Akihiro SUGIMOTO, Masahiko SHIZAWA
Two Plane Structures and Motions from Point Correspondences in Two Images
Abstract:This paper provides a study on the structure-from-motion problem under the conditions
that two planes independently and rigidly move in three dimensions and that, for given two
perspective images, the correspondences of points in the planes are known. A typical approach
to this problem involves segmenting the images into regions each of which has only one plane,
and then determining the normal vector of each plane and the motion of the plane. In this
paper, however, we take a different approach where the images need not be segmented: we
directly handle the images in which two planes exist. We show that we can generally determine
the normal vectors of the two planes and the two motions when we observe 17 points, where
the tensor product of two transformation matrices and its decomposition play the central role.
We first determine the tensor product and then decompose it into the two transformation matrices.
Here the tensor product is expressed as a pair of its symmetric part and its alternating
part. We also clarify the cases where the tensor product cannot be determined. It is shown
that when the two planes share the same rotation, we cannot determine the alternating part
if the two normal vectors are parallel or the two translation vectors are parallel. Furthermore,
we show that unless at least four points exist in each plane such that no three of them are
collinear, we cannot determine the symmetric part; whereas at least seven points are needed
in each plane to determine the alternating part.
Key Words: structure from motion, planes, transformation matrix, tensor product, symmetrization,
alternization, critical conditions.