An efficient algorithm for finding the centers of conics and quadrics in noisy data

We present an efficient algorithm for finding the center of conics and quadrics of known parameters in noisy or scarce data. The problem arises in applications where a conic or quadric of known parameters, such as a circle of known radius, is extracted from a scene or part. Although the original problem is nonlinear and usually requires an iterative method for its solution, we reduce it to the well-known problem of minimizing a nonhomogeneous quadratic expression on the unit sphere. In the case of closed conics and quadrics, such as circles, ellipses, spheres, and ellipsoids, we obtain the solution in just one iteration, and no starting estimate is required. For hyperbolas and hyperboloids, we describe the Gauss Seidel algorithm, for which we give a Lyapunov type proof of convergence. Furthermore, every iteration of this algorithm satisfies all constraints.