发表机构
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对现有最优传输模型无法识别分布间几何变换的局限，提出带地标约束的耦合最优传输框架，整合传输计划与形变场优化，通过有限元算法验证其在形状匹配中的有效性。
AI 中文摘要
现有的最优传输（OT）模型主要通过最小化规定的传输成本或失真，来寻求分布之间的OT映射或计划。然而，仅最小化传输成本或失真可能无法识别两个分布之间具有几何意义的变换。为解决这一局限，本文提出一种新颖的耦合OT框架，该框架利用少量带注释的地标来指导恢复控制分布变换的潜在形变。耦合OT框架将传输计划和形变场的优化整合为统一模型，其中由地标引导的形变场与由成本驱动的传输计划通过互一致性约束耦合。因此，形变由带注释的地标和成本驱动的分布匹配共同决定。所提框架在地标配准与基于传输的分布匹配之间建立了原则性联系，能够从稀疏几何监督中恢复传输映射。我们在一般变分设定中证明了所提模型的适定性，并开发了基于有限元的数值算法进行计算，对其收敛性进行了系统分析。所提方法的实际有效性在形状匹配中得到验证。
英文摘要
Existing optimal transport (OT) models primarily seek an OT map or plan between distributions by minimizing a prescribed transport cost or distortion. However, minimizing transport cost or distortion alone may fail to identify a geometrically meaningful transformation between the two distributions. To address this limitation, this paper proposes a novel coupled OT framework that leverages a small number of annotated landmarks to guide the recovery of an underlying deformation governing the distribution transformation. The coupled OT framework integrates the optimization of the transport plan and the deformation field into a unified model, where the landmark-guided deformation field and the cost-driven transport plan are coupled through a mutual-consistency constraint. As a result, the deformation is jointly determined by the annotated landmarks and cost-driven distribution matching. The proposed framework provides a principled connection between landmark-based registration and transport-based distribution matching, enabling the recovery of transport maps from sparse geometric supervision. We establish the well-definedness of the proposed model in a general variational setting and develop a finite-element-based numerical algorithm for computation whose convergence properties are systematically analyzed. The practical effectiveness of the proposed approach is verified in shape matching.
