On Some Properties of Calibrated Trifocal Tensors
This work addresses a foundational problem in computer vision for researchers and practitioners in multi-view geometry, providing incremental theoretical insights into algebraic constraints for calibrated trifocal tensors.
The paper tackles the problem of characterizing calibrated trifocal tensors in three-view geometry by introducing the trifocal essential matrix as a generalization of the bifocal essential matrix, proving necessary and sufficient conditions for it, and deriving 15 quartic and 99 quintic polynomial constraints for calibrated trifocal tensors, with the quartic constraints shown to be sufficient in real cases.
In two-view geometry, the essential matrix describes the relative position and orientation of two calibrated images. In three views, a similar role is assigned to the calibrated trifocal tensor. It is a particular case of the (uncalibrated) trifocal tensor and thus it inherits all its properties but, due to the smaller degrees of freedom, satisfies a number of additional algebraic constraints. Some of them are described in this paper. More specifically, we define a new notion --- the trifocal essential matrix. On the one hand, it is a generalization of the ordinary (bifocal) essential matrix, and, on the other hand, it is closely related to the calibrated trifocal tensor. We prove the two necessary and sufficient conditions that characterize the set of trifocal essential matrices. Based on these characterizations, we propose three necessary conditions on a calibrated trifocal tensor. They have a form of 15 quartic and 99 quintic polynomial equations. We show that in the practically significant real case the 15 quartic constraints are also sufficient.