International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
surface inspections, for example in the measurement of antenna
reflectors or moulded components.
The target plane determination process was implemented and
evaluated in Australis, a photogrammetric software package for
off-line VM (Fraser & Edmundson, 2000; Photometrix, 2004).
2. TARGET PLANE DETERMINATION
The proposed target plane determination process can be
subdivided into two stages. First the ellipse parameters of the
circular target are computed in each image. In the next stage the
actual elements of the circular target (target plane normal and
radius) are determined using the computed ellipse information
and the exterior orientation (EO) of the images. Considering
mathematical rigor, this stage should be performed inside the
bundle adjustment since object points and the EO of the images
are correlated (within the bundle). However, this fact may be
neglected because the eccentricity error has only a very small
effect on the parameters of the bundle adjustment. Hence, it is
justified to compute the target elements only once when the
bundle is near convergence. Then the final iterations are
computed considering the eccentricity corrections.
In the following, knowledge of least-squares formulations in
bundle adjustment is assumed and therefore only the
fundamental observation equations for the adjustment models
are derived.
2.1 Ellipse Parameter Determination by 2D Gaussian
Distribution Fitting
The determination of the ellipse parameters is a delicate
problem since targets are typically only a few pixels in
diameter (see Figure 2). State-of-the-art VM systems only
determine the centre of the imaged target, mostly via by the
well-known intensity-weighted centroiding approach:
Xj
s) EEUU «
Yo SN e
i=l j=1
Here, Xo, Yo are the centroid coordinates, Xj, yj are the pixel
coordinates and g; are the grey values within a window of
dimension n x m. It should be mentioned, that a careful
thresholding process needs to be performed before the actual
centroiding, to remove disturbing background noise. Equation 1
points out clearly that as much pixel. information as possible is
used to compute the target centre. This consideration is also
accounted for in the determination process for the ellipse
parameters.
The idea of using the 2D Gaussian distribution to find the
centre of gravity of a 2D object can be found in literature quite
often. However, a visual analysis of the Gaussian distribution
(bell curve) and intensity images of real targets (Figure 2)
proves that the Gaussian distribution fits well to small targets
only. Bigger targets have an intensity plateau, which cannot be
described by the Gaussian distribution.
P ar
LIL]
GG
de
(ZS
Figure 2: Typical target image in VM and its intensity image.
As it turns out, the cumulative Gaussian distribution (CGD) is
an ideal base function (Figure 3) for the designed target
function, which allows a description of targets with the
aforementioned intensity plateau.
15 À —05 9" 0.5 1 15
Figure 3: Cumulative Gaussian distribution (CGD).
The CGD is defined by
Qx)= € [Gx =
1 : 1
-2 visa” m 2 \ 20 /
where G(x) is the Gaussian distribution, cis the corresponding
standard deviation and 4/ the expectation. Substituting x by
fx 1) in Equation 2 leads to the ID function shown in Figure
4. The next step is to substitute x by an implicit ellipse
equation, which results in the sought-after 2D function T.
x (cos sing) x—c, (3)
y i —sind cosdAy-—c,
B= x + + -] (4)
a b
T(s, B, c, c,,a b, 4,0, 20) 5 s- O(- E)« f (5)
Equation 3 describes a transformation, the use of which within
the implicit ellipse equation 4 allows the interpretation of c, and
c, as the centre of the ellipse and ¢ as the bearing of the semi
major axis.
DAL m te
08 AN
/ N
/ V
/ 04 \
/ N
/ 0.2 \
7 N
7 Ng
6 ATS U 1.5
Figure 4: Target function derived from the CGD.
pe
ta
hc
The
per
pix
unl
and
par
fon
con
wit!
