Newton geodesic optimization on special linear group

The Riemannian exponential map on a noncompact Lie Group, which is determined by a Riemannian metric, is different from the Lie group exponential map determined by one-parameter subgroups. The Riemannian exponential map which represents the geodesic of the optimal transformation is obtained in terms of the minimal geodesic equation on SL(n, R). Generally, the Newton optimization method on Lie group is independent of the connection but with the one-parameter group. Based on the parameterization of the manifold with the Riemannian exponential map, we propose an intrinsic Newton optimization method on special linear group and prove its locally quadratic convergence to critical point of the cost function. Our approach is slightly superior to the counterpart based on Lie group exponential map. We demonstrate this by an image registration example.