Zmc Z 
Observe that for m = 1 the relation H,9 = H», holds. In 
many practical cases the roll angle y equals zero, so that 
nx = 0 holds and the homography (6) becomes 
—l 0 0 
Hni-—Z cemHnyng 
  
Z f£ 
Hnynz Hnz — 2 
Zme Z 
  
0 
as a common specialization. In this case only the principal 
distance or the scale factor is determinable. 
2.2.2 Decomposition of H. Parameter estimation re- 
quires approximation values for the unknown calibration 
parameters. These values can be deduced by a direct es- 
timation of the eight parameters of the common homogra- 
phy (3), if a real-valued decomposition according to (5) is 
available. 
(1) Intrinsic camera parameters. From eq. (6) the prin- 
cipal distance and the scale difference are c = / H13/ H3 
and m = 4/H21/H12,, but for the frequent case of the 
roll angle 4 — 0 the element H3; becomes zero. In this 
case the principal distance must be computed via c — 
V H3;/ H3; /m with the known scale factor m. 
(2) Exterior orientation. Once the intrinsic parameters 
c and, where applicable, m have been determined, the mo- 
tion matrix M = K7'HK can be computed with K = 
1069 
Diag(e, mc, 1). The eigenvalue-eigenvector-decomposi- 
tion of M has three real-valued eigenvectors e;, i — 1,2,3, 
with two identical real-valued eigenvalues A2 — A3 and an 
individual eigenvalue Ai. The normal vector of the plane 
results from the eigenvectors 
n =e; = N(e; x e3) 
and the ratio of camera and object height from the eigen- 
values: 
H A A1 — À3 HE Al — Ag 
Z Ar Ài 
Note that the solution is unambiguous except for a com- 
mon sign of c and nz and the sign of n. But the require- 
ment ny > 0 is reasonable for most camera installations. 
With the orientation parameters determined in this manner 
we are able to measure the object height and position. 
2.3 3D Object Measurement 
Similar formulas for the computation of the height of an 
object have been developed independently in (Criminisi, 
2001) and (Renno et al., 2002, Jones et al., 2002) — on the 
one hand geometric and on the other hand more algebraic. 
Below the equivalence of both is shown. 
We start from the formulation of the transformation (3) as 
a condition 
S(x/)Hx; = 0. (7) 
With the vertical vanishing point v' — Kn (the fixed point 
of the transformation) and the horizon line (fixed line) I = 
Kn (Hartley and Zisserman, 2000) in (5) the condition 
(7) leads to the formula developed in (Criminisi, 2001) 
Z Sex 
x ISGOVI 
  
n= - 
by taking the norm of the condition. With the directions 
H c 
Mi: = N(KC!x/) the homography for directions m7 = 
Mm. can be expressed 
and for the object height the second expression results 
__Z__|IS(m/)m!] 
nmi [Sma] 
(8) 
The position of the object on the plane results from sub- 
stituting the angular distance \, — —Z/(n'm/) from 
the projection center to the foot point X ' into the point- 
direction-form 
YX =X VY 20 =Z+YR'm, (9) 
for which Z; — 0 holds. 
The formulas (8) and (9) provide the basis for the object 
measurement. The calibration procedure is described in 
the following section.