  In this paper, we formulate a generic non-minimal solver using the existing
tools of Polynomials Optimization Problems (POP) from computational algebraic
geometry. The proposed method exploits the well known Shor's or Lasserre's
relaxations, whose theoretical aspects are also discussed. Notably, we further
exploit the POP formulation of non-minimal solver also for the generic
consensus maximization problems in 3D vision. Our framework is simple and
straightforward to implement, which is also supported by three diverse
applications in 3D vision, namely rigid body transformation estimation,
Non-Rigid Structure-from-Motion (NRSfM), and camera autocalibration. In all
three cases, both non-minimal and consensus maximization are tested, which are
also compared against the state-of-the-art methods. Our results are competitive
to the compared methods, and are also coherent with our theoretical analysis.
The main contribution of this paper is the claim that a good approximate
solution for many polynomial problems involved in 3D vision can be obtained
using the existing theory of numerical computational algebra. This claim leads
us to reason about why many relaxed methods in 3D vision behave so well? And
also allows us to offer a generic relaxed solver in a rather straightforward
way. We further show that the convex relaxation of these polynomials can easily
be used for maximizing consensus in a deterministic manner. We support our
claim using several experiments for aforementioned three diverse problems in 3D
vision.
