In case of errorfree observations the object line, the
observed image ray p from the perspective center L to any
point P on the line and the vector between the perspective
center L and the line center point C form a plane.
L
Fig. 2: Line representation in object and image space
This coplanarity relationship can be expressed by the scalar
triple product
[p B (C-L)] =0 (1-4)
The expression is free of any nuisance parameters. One
equation allows the description of one observed point (x,y).
The linearization of (1-4) with respect to the parameters C
and B to be determined in a space intersection gives
9 _ ves T —60F T
op [(C-L) xp), IC [pxß]
(1-5)
According to (Mulawa, 1988) the coplanarity relationship
is very stable with respect to the initial approximations.
Line determination is not possible, when two cameras are
used and the line falls on an epipolar plane of the cameras.
1.3 Model with conditions and constraints for least
squares adjustment
The model with conditions and constraints is chosen for a
least squares adjustment of C and D. The model is applied
in order to handle the implicit observation equation and the
two constraints. This paper will present the model only in
a rough way, because it is already detailed described in
(Mikhail, 1976) and (Mulawa, 1988).
The covariance matrix X of observations is usually scaled
by the a priori reference variance o^. In the adjustment the
scaled version Q is used.
A
c?
Q- —E (1-6)
The linearization of the non linear condition equation F is
done by the Taylor Series expansion up to the first order.
er oF dF
F(1,&) = F(l,x,) * al A, t z^ (4-7)
= —f + Av + BA
1 = [x,.y,.-X,¥,] = vector of image observations
v = residuals, approximation that v — A,
x = unknown parameters C,, C,, C,, B, D, D,
A = superscript referring to estimated values
X, = current approximations for parameters
A = matrix of derivates with respect observations
B = matrix of derivates with respect unknowns
A, = corrections to parameters
f = coplanarity value = current value of condition equation
= number of image observations
= number of condition equations = %n
number of parameters = 6
mos
|
The linearized form of the condition equation is written
Aa Vl + Bou Aul = f. (1-8)
As weights for the condition equations F; - including both
observations x, and y; - the matrix W, is introduced as
Q. = A Q AT, Ww, = Q." (1-9)
The matrix Q, has diagonal structure and its elements are
here called ’pseudo weights’. In the value of a pseudo
weight the individual weights of the coordinates and the
local geometry described by the matrix A are involved.
The linearized form of the two constraint equations is
Ca A, = 85,1 (1-10)
0° 0 0 2h, 25, 2P, x HER
B. 8,5, 6 o C6, .|09-£
where s = number of constraint equations = 2
The least squares technique is based on minimizing a
quadratic form. It leads to the normal equations
Ae LBW
RS
c
Bw B.C
C 0
(1-11)
For building up the matrices B'W,B and B'W.f in a
computational efficient way the summation accumulation
algorithm suggested by (Mulawa, 1988) is used. It is based
on the diagonal structure of the A and Q matrices, so that
a pointwise partitioning of the data and the matrices is
possible. One update step consists of calculating the
matrices or vectors A, We, B and f only due to one
condition equation. The complete normal equation is the
sum of all those pointwise calculated values.
The redundancy of this model is r = c - (u - s). Then the
a posteriori reference variance can be computed by
fT W. f
r T
a2 vIWyv
0° = =
(1-12)
Here the second term of this expression was used because
the residuals v are not computed. All considerations about
quality and outlier detection are done with the coplanarity
values f. The coordinates x,y are not any longer handled as
single observations. The coplanarity values offer the
treatment of a point as pseudo observation’ with a standard
deviation expressed in the inverse pseudo weight and the
coplanarity value as residual.
