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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
where C is the projection center, el and e2 are arbitrary
orthogonal vectors within the circle plane, M is the midpoint of
the circle and r the circle radius. Using the well known
coplanarity equation
f x
R
: Hi
y |=R-(X-C)=R ui
R
2 (x = C)
. ;
the cone can be transformed into image space by
x=A-r-R-(cosa-e, +sina-e,)+1-R-(M-C) (12)
The intersection of Equation 12 with the image plane is simple.
The plane is defined by z = -c. Hence, À can be obtained from
€ (13)
A-—
R,-(r:cosa-e, *r:sina-e; « M- C)
and the coordinates of the intersection figure follow as
R,:(r:cosa:e, * r-sina-e; - M- C) R,:v
Xz-—C m—C
R,-(r-cosa-e +r-sina-e, +M-C) R,-v (14)
R, -(r-cosa-e, +r-sina-e, +M-C) R,v
y=-c—== m
R,-(r:cosa-e, *r:sina-e, - M- C) R,-v
If Equations 14 can be transformed so that the parameter a is
eliminated, the implicit form of the ellipse equation (15) is
found. The derivation is lengthy and only the final form of the
implicit parameter is presented here:
2 y
a-x>+b-xy+c-y +d-x+e-y—-1=0 (15)
2 2 2 2
rm, -n,-n
a= 1 = 21
d
be r^ qm nac. n Wi
d
2 2 2 2 (16)
yf mh. ny
Cm M Z2
d
„2 l^ -
Bal ol HH Ha Ps — May Has
d
PR, ity, Hg
ez2:c BEER
d
where the following substitutions have been used:
m zR,e, R,e,-R, e, R,e,
m,zR,e, R,e, -R,e,-R,-e,
m,-zR,e,R,e;-R, e, R,.e,
n,-R, eR, (M-C)-R, e,- R, (M-C)
n,-R,e,R,(M-C-R,e,R,-(M-C) (17)
R,-(M-CO)-R,-e,- R,- (M-C)
n,-R,e,R,(M-CL-R, e, R, (M-C)
n4-R, e, R,(M-C)-R, e; R, (M-C)
n,-R, e, R, (M-O)-R, e, R,-(M-C)
= 2 2 2e 2
dzcln.tm.-r-:m,
n;=R,-e-
Thus, a description of the implicit ellipse parameters dependent
on the circle parameters is found. For the target plane
adjustment, linearisation with respect to the circle parameters
(M, r, el and e2) is required.
The chosen parameterisation of the circular target by midpoint
M, radius r and circle plane vectors el and e2 consists of 10
elements. However, a circle in 3D can generally be described
by 6 parameters, i.e. midpoint, radius and two rotation angles.
Hence, the selected parameterisation has 4 degrees of freedom
which have to be eliminated within the adjustment. One
possibility is to introduce constraints. This was implemented in
the developed process in Australis. There, the length of the
vectors el and e2 was fixed to 1 and the orthogonality of el and
e2 was secured. The final constraint prevented el and e2 from
rotating within the plane of the circle.
Since the observations for this adjustment were created by a
previous adjustment, the previously computed covariance
information can be incorporated as well. Tests have shown that
this is an absolute requirement for the computation of an
accurate target plane since the implicit ellipse parameters have
different scales and different accuracies (for introducing full
variance/covariance matrices, see Mikhail et al. 1996).
Finally, it should be mentioned that the first adjustment delivers
explicit ellipse parameters whereas the target plane adjustment
takes implicit ellipse parameters. The required conversion of
the parameters and of the covariance matrix is shown in the
appendix.
3. QUALITY ANALYSIS OF THE TARGET PLANE
DETERMINATION PROCESS
For the quality analysis of the developed target plane
determination process an object with a precisely known surface
shape was surveyed. The test field was placed on a calibration
table which had a machine-levelled surface. The table was
kindly made available by Boeing Australia Limited at a factory
in Melbourne. The main aim of the test field was to investigate
the target size effects and the accuracy of determining the target
normal.
The targets of the test field were arranged in a 4 by 4 grid, and
four different target sizes were used (3, 5, 6.4 and 9.4 mm).
Thus, the test field consisted of 64 inspection targets and some
additional system-required targets (Figure 8).
Figure 8: Test field on machine-levelled surface.
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