Spherical surface parameterization is a fundamental tool in geometry processing and imaging science. For a genus-0 closed surface, many efficient algorithms can map the surface to the sphere; consequently, a broad class of task-driven genus-0 mapping problems can be reduced to constructing a high-quality spherical self-map. However, existing approaches often face a trade-off between satisfying task objectives (e.g., landmark or feature alignment), maintaining bijectivity, and controlling geometric distortion. We introduce the Spherical Beltrami Differential (SBD), a two-chart representation of quasiconformal self-maps of the sphere, and establish its correspondence with spherical homeomorphisms up to conformal automorphisms. Building on the Spectral Beltrami Network (SBN), we propose a neural optimization framework BOOST that optimizes two Beltrami fields on hemispherical stereographic charts and enforces global consistency through explicit seam-aware constraints. Experiments on large-deformation landmark matching and intensity-based spherical registration demonstrate the effectiveness of our proposed framework. We further apply the method to brain cortical surface registration, aligning sulcal landmarks and jointly matching cortical sulci depth maps, showing improved task fidelity with controlled distortion and robust bijective behavior.

翻译：球面参数化是几何处理与成像科学中的基础工具。对于零亏格闭曲面，现有多种高效算法可将其映射至球面；因此，一大类任务驱动的零亏格映射问题可转化为构建高质量的球面自映射。然而，现有方法往往需要在满足任务目标（如地标或特征对齐）、保持双射性以及控制几何畸变之间进行权衡。本文提出球面Beltrami微分（SBD）——一种球面拟共形自映射的双图表示，并建立了其与球面同胚（模去共形自同构）的对应关系。基于谱Beltrami网络（SBN），我们提出神经优化框架BOOST，该框架通过半球球极投影图优化两个Beltrami场，并利用显式的接缝感知约束保证全局一致性。在大形变地标匹配与基于强度的球面配准实验中验证了所提框架的有效性。我们进一步将该方法应用于大脑皮层表面配准，在实现脑沟地标对齐的同时联合匹配皮层沟深图，在可控畸变与鲁棒双射行为下展现出更优的任务保真度。