  In computer vision, camera pose estimation from correspondences between 3D
geometric entities and their projections into the image has been a widely
investigated problem. Although most state-of-the-art methods exploit low-level
primitives such as points or lines, the emergence of very effective CNN-based
object detectors in the recent years has paved the way to the use of
higher-level features carrying semantically meaningful information. Pioneering
works in that direction have shown that modelling 3D objects by ellipsoids and
2D detections by ellipses offers a convenient manner to link 2D and 3D data.
However, the mathematical formalism most often used in the related litterature
does not enable to easily distinguish ellipsoids and ellipses from other
quadrics and conics, leading to a loss of specificity potentially detrimental
in some developments. Moreover, the linearization process of the projection
equation creates an over-representation of the camera parameters, also possibly
causing an efficiency loss. In this paper, we therefore introduce an
ellipsoid-specific theoretical framework and demonstrate its beneficial
properties in the context of pose estimation. More precisely, we first show
that the proposed formalism enables to reduce the pose estimation problem to a
position or orientation-only estimation problem in which the remaining unknowns
can be derived in closed-form. Then, we demonstrate that it can be further
reduced to a 1 Degree-of-Freedom (1DoF) problem and provide the analytical
derivations of the pose as a function of that unique scalar unknown. We
illustrate our theoretical considerations by visual examples and include a
discussion on the practical aspects. Finally, we release this paper along with
the corresponding source code in order to contribute towards more efficient
resolutions of ellipsoid-related pose estimation problems.
