AI 中文总结
该研究统一Kendall地标形状空间与带微分同胚群右不变Sobolev度量的地标构形空间，定义屏蔽弹性算子实现所需几何，可避免地标碰撞并保留形状不变性，用于求解匹配与测地线计算问题。
AI 中文摘要
我们提出了Kendall地标形状空间与带微分同胚群上右不变Sobolev度量的地标构形空间的统一框架，前者针对欧氏几何的地标构形消除了刚体运动并固定了尺度。所得新地标形状空间兼具两种方法的核心特性：导出度量的正则性可避免地标碰撞；该度量定义于环境空间，与地标数量无关；保留了局部刚体变换，消除了全局刚体运动并固定了尺度。为实现这一点，我们定义了一类特殊的Sobolev型算子——屏蔽弹性算子，其零空间恰好由刚体运动构成；我们证明该算子可导出所需几何结构，并给出了数值求解匹配问题与计算测地线的方法。该构造使应用中可使用带足够正则度量的地标构形空间，同时保留Kendall形状空间标志性的形状不变性。
英文摘要
We present a unification of Kendall's landmark shape spaces, where rigid motions are factored out and scale fixed on landmark configurations equipped with Euclidean geometry, with landmark configuration spaces carrying Riemannian metrics descending from right-invariant Sobolev metrics on the diffeomorphism group. The resulting new landmark shape spaces achieve the defining properties of both approaches: The regularity of the descending metric prevents landmarks from colliding, the metric is defined in the ambient space independent of the number of landmarks, local rigid transformations are preserved, global rigid motions are removed, and scale fixed. To achieve this, we define a particular Sobolev-type operator, the screened elasticity operator, whose null-space consists exactly of the rigid motions, we show how this operator descends to achieve the desired geometry, and we present approaches to solving matching problems and computing geodesics numerically. The resulting construction allows the use of landmark configuration spaces with sufficiently regular metrics in applications while retaining the shape invariances that are a hallmark of Kendall's shape spaces.
