Fast and Robust Algorithms for Harmonic Energy Minimization on Genus-0 Surfaces

Surface harmonic map between genus-0 surfaces plays an important role in applied mathematics and engineering, with applications in medical imaging and computer graphics. Previous work [1] introduces a variational approach for computing surface harmonic maps. It obtains global conformal parameterizations of genus-0 surfaces through minimizing the harmonic energy. Two weaknesses of this approach are addressed in this paper: the slow speed of its gradient descent iterations and the presence of undesired parameterization foldings when the underlying surface has long sharp features. This paper proposes an algorithm that significantly accelerates the harmonic map computation and a method that helps this algorithm produce global conformal parameterizations without foldings. These are achieved by applying recent results of optimization on manifolds and taking advantages of the weighted Laplace-Beltrami eigen-projection. Experimental results show that the proposed approaches compute genus-0 surface harmonic maps much faster than the existing algorithm in [1] and the results contain no foldings.