International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004
2. CAMERA CALIBRATION
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,, y») 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 center, 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:
S n, 7 Xo) tr (Y, -Y)-n(Z,-72Z4) -
—€ E ; : : = = + Ax
ha (X = Xo) +m(Y, 7 Yo) e rZ, Zo)
a "P
y, y,-€ C ia : EU E os e Zl Ay e:
nz (X, E Xo) + m (Y, = Yo) Hs 3(Z, = Zo)
Where:
x, and y, are the image coordinates
X,4,Y,andZ, are the ground coordinates
Ax and Ay are compensations for the deviations from
collinearity
are the IOP of the camera
are the ground coordinates of the exposure
station (perspective center)
Ili, Fos...» K33 are the elements of a rotation matrix that are a
function of c, ÿ and x
Xp» V, and c
Zz
Xo» Yo, Zo
Potential sources of the deviation from collinearity are the
radial lens distortion, de-centric lens distortion, atmospheric
refraction, affine deformations and out-of-plane deformations
(Fraser, 1997). These distortions are represented by explicit
mathematical models whose coefficients are called the
distortion parameters. The relative magnitude of these
distortions is an indication of the condition and quality of the
camera.
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. As mentioned earlier, 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 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. 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. Corresponding lines in the image space can be easily
extracted using image-processing techniques such as image
resampling and application of edge detection filters. Moreover,
automation of the linear feature extraction process can be a
reliable and time-saving approach. For camera calibration
purposes, object space straight lines will project into the image
space as straight lines in the absence of distortion. Therefore,
deviations from straightness in the image space can be modelled
and attributed to various distortion parameters in a near
continuous way along the line.
Several approaches for the representation and utilization. of
straight lines have been proposed in literature and all suffer
from a few drawbacks. In these approaches, the IOP estimation
follows a sequential procedure (Brown, 1971; Guogqing et al,
1998; Prescott and McLean, 1997; and Heuvel, 1999). First,
linear features are used to derive an estimate of the radial and
de-centric lens distortions, which is then followed by a
traditional calibration to determine the principal distance and
principal point coordinates. The estimated parameters in the
calibration will be contaminated by uncorrected systematic
errors such as affine deformations, which are not compensated
for during the first step. Another approach by Bräuer-Burchardt
and Voss (2001) assumed that distorted lines can be modelled
as circular curves, which might not always be the case. Chen
and Tsai (1990) introduced another method that requires the
knowledge of the parametric equations of the object space
straight lines, which mandates additional fieldwork.
In this research, Habib et al (2002-a, 2002-b) proposed a
calibration test field consisting of straight lines that are
represented by two points along the line in the object space.
Acquired imagery over the test field is used in a bundle
adjustment with self-calibration procedure to simultaneously
estimate the IOP of the implemented camera and the EOP of the
exposure stations. For a detailed explanation of the bundle
adjustment procedure, the representation, selection and optimal
configuration of straight lines in imagery, and the automated
linear feature extraction process, refer to Habib et al (2002-b)
and Habib et al (2004). Once the calibration procedure has been
carried out, the IOP of the camera that are derived from two
different calibration sessions can be inspected.
3. STABILITY ANALYSIS
The desired outcome of stability analysis is to determine
whether two sets of IOP are equivalent to each other. The
following sections describe possible approaches for comparing
two IOP sets to analyze camera stability.
3.1 Statistical Testing
The statistical properties of two IOP sets can be described by an
assumed normal distribution, which has a mean of the true IOP
(IOP4) of the implemented camera. For stability analysis, a null
hypothesis (H,) can be tested for possible rejection under the
assumption that the two IOP sets are equivalent. Accepting the
null hypothesis simply affirms that there is no significant
difference between the two IOP sets and the internal
characteristics of the camera are stable. Assuming that the two
IOP sets are uncorrelated and that the true IOP of the camera
does not change between the two calibration sessions, the null
hypothesis is:
Hoi IOP, = IOPT or Ho: e= IOP, IOP = (0, S1 ER Zu)
Where: IOP, and IOP, are the estimated IOP sets from the two
calibration sessions, and X, and X are the corresponding
variance-covariance matrices.A test statistic (T), which is used
to determine whether or not the null hypothesis is rejected,
follows a y” distribution with degrees of freedom that is equal
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