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Abstract:
2D-3D pose estimation means to estimate the relative position and orientation
of a 3D object with respect to a reference camera system. This work has its
main focus on the theoretical foundations of the 2D-3D pose estimation problem:
We discuss the involved mathematical spaces and their interaction within higher
order entities. To cope with the pose problem (how to compare 2D projective
image features with 3D Euclidean object features), the principle we propose is
to reconstruct image features (e.g. points or lines) to one dimensional higher
entities (e.g. 3D projection rays or 3D reconstructed planes) and express
constraints in the 3D space. It turns out that the stratification hierarchy
[11] introduced by Faugeras is involved in the scenario. But since the
stratification hierarchy is based on pure point concepts a new algebraic
embedding is required when dealing with higher order entities. The conformal
geometric algebra (CGA) [24] is well suited to solve this problem, since it
subsumes the involved mathematical spaces. Operators are defined to switch
entities between the algebras of the conformal space and its Euclidean and
projective subspaces. This leads to another interpretation of the
stratification hierarchy, which is not restricted to be based solely on point
concepts. This work summarizes the theoretical foundations needed to deal with
the pose problem. Therefore it contains mainly basics of Euclidean, projective
and conformal geometry. Since especially conformal geometry is not well known
in computer science, we recapitulate the mathematical concepts in some detail.
We believe that this geometric model is useful also for many other computer
vision tasks and has been ignored so far. Applications of these foundations are
presented in Part II [36].
