]
|
|
i
|
|
If the two lines should be parallel then (12) becomes
Idil Idi4l (1 -(e+f+g)) = 0 (13)
and for perpendicular lines we get
| d; | | d;41 | (e+f+g) = 0 (14)
Two lines will intersect in a point, if the coplanary
constraint
CS) -51), dy, dy | = 0 (15)
is fullfilled. The last constraint is the intersection of
more than two lines in a point. The coplanarity
constraint (15) is not sufficient when three or more
lines have to intersect in one point. À point P on a line
is described by
P= 5; + ti di (16)
where i = 1,..,n and n is the number of lines inter-
secting in point P. To describe the point P common to
all lines, (n-1) conditions are necessary:
$1 * 4 dq S, * t d»
S1 + ty dj = S3 + ts d3
" (17)
S1 + t dy s Si * tj d,
The main difference between the conditions in (17)
and all others, is the introduction of the new para-
meters t;. The scalars t; locate the intersection point P
on each line L,. Point S2f(6,6,r) and direction
d -f(6, 9, y).
3. ADJUSTMENT PROCESS
For the calculation of the four line parameters a mini-
mal number of four measured image points in at least
two images are necessary. Normally much more than
these four points will be measured. This leads to an
overdetermined problem, which can be solved by a
general least square adjustment method. The copla-
general least square adjustment method. The copla-
narity relationship (7) gives one condition equation for
each measured image point. This relationship is non-
linear and a function of the measured image coor-
dinates x y of each image line point and the four
unknowns 6, 6, r, y :
F = f[SCö,oö,r), d(8,0,y), m(x,y)] (19
672
Equation (18) is linearised by a Taylor serie using only
the linear terms. The linearised coplanarity condition
looks
Fo + (=) dx + (S dy + E dô
+ Goo * (Se: * e m (19)
Expressing the unknown residuals (dx, dy) of the
measurements by vector Al, the parameter
corretions ( dó , d$, dr, dy ) by vector Ax and using a
matrix formulation results in
BA A Ax +wy = 0 (20)
where
yi RUES
B = x , ay ) ,
AB (OF FE
e T tm]
t o
WB i tA x;
o
Vector 1 contains the measured image coordinates and
vector x the approximated parameters. Assuming that
the measurements are normal distributed and not cor-
related a diagonal weight matrix P is introduced. The
introduction of geometric constraints, concerning
separate lines investigated in chapter (2.3), leads to an
expansion of the least square model (20) with
Ar Axi. wy = 0 (21)
Finally we get
BA ALAVA wıl= 0 (22a)
A, Ax ++ Wa: =: 0 (22b)
. L ; .
The matrix A, include the derivations of one or
several of the equations from chapter 2.3 with respect
to dó, d$, dr, dy. The solution of this adjustment
problem is
ee Mut JJ. m —-» | ] ^1 ^ bue Pee ad eed A
