Reconstructing Ellipsoids from Projections

Abstract

In this paper we examine the problem of reconstructing a (possibly dynamic) ellipsoid from its (possibly inconsistent) orthogonal silhouette projections. We present a particularly convenient representation of ellipsoids as elements of the vector space of symmetric matrices. The relationship between an ellipsoid and its orthogonal projections in this representation is linear, unlike the standard parameterization based on semiaxis length and orientation. This representation is used to completely and simply characterize the solutions to the reconstruction problem. The representation also allows the straightforward inclusion of geometric constraints on the reconstructed ellipsoid in the form of inner and outer bounds on recovered ellipsoid shape. The inclusion of a dynamic model with natural behavior, such as stretching, shrinking, and rotation, is similarly straightforward in this framework and results in the possibility of dynamic ellipsoid estimation. For example, the linear reconstruction of a dynamic ellipsoid from a single lower-dimensional projection observed over time is possible. Numerical examples are provided to illustrate these points.