Optimization without Minimization Search: Constraint Satisfaction by Orthogonal Projection with Applications to Multiview Triangulation

Abstract

We present an alternative approach to what we call the “standard optimization”, which minimizes a cost function by searching a parameter space. Instead, our approach “projects” in the joint observation space onto the manifold defined by the “consistency constraint”, which demands that any minimal subset of observations produce the same result. This approach avoids many difficulties encountered in the standard optimization. As typical examples, we apply it to line fitting and multiview triangulation. The latter produces a new algorithm far more efficient than existing methods. We also discuss the optimality of our approach.