Optimization without minimization search: Constraint satisfaction by orthogonal projection with applications to multiview triangulation

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 de-mands that any minimal subset of observations produce the same result. This approach avoids many difficulties encountered in the standard opti-mization. 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.