We propose a new Riemannian gradient descent method for computing spherical area-preserving mappings of topological spheres using a Riemannian retraction-based framework with theoretically guaranteed convergence. The objective function is based on the stretch energy functional, and the minimization is constrained on a power manifold of unit spheres embedded in 3-dimensional Euclidean space. Numerical experiments on several mesh models demonstrate the accuracy and stability of the proposed framework. Comparisons with two existing state-of-the-art methods for computing area-preserving mappings demonstrate that our algorithm is both competitive and more efficient. Finally, we present a concrete application to the problem of landmark-aligned surface registration of two brain models.

翻译：我们提出一种新的黎曼梯度下降方法，用于计算拓扑球面的球面保面积映射。该方法基于黎曼收缩框架，并具有理论保证的收敛性。目标函数采用拉伸能量泛函，优化过程受限于嵌入三维欧几里得空间的单位球面幂流形。在多个网格模型上的数值实验证明了该框架的精度与稳定性。与两种现有最先进的保面积映射方法相比，我们的算法兼具竞争力与更高效率。最后，我们将该方法具体应用于两个脑模型的标志点对齐表面配准问题。