ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002
procedure. The suggested mathematical model for incorporating
straight lines in bundle adjustment with self-calibration is
described in section 3. Then, we present experimental results
obtained from the suggested approach using real data. Finally,
conclusions and recommendations for future work are
summarized in section 5.
2. SELF-CALIBRATION: BACKGROUND
The main objective of photogrammetry is to inverse the process
of photography. When the film or CCD array inside the camera
is exposed to light, light rays from the object space pass through
the camera perspective centre (the lens) until they hit the focal
plane (fim or CCD array) producing images of the
photographed objects. Light rays from the object points to the
corresponding image points passing through the lens are
assumed to be straight. This assumption is known in the
photogrammetric literature as the collinearity model (Kraus,
1993).
During camera calibration, we determine the IOP, which
comprise the coordinates of the principal point, the principal
distance, and image coordinate corrections that compensate for
various deviations from the collinearity model. There are four
principal sources of departure from collinearity, which are
“physical” in nature (Fraser, 1997). These are the radial lens
distortion, decentric lens distortion, image plane unflatness, and
in plane image distortion. The net image displacement at any
point is the cumulative influence of these perturbations. The
relative magnitude of each one of these perturbations depends
very much on the quality of the camera being employed.
2.1 Distortion models
Radial lens distortion is usually represented by polynomial
series (Equation 1). The term K, alone will usually suffice in
medium accuracy applications. The inclusion of K, and Kj
terms might be required for higher accuracy and wide-angle
lenses. The decision as to whether to incorporate one, two or
three radial distortion terms can be based on statistical tests of
significance.
Ax pp = Ky (r? =D)x+ Ky (r* Dx Ks (r* -D)x
2 4 6 (D
Ayup ^ Ky(r -Dyt K(7' -DytKS,(G -Dy
where
pax, +(y=v,)"
and K,, K; and K; are the radial lens distortion parameters.
A lack of centring of the lens elements along the optical axis
gives rise to the second category of lens distortion, namely,
decentric distortion. The misalignment of the lens components
cause both radial and tangential distortions which can be
modelled by correction equations according to (Brown, 1966)
as follows:
AXprp 7 P(r’ +2x2)+ 2Pxy 2)
Ay prp = P,(r? + 2y%)+ 2P xy
where P, and P; are the decentric lens distortion parameters.
Systematic image coordinate errors due to focal plane
unflatness can limit the accuracy of photogrammetric
triangulation. Radial image displacement induced by focal plane
unflatness depends on the incidence angle of the imaging ray.
Narrow angle lenses of long focal length are much less
influenced by out-of-plane image deformation than short focal
length and wide-angle lenses. To compensate for focal plane
unflatness, the focal plane needs to be topographically
measured. Then, a third- or fourth-order polynomial can model
the resulting image coordinate perturbations. In this work, the
effect of focal plane unflatness is assumed to be very small and
will be ignored.
In-plane distortions are usually manifested in differential
scaling between x and y image coordinates. In addition, in-
plane distortions might introduce image axes non-orthogonality.
Those distortions are usually denoted affine deformations and
can be mathematically described by Equation 3. One should
note that affine deformation parameters, which are correlated
with other IOP and EOP are eliminated (for example, shifts are
eliminated since they are correlated with the principal point
coordinates).
AY ap = AY
where A, and A, are the affine distortion parameters.
(3)
2.2 Traditional approach of camera calibration
In traditional camera calibration, convergent imagery is
acquired over a test field. Together with tie points, a large
number of control points are measured in both image and object
space. The extended collinearity equations (Equations 4 and 5)
are used in bundle adjustment with self-calibration to solve for:
e Ground coordinates of tie points.
e EOP of the involved imagery.
e IOP ofthe involved camera(s).
oa X rre Y en zoe (4)
xX, =X
pt NAT TE, ZZ)
y =y c Xr mU EE Pa) à dy (5)
: 7 n(X,- X) try(Y,- Y) tn, Z2.)
where:
Ax 2 Axgrp + AXpzp + AX ap + Aye
Ay = AVrzp * AVpip + AV ap + Mere »
(x) are the observed image coordinates of image
point a,
(X ,,Y4,Z,) are the ground coordinates of object point A,
(Xp: Va) are the image coordinates of the principal point,
c is the camera constant (principal distance),
(X,,Y,,Z,) are the ground coordinates of the perspective
centre, and
(ij.73,) are the elements of the rotation matrix that
depend on the rotation angles (0,0, K) .
3. SUGGESTED APPROACH
3.1 Mathematical Model
Before discussing the mathematical model, it should be noted
that we want to incorporate overlapping images with straight
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