juality of the lithog-
phic mask has been
itor which yields an
nstrated in section 5
Xf my calibration.
ed that for sufficient
ages of four views of
ew have to be ac-
ation of the grating
ever, for my calibra-
proximate rotations.
the Bundle Adjust-
on is not concentric
the needed motions
the shelf micrometer
on
ge
' matches?
pt uncertain matches?
age
» set
Stop Data Snooping?
: no
Parameter set ok?
and user interaction
processes, the data
libration procedure.
ion steps that run
the algorithm con-
xes. The first block
easuring the precise
zes. The processes
eters of the imag-
| properties of SLM
:ly new photogram-
linate acquisition is
| model of the Bun-
on 3.
| marks in figure 2,
es in the center of
ed to the user. The
ot or reject the im-
the illumination and
1996
the sharpness are appropriate. The three optional user in-
teractions (accepting uncertain matches, supervision of data
snooping and refinement of the parameter set) may be valu-
able to improve the quality of the final results. They can be
omitted in situations where fast and automatic calibration is
more important than results of highest quality.
3 MATHEMATICAL FRAMEWORK
To provide the context for sections 4 and 5 the mathematical
framework is outlined very briefly. An extended version of
this discussion can be found in [Danuser and Kiibler 1995]
and for the derivations | refer to [Danuser 1995].
The Bundle Adjustment involves four types of observation
equations. The relation between the image coordinate ob-
servations and the 3D position of the target points on the
calibration standard is introduced with
oo
( Y = gue e. [1 — > ^] + SE”
3,
k=1
— M'Y R view view, view (So : Zi + zo,2)
view view view view view view
Un óc (Ki 9 £39 9 £1 ‚Po ‚E,a .8 ) (1)
The index i runs over all target points. €) is the rotation angle
of the grid with respect to the object coordinate system. My
calibration procedure is based on four rotation angles, each
approximately 90° apart. £ is the observed image coordinate
vector. Its index view specifies whether the point is observed
in the left or right image. The corresponding 3D position
£; is defined with respect to a coordinate system rotating
together with the calibration grid. Note that through the for-
mulation of (Sq + Zi + 0,0) a non concentric rotation of the
calibration grid in the superior fixed object coordinate system
is introduced. The translation between the grid coordinate
system and the object system is defined by xo,n=o = 0.
Equation (1) incorporates the weak perspective situation in
microscopy. The nonlinearity of perspective imaging there-
fore disappears and a perspective distortion term S oF
is introduced, instead. Its magnitude depends on the lateral
and vertical position of the target point with respect to a per-
spective distortion free point. The latter's image coordinates
are given by £3". The expansion of the distortion series is
stopped when 92* (EVE —€5°V) < diag(Qee) * which holds
in most cases for k = 2. This approach guarantees that the
inaccuracy of the weak perspective model becomes smaller
than the accuracy of the image coordinates.
M"*¥ symbolizes the magnifications of the left and right
images, respectively. R'** is an orthonormal matrix describ-
ing the orientation of the image coordinate system with re-
spect to the object coordinate system. The choice of the
primary, secondary and tertiary angle has been adapted to
the microscopic situation retaining the technical meaning for
PY, we, kV in standard photogrammetry.
The distortion function J contains the well known radial
and decentering distortion for each view. Its coefficients are
— according to the technical terms in photogrammetry —
Kj'e*, Kyiew, pyiew Py”. The terms s"i** and a"i** com-
pensate for scale and shear distortion in the image coordinate
frames. The CMO distortion function §;(E) has been derived
to particularly compensate for geometric deformations origi-
nating from the non paraxial imaging (see figure 1). Even in
! Qee is the cofactor matrix of the image coordinates.
the case of perfect lenses, non paraxial optics leads to image
deformations. To get highly accurate vertical positions, the
introduction of this term is of utmost importance, as the-
oretically demonstrated in [Danuser and Kübler 1995] and
empirically verified in section 5. A remarkable behavior of
this distortion type is that the very same parameter E is in-
volved in the distortion model of the left and the right view.
Therefore, in contrast to macroscopic photogrammetry, the
two image spaces of the stereo rig are closely related to each
other. In particular, this renders a separate calibration of the
left and the right imaging function impossible.
The coordinate residuals e,view may origin either from noise
i,
in the image coordinate measurements, from unmodeled Sys-
tematic errors or from the position errors of the control points
on the photo lithographic grating. An adequate error model
of the control point precision has to be introduced:
Lai + eri g with Qu (2)
To get reliable results the relation between Qee and Qe,e;
must be very thoroughly determined. This can be achieved
by dividing the target points into a set of control points and
a set of check points. The empirical RMS of the check points
contains valuable information to define Quid:
The full parameter vector p to be estimated in the Bundle
Adjustment includes p’,
/ view view view view
Puede, MES.
Kew Kew view view E view view jt
1 32412 ui 442 4, ,8
all the grid rotation angles €2, the corresponding translation
vectors zo,o and all the coordinates Z' of the target points
on the calibration grid.
The observation equations (3) and (4) allow me to influence
the estimation of the parameter sub-vector p'.
p with Qi, t, (3)
Bp with Q.. (4)
Ly + ZA
€c
With the equations of type (3), | can gradually turn on and
off the estimation of a certain parameter. With those of type
(4), | can specify similarities between parameters of the same
type in the left and right view, where B is the corresponding
similarity matrix.
A Least Squares approach using equations (1) to (4) is im-
plemented as in classical Bundle Adjustment. Data Snooping
is employed to efficiently detect and eliminate outliers in the
image coordinate observations. Each individual image coor-
dinate residual is compared with the corresponding diagonal
element of the residuals covariance matrix Q,,. This is a well
known standard procedure for blunder detection in the pho-
togrammetric Bundle Adjustment. Due to the limited depth
of field the estimation process suffers from a considerable nu-
merical weakness. Simulations in [Danuser and Kiibler 1995]
turned out that gross errors in the image coordinates can
cause serious numerical oscillations in the parameter space.
Such oscillations indirectly affect the quality of blunder de-
tection, too. To solve this problem, as many erroneous image
coordinate observations as possible must be detected before
entering the Bundle Adjustment. An expensive but fully au-
tomatic algorithm has been implemented to achieve this de-
manding task. The next section describes the most important
steps in this procedure.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996
