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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004
to the rank of the matrix - X, * Xj (Koch, 1999). It is computed
as:
Tzel(X,-Xyy!e
The acceptance or rejection of the test statistic will partly
depend on the assumed level of significance, which is the fixed
probability of rejecting a true null hypothesis. Assuming a
certain level of significance, if the computed value is greater
than the critical value (T,) of the test statistic (i.e., T > T,), the
null hypothesis is rejected and hence, the two IOP sets are
deemed to be significantly different from each other.
Statistical testing for the purposes of evaluating camera stability
includes a number of assumptions that make it impractical to
use. It assumes a normal distribution for the estimated IOP
without any biases; it assumes that the variance-covariance
matrices associated with the IOP sets are available; and it does
not take any possible correlation between IOP and EOP into
consideration. Furthermore, Habib and Morgan (2004)
demonstrated that statistical testing generally gives pessimistic
results for stability analysis even though the two sets of IOP
may be similar from a photogrammetric point of view. Lastly,
the differences in IOP should be evaluated by quantifying the
discrepancy between bundles of light rays, defined by the two
IOP sets, in terms of the dissimilarity of the reconstructed
object space. This will provide a more meaningful measure of
the differences between the [OP sets. Due to these shortcomings
of statistical testing, two alternative techniques for evaluating
camera stability are utilized in this research and explained in
the next section.
3.22 Similarity of Reconstructed Bundles
In this research, two methods for evaluating the similarity are
used. One method is a comparison that is confined to the image
space and the other is an object space comparison.
3.2.11 Image Space Comparison
In this method, two IOP sets define two bundles of light rays
that share the same perspective center, Figure I. The degree of
similarity between these bundles can be evaluated by
computing the mean spatial angle (angular offset) between
conjugate light rays, while assuming that the image coordinate
systems associated with the two bundles are parallel to cach
other.
m Ele ]
Bundle II
© Original Grid Vertices
@ Distortion-free Grid Vertices
Figure 1 — Two bundles with same perspective center and parallel image
coordinate systems
The steps to derive a quantitative measure for the degree of
similarity between the two bundles can proceed as follows:
i. Define a synthetic regular grid in the image plane. The user
can specify the size of the grid cells and the extent of the
grid with respect to the image size. The extent of the grid
should cover the entire imiage (i.e., 10095 of the image).
i. Remove various distortions at the defined grid vertices
using the involved IOP from two calibration sets.
ill. Assuming the same perspective center, define two bundles
of light rays using the principal distance, principal point
coordinates and distortion-free coordinates of the grid
vertices.
iv. Compute the spatial angle between conjugate light rays
within the defined bundles.
v. Derive statistical measures (i.e., the mean and standard
deviation) describing the magnitude and variation among
the estimated spatial angles.
The above methodology for comparing the reconstructed
bundles assumes the coincidence of the optical axes defined by
the two IOP sets. However, stability analysis is concerned with
determining whether the reconstructed bundles coincide with
each other regardless of the orientation of the respective image
coordinate systems. Therefore, there might be a unique set of
three rotation angles (c, q, x) that can be applied to the first
bundle to produce the second one while maintaining the same
perspective center, Figure 2.
Ju
P.C. (0,0,0)
Aram, s)
Figure 2 — Image Space Comparison where bundles are rotated to
reduce the angular offset
As shown in Figure 2, (xy, yy -cj) and (xg, yi, -ci) are the three-
dimensional vectors connecting the perspective center and two
conjugate distortion-free coordinates of the same grid vertex
according to IOP; and IOPy, respectively. To make the two
vectors coincide with each other, the first vector has to be
rotated until it is aligned along the second vector. This
coincidence of the two vectors after applying the rotation angles
can be mathematically expressed as:
Xu *,
Yr i= 4 Rod, yr, 2)
— Cy = :
To eliminate the scale factor (4), the first two rows in Equation
2 are divided by the third one after multiplication with the
transpose of the rotation matrix to give the following equations:
= DaXg tUa yy T3 Cr
f Xy ave Tails
; D, Xy t» Ygj 76g
nt sn T'as Yn “1x3 CN
Equation 3 represents the necessary constraints for making the
two bundles defined by IOP; and IOP; coincide with each other.
The rotation angles (w, ¢, x) are estimated using a least squares
adjustment. The variance component (6.2), the variance of an
observation of unit weight, resulting from the adjustment
procedure represents the quality of the coincidence between the
two bundles after applying the estimated rotation angles.
Assuming that (xi, yj) in Equation 3 are the observed values, the
corresponding residuals represent the spatial offset between the
two bundles, after applying the rotation angles, along the image
plane defined by the first IOP set. Therefore, assigning a unit
weight to all the constraints resulting from various grid vertices
