opposite thesis, it does not require knowledge about
such parameters and relies instead on invariant
properties derived directly from the overlapping
imagery. Potential gain may be expected from analyzing
these two different theories, establishing their
relationships, and seeking a hybrid approach which
maximizes the contribution of each. A hybrid approach
may lead to improved techniques for object
reconstruction with rigorous propagation of quality
measures for a variety of imaging systems.
3.2 Invariance Applications in IU/CV
The central theme of CV is to achieve human level
capability in the extraction of information from imagery
for such applications as object recognition, navigation,
and object modeling (Hartley, 1993). By contrast, the
primary goal of photogrammetry is accurate
reconstruction of 3D object from overlapping imagery.
Thus, object model construction is a common goal of
both IU/CV and photogrammetry, in which invariance
plays a role. Other IU/CV applications of invariance
include (Zisserman, 1995); (1) Image and object featurc
transfer for 2D objects; (2) Model based object
recognition: given a perspective image of a scene, the
task of model based vision is to identify which objects if
any, from the model library, are in the scene; (3)
Epipolar Geometry: a point in one image determines a
line in the other on which the corresponding point must
lie. This reduces the correspondence (matching)
problem to 1D, rather than 2D search. (Used also
extensively in photogrammetry); (4) Transfer (image
transfer for 3D objects): given two images of a 3D
structure, points in a new image are determined, given
only a small number of point correspondences. This is
accomplished without reconstructing the 3D structure,
nor knowing the camera parameters or motion; (5) 3D
structure recovery (3D object reconstruction):
recovering non-euclidean 3D structure given only
corresponding image points in a stereo pair of views.
Using control points, the object is reconstructed in 3D
euclidean space. (Main application in photogrammetry).
33 Photogrammetric Analysis of Invariance
Invariance is based on the same mathematical principles
as photogrammetric theory. Therefore, one would
expect that invariance techniques would have equivalents
in photogrammetry. | Such techniques, which we
analyzed, include point- and line-based image and object
transfer for 2D planar objects (Barakat, 1994).
Invariance yields equations of straight lines the
intersections of which give the positions of the points to
be transferred. For non-redundant 4-point invariance,
the sequence of points used yields line pairs of different
geometric strengths. In redundant cases, using different
538
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
point sequences to form linear condition equations
results in least squares estimates which are different for
both the positions and their quality. Corresponding
photogrammetric techniques (which implement
projective transformation between planes) based on
point and line features, on the other hand, provide
unique estimates and covariances for both non-
redundant and redundant cases. A refined least squares
approach, for which the linear invariance equations
become non-linear, appears to alleviate the non-
uniqueness problem.
Next, point-based image invariance is investigated for
three-dimensional objects in multiple images; in
particular the use of the fundamental matrixto transfer
images from two photographs to a third. Introducing
the constraint of zero determinant on the fundamental
matrix stabilizes the solution, which otherwise leads to
widely varying results. Accurate recovery of F is quite
critical as will be discussed also in object reconstruction
in the following section.
3.4 Object Reconstruction By Invariance
In the derivation of invariance relationships for image
transfer, object coordinates are eliminated and the
image acquisition parameters are usually lumped
together and replaced by other nonphysically significant
parameters such as the fundamental matrix. In an
alternative derivation, algebraic elimination of the
camera orientation parameters from the equations
results in invariant coordinates of the object points.
These coordinates are identical from any two images of
the object, provided that 5 control points, not any four
of which lie in a plane, are identified in both images.
The 3-D object is, then, reconstructed from the
invariant coordinates using a cross-ratio of determinants
in a similar approach to the 2-D (planar object) case.
According to Barrett (Barrett, 1994) the method is
explained as follows. Two points are selected, e.g. P,,
and P,, and the line passing through them becomes the
"spine" of a "pencil" of three planes; P,P,P,, P,P,P,,
and P P,P, as shown in the figure. For any other
general object point, P, a fourth plane in this pencil is
constructed, P, P,P. Then, the cross-ratio of these four
planes is computed as the first invariant coordinate of
P; C,(P). The procedure is repeated for two other
choices of the "spine" of the pencil, e.g, P,P, and
P,P,. The resulting set of cross-ratios of planes;
C,(P), C,(P),C;(P) provides invariant properties of the
three planes in space hinged on the spines P, P,,P,P,
and P,P,. These three planes intersect at the general
point P, whose object coordinates are thus calculated.
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