An Effective Rigidity Constraint for Improving RANSAC in Homography Estimation - Advanced Concepts for Intelligent Vision Systems - page 208

Fig. 3. All possible rotation-invariant oriented closed paths passing through four points

2 and the
solution is then projected back into the Euclidean plane of an image. But as the
projective plane

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are evaluated using homogeneous coordinates in the projective plane

2 is two-sheeted and not simply connected, the result can be
very surprising once projected into the simply-connected Euclidean plane.
Given that a homography exists for any four-correspondences set, it is im-
possible for RANSAC to reject degenerated homographies when the number of
correspondences consistent with them equals or is greater than the number of
inliers. A solution could come out from the ability to decide, from each sample
of correspondences to be evaluated by RANSAC, whether a particular sample
can possibly lead to a rigid-body transformation or not.

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3.2 Toward a Rigidity Constraint for Improving RANSAC

In the previous analysis of some invalid homographies, it was shown that when
considering the shapes formed by the four point correspondences, before and
after the projective transformation, modifying the relative order of the corners
of a shape or changing its convexity leads to a non-rigid-body transformation.
In other words, for a homography to correspond to a rigid-body transformation,
regardless of any rotation or relative distance variation between the corners, a
fully convex shape has to be related to a fully convex shape and a shape with one
concavity has to be related to another shape with one concavity. Additionally,
in the case of shapes with one concavity, the point correspondences have to be
correctly ordered for the concavities to be related by the same correspondence.
In a first approach, the basic idea of checking for a rigid-body transformation
from a randomly chosen set of four-correspondences, could be to order the point
correspondences and link them together to form either a fully concave shape or
a shape with one concavity, given that with four points, it can only be one or
the other. Then, it has to be verified whether the shape correspondence pre-
serves the convexity, and in the case of a concavity in the shapes, whether the
concavity is related by the same correspondence. But this approach would be by
far too complex. A better solution consists of keeping the initial random order
of the point correspondences and of finding a criterion which makes it possi-
ble to confirm whether the shape transformation is consistent with a rigid-body
transformation or not. In order to do so, it is possible to consider, for both the