International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
3 REALISATION
3.1 Calibration Procedure
The proposed calibration procedure consists of two stages:
After the initial calibration with objects of equal and
known heights, the parameters can be checked and — if
needed — updated in the continuous operation phase with
new objects of unknown height:
(1) Initial Calibration. After the installation of the cam-
era the foot and head points of the objects have to be mea-
sured. Depending on the specific calibration object this
can be done manually or with the help of feature extrac-
tion. The observed heights may not be arranged on a sin-
gle straight line in the object space (cf. section 3.3, de-
terminability of the parameters). While the height of the
objects has to be known, the height of the camera Z may
be introduced as an unknown parameter or — if accessible
— as a measured quantity. Approximation values for the
unknown parameter can be determined as described below
or by a rough guess, e. g. for the roll angle zero is always
a good assumption.
(2) Parameter Update. For the continuous operation we
assume that the height of the camera does not change,
while the other parameters may vary due to environmen-
tal influences, for instance temperature. For every new
scene t an unknown height 7, is introduced into the ad-
justment procedure. Since the unknown object heights H,
can vary, relinearisation with few iterations is advisable for
every new scene — slightly increasing the computing time.
At the same time the measurements yield the position and
height of the objects for each image. Furthermore, the ad-
justment provides the average height for every type of ob-
ject.
3.2 Approximation Values
Minimizing algebraic distances. The transformation
parameters can possibly be determined without the knowl-
edge of approximation values (Hartley and Zisserman,
2000). With the projective transformation x7 = Hx; writ-
ten in homogeneous coordinates
uw’ à: ban? u/
ot nl Cf v'
wu" q ji w
with x = uuu) e cuiu lj and x" s
(u^, v" ap") T 2 (25,1, 1)! we first of all get the con-
straints between the image coordinates and the homogra-
phy elements
u’ (gu. 4- hv; --i) —w; (au; 4-bv; 4-cw;) — 0 (10)
v/ (gu, -hv, ti) —w; (du, 4-ev; 4 fw;) = 0. (11)
In compact form a] ;h=0 and al; h —0 with the 9-vectors
n.r SE HAT
i T ——— UI 8
a, SM 0 1 274)
1070
and the unknown parameters h — (a,b, c, d.e, f. g. h. i)T
we get the homogeneous equation system Ah = 0. The
right eigenvector of A for the smallest eigenvalue A, is a
good estimation for h. With the singular value decomposi-
tion A— UDV' the solution is
h.m Vg with Kui. (12)
For numerical reasons a conditioning of the problem is ad-
visable.
Enforcing the homology constraints. The estimation
(12) of H does not possess the properties of a planar homo-
logy presented in section 2.2.1. Therefore, a least squares
adjustment can be done assuming the elements h =vec(H)
as 1. i. d. observations. The explicit model of this observa-
tion process reads
h—f(co,4,Z) wih XO -o2 (13)
with the a priori covariance matrix oA of the observa-
tions and the unknown variance factor o2. The solution
H minimizes the Frobenius norm |H All. Approxima-
tion values are taken from the decomposition explained in
section 2.2 2.
Although the solution h fulfills the constraints of the pla-
nar homology, it is still an approximation since potential
individual weights of the observations have not been taken
into consideration. Therefore, a subsequent stringent ad-
justment is necessary.
3.3 Parameter Estimation
Determinability of the Parameters. If the pitch angle a
is zero or 90? — i. e. the viewing direction is horizontal or
towards the nadir — the element nz of the normal vector
(1) becomes zero. In this case the 2D-homography (6) de-
generates to a 1D-homography and the principal distance c
is not determinable. If the pitch angle is approximate zero
or 7, the determination of the parameters is very weak. In
this case prior information about the parameters has to be
provided. This can easily be done by introducing these
values as additional, fictitious observations into the adjust-
ment process explained in the following.
One critical arrangement of the calibrating objects can be
observed: if the foot and head points in the image are
collinear, the homography degenerates and the parameters
are not determinable. Thus not all objects may be situated
on a single straight line.
Adjustment Model. For the calibration phases (initial
and update) the general non-linear model
; (0) _ 2p-1 14
g(l.p) — 0 with Xj'-ojP; (14)
with the constraints between the observations /, the param-
eters p and the a priori covariance matrix of the observa-
tions x0 is arranged, cf. for instance (Mikhail, 1976).
The constraints of the model are the eqs. (10) and (11).
For technical convenience with p = (c, a, y, Z, H)" five
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