The Geometry of Dynamic Scenes - On Coplanar and Convergent Linear Motions Embedded in 3D Static Scenes

In this paper, we consider structure and motion recovery for scenes consisting of static and dynamic features. More particularly, we consider a single moving uncalibrated camera observing a scene consisting of points moving along straight lines converging to a unique point and lying on a motion plane. This scenario may describe a roadway observed by a moving camera whose motion is unknown. We show that there exist matching tensors similar to fundamental matrices. We derive the link between dynamic and static structure and motion and show how the equation of the motion plane (or equivalently the plane homographies it induces between images) may be recovered from dynamic features only. Experimental results on real images are provided, in particular on a 60-frames video sequence

On Degeneracy of Linear Reconstruction from Three Views: Linear Line Complex and Applications

This paper investigates the linear degeneracies of projective structure estimation from point and line features across three views. We show that the rank of the linear system of equations for recovering the trilinear tensor of three views reduces to 23 (instead of 26) in the case when the scene is a Linear Line Complex (set of lines in space intersecting at a common line) and is 21 when the scene is planar. The LLC situation is only linearly degenerate, and we show that one can obtain a unique solution when the admissibility constraints of the tensor are accounted for. The line configuration described by an LLC, rather than being some obscure case, is in fact quite typical. It includes, as a particular example, the case of a camera moving down a hallway in an office environment or down an urban street. Furthermore, an LLC situation may occur as an artifact such as in direct estimation from spatio-temporal derivatives of image brightness. Therefore, an investigation into degeneracies and their remedy is important also in practice

Tensor Algebra: A Combinatorial Approach to the Projective Geometry of Figures

This paper explores the combinatorial aspects of symmetric and antisymmetric forms represented in tensor algebra. The development of geometric perspective gained from tensor algebra has resulted in the discovery of a novel projection operator for the Chow form of a curve in P3 with applications to computer vision

The Geometry of Projective Reconstruction I: Matching Constraints and the Joint Image

Circulated in 1995. Accepted subject to revision to IJCV in 1995, but never completedThis paper studies the geometry of perspective projection into multiple images and the matching constraints that this induces between the images. The combined projections produce a 3D subspace of the space of combined image coordinates called the joint image. This is a complete projective replica of the 3D world defined entirely in terms of image coordinates, up to an arbitrary choice of certain scale factors. Projective reconstruction is a canonical process in the joint image requiring only the rescaling of image coordinates. The matching constraints tell whether a set of image points is the projection of a single world point. In 3D there are only three types of matching constraint: the fundamental matrix, Shashua's trilinear tensor, and a new quadrilinear 4 image tensor. All of these fit into a single geometric object, the joint image Grassmannian tensor. This encodes exactly the information needed for reconstruction: the location of the joint image in the space of combined image coordinates