  Surface parameterizations are widely applied in computer graphics, medical
imaging and transformation optics. In this paper, we rigorously derive the
gradient vector and Hessian matrix of the discrete conformal energy for
spherical conformal parameterizations of simply connected closed surfaces of
genus-$0$. In addition, we give the sparsity structure of the Hessian matrix,
which leads to a robust Hessian-based trust region algorithm for the
computation of spherical conformal maps. Numerical experiments demonstrate the
local quadratic convergence of the proposed algorithm with low conformal
distortions. We subsequently propose an application of our method to surface
registrations that still maintains local quadratic convergence.
