spatial distribution, which is considered as one of the most
difficult and time-consuming tasks for architects (Lerma et al.,
2000).
In summary, photogrammetry offers a rapıd and accurate
method of acquiring three-dimensional information regarding
cultural monuments. Combining the measurements obtained
from the photogrammetric record and 3D CAD models offer the
means to recreate historic environments. This facilitates the
generation of accurate digital records of historical and
archaeological objects, while reducing the overall costs.
Before using photogrammetric techniques for the recording and
documentation of cultural heritage, factors that have an impact
on recording accuracy and archiving efficiency have to be
discussed: namely, metric characteristics of the camera,
imaging resolution, and requirements of the bundle adjustment
procedure (Chong et al.,2002). The next section will describe a
mathematical model that incorporates straight lines in a self-
calibration and bundle adjustment procedure for accurate
estimation of the interior orientation parameters. This is a
necessary prerequisite for accurate and reliable 3D-
reconstruction.
3. MATHEMATICAL MODEL
The purpose of camera calibration is to determine numerical
estimates of the IOP of the implemented camera. The IOP
comprises the focal length (c), location of the principal point
(x, yj) and image coordinate corrections that compensate for
various deviations from the assumed perspective geometry. The
perspective geometry is established by the collinearity
condition, which states that the perspective centre, the object
point and the corresponding image point must be collinear. A
distortion in the image signifies that there is a deviation from
collinearity. The collinearity equations, which define the
relationship between image and ground coordinates of a point in
the image, are:
Ze nO, 7 Xo) t rj, 7 Yo) (Z4 7 Zo) SAY
X a = X p y r 7 3 r
ni(X 47 Xo) (Y, = Ko) ll = La) (1)
Ya Ya -C naX ,— X9) a. = To) + (24-20) +Ay
n3GX, 7 Xo) * r34(Y,- Yo) * (Z4 7 Zo)
where
Xn yu are the observed image coordinates of image point a
X, Y,.Z: are the ground coordinates of object point A.
X» yp are the image coordinates of the principle point
e: is the camera constant ( principle distance)
Xo» Yo, Zo: are the ground coordinates of the perspective centre,
f,y__f33: are the elements of the rotation matrix that are a
function of (w,ÿ,x)
Ax,Ay : are compensations for the deviations from
collinearity.
Potential sources of the deviation from collinearity are the
radial lens distortion, decentric lens distortion, atmospheric
refraction, affine deformations and out-of-plane deformations.
These distortions are represented by explicit mathematical
models whose coefficients are called the distortion parameters
such as K;, K,, K; for radial lens distortion; P,, P5, P, for
decentric lens distortion; and A;, A, for affine deformation.
The relative magnitude of these distortions is an indication of
the condition and quality of the camera.
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
In order to determine the IOP of the camera, including the
distortion parameters, calibration is done with the use of control
information. in the form of a test field. In a traditional
calibration test field, numerous control points are precisely
surveyed prior to the calibration process. Image and object
coordinate measurements are used in a bundle adjustment with
self-calibration procedure to solve for the IOP of the involved
camera, EOP of the imagery and object coordinates of the tie
points. Establishing a traditional calibration test field is not a
trivial task and it requires professional surveyors. Therefore, an
alternative approach for camera calibration using an easy-to-
establish test field comprised of a group of straight lines as well
as some tie points is implemented in this research.
Object space straight lines prove to be the least difficult and
most suitable feature to use for calibration. They are easy to
establish in a calibration test field. Corresponding lines in the
image space can be easily extracted using image-processing
techniques such as image resampling and application of edge
detection filters. Automation of the extraction process can be a
reliable and time-saving approach in camera calibration.
Furthermore, linear features, which essentially consist of a set
of connected points, increase the system redundancy and
consequently enhance the geometric strength and robustness in
terms of the ability to detect blunders.
As shown in Figure 1, for a frame camera, a straight line in the
object space will be a straight line in the image space in the
absence of distortions. A deviation from straightness in the
image space is a function of the distortion parameters.
x
rz \ XYZ.
orgy
‘x, ~ x, - distortions x |
- V, 2 R(9 ,$ .& ) ye —», - distortions $
-C
X,
ii E
Z,
°
= X,
yi
5%
Figure 1. Perspective transformation between image and object
space straight lines
Object space straight lines are incorporated in the calibration
procedure by representing them with any two points along the
line such as points 1 and 2 in Figure 1. These points are
monoscopically measured (i.e., there is no need to identify
conjugate points in overlapping images) in one or two images
within which this line appears. In the image space, the lines are
defined by a sequence of intermediate points such as point 3 in
Figure 1. Therefore, the distortion at each point along the line
can be independently modelled. In order to restrict the points to
form a straight line, a mathematical constraint is adopted to
establish the perspective relationship between image and object
space lines. The underlying principle in this constraint is that
the vector from the perspective centre to any intermediate
image point along the line lies on the plane defined by the
perspective centre of that image and the two points defining the
straight line in the object space. This constraint is expressed as
follows:
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