Many genus-0 surface mapping tasks such as landmark alignment, feature matching, and image-driven registration, can be reduced (via an initial spherical conformal map) to optimizing a spherical self-homeomorphism with controlled distortion. However, existing works lack efficient mechanisms to control the geometric distortion of the resulting mapping. To resolve this issue, we formulate this as a Beltrami-space optimization problem, where the angle distortion is encoded explicitly by the Beltrami differential and bijectivity can be enforced through the constraint $\|μ\|_{\infty}<1$. To make this practical on the sphere, we introduce the Spherical Beltrami Differential (SBD), a two-chart representation of quasiconformal self-maps of the unit sphere $\mathbb{S}^2$, together with cross-chart consistency conditions that yield a globally bijective spherical deformation (up to conformal automorphisms). Building on the Spectral Beltrami Network, we develop BOOST, a differentiable optimization framework that updates two Beltrami fields to minimize task-driven losses while regularizing distortion and enforcing consistency along the seam. Experiments on large-deformation landmark matching and intensity-based spherical registration demonstrate improved task performance meanwhile maintaining controlled distortion and robust bijective behavior. We also apply the method to cortical surface registration by aligning sulcal landmarks and matching cortical sulcal depth, achieving comparative or better registration performance without sacrificing geometric validity.

翻译：许多亏格为0的表面映射任务，如地标对齐、特征匹配和图像驱动的配准，可以通过初始球面共形映射简化为优化具有可控失真的球面自同胚映射。然而，现有工作缺乏有效机制来控制所得映射的几何失真。为解决这一问题，我们将其表述为Beltrami空间优化问题，其中角度失真由Beltrami微分显式编码，而双射性可通过约束$\|μ\|_{\infty}<1$来保证。为使该方法在球面上实用，我们引入了球面Beltrami微分（SBD），这是单位球面$\mathbb{S}^2$拟共形自映射的一种双图表示，并结合跨图一致性条件，从而产生全局双射的球面形变（模去共形自同构）。基于谱Beltrami网络，我们开发了BOOST，这是一个可微优化框架，通过更新两个Beltrami场来最小化任务驱动的损失，同时沿接缝正则化失真并强制执行一致性。在大变形地标匹配和基于强度的球面配准上的实验表明，该方法在保持可控失真和鲁棒双射行为的同时，提高了任务性能。我们还将该方法应用于皮层表面配准，通过对齐脑沟地标并匹配皮层沟深，在不牺牲几何有效性的前提下，取得了相当或更好的配准性能。