AI 中文总结
本研究针对计算解剖学中配准时高维变形吸收低维分量的问题，提出基于变流形一阶变分的耦合得分，将其作为惩罚项用于配准，可解耦变形并在数值实验中展现出良好效果。
AI 中文摘要
在计算解剖学中，分析形状群体间的形态变异性通常需要结合结构化运动和无约束微分同胚的多分量变形模型。然而，配准过程中存在一个主要挑战：高维变形往往会吸收低维分量，改变真实的几何变异性，阻碍准确的统计分析。为解决该问题，我们提出一种新型耦合得分，用于在配准过程中解耦不同的变形模式。该得分采用变流形的一阶变分定义，变流形是向量场对形状的无穷小作用的变流形表示。所提得分量化了给定向量场对形状的作用可被另一向量场子空间复制的程度。我们对该耦合得分进行了理论分析，阐明其在有限维空间上的行为。最后，我们将该得分作为惩罚项整合到配准问题中。数值实验表明，其在多种场景中具有效率，例如强制或阻止特定运动，以及通过强制结构化方向先验迭代校正复杂匹配场景。
英文摘要
In computational anatomy, analyzing morphological variability across shape populations often requires multi-component deformation models that combine structured motions and unconstrained diffeomorphisms. However, a major challenge arises during the registration process, as high-dimensional deformations tend to absorb lower-dimensional components, altering the true geometric variability and preventing accurate statistical analysis. To address this issue, we introduce a novel coupling score designed to decouple distinct deformation modes during registration. This score is defined using first variation of varifolds which is a varifold representation of the infinitesimal action of vector fields on shapes. The proposed score quantifies the extent to which the action of a given vector field on a shape can be replicated by another subspace of vector fields. We provide a theoretical analysis of this coupling score, illustrating its behavior on finite-dimensional spaces. Finally, we integrate this score as a penalization term in registration problems. Numerical experiments illustrate its efficiency in various use cases such as enforcing or preventing specific motions, and iteratively correcting complex matching scenarios by enforcing structured directional priors.
