Coherent Point Drift (CPD) is a representative probabilistic framework for unsupervised non-rigid point set registration. Its standard non-rigid M-step, however, relies on a point-indexed Gaussian-kernel system whose size grows with the number of moving points, making deformation estimation computationally heavy for large point sets and difficult to control in complexity during registration. To address these limitations, we propose Analytic-CPD, a new unsupervised non-rigid registration framework that gives CPD a structured analytic reformulation. Analytic-CPD preserves the CPD posterior correspondence layer, but lifts the M-step from point-indexed kernel displacement estimation to structured analytic mapping estimation. By coupling the Gaussian-mixture posterior mechanism of CPD with Structured Analytic Mappings (SAM), the method obtains a deformation model whose coefficient dimension is governed by the ambient dimension and analytic order rather than by the number of moving points. More importantly, deformation estimation is organized over an interpretable hierarchy of analytic function spaces, so the analytic order can be increased progressively as posterior correspondences become more reliable. We implement this idea through an increasing-degree continuation strategy with decreasing stage lengths: low-order analytic maps first stabilize the posterior correspondence structure, while higher-order modes later refine nonlinear residual deformation. Experiments on controlled model-matched, smooth model-mismatch, and registered human-shape data demonstrate the effectiveness and favorable accuracy--efficiency performance of Analytic-CPD.

翻译：相干点漂移（CPD）是一种具有代表性的无监督非刚性点集配准概率框架。然而，其标准非刚性M步骤依赖于点索引高斯核系统，该系统的大小随移动点数量增长，使得大点集的形变估计计算量巨大，且配准过程中复杂度难以控制。为解决这些局限，我们提出Analytic-CPD——一种新的无监督非刚性配准框架，为CPD赋予结构化解析重构。Analytic-CPD保留了CPD的后验对应层，但将M步骤从点索引核位移估计提升为结构化解析映射估计。通过将CPD的高斯混合后验机制与结构化解析映射（SAM）耦合，该方法获得的形变模型系数维度由环境维度和解析阶数而非移动点数量决定。更重要的是，形变估计在可解释的解析函数空间层次结构中有序组织，因此可随着后验对应关系更可靠而逐步提高解析阶数。我们通过递减阶段长度的递增阶数延续策略实现这一思想：低阶解析映射首先稳定后验对应结构，高阶模式随后精化非线性残差形变。在受控模型匹配、光滑模型失配及配准人体形状数据上的实验证明了Analytic-CPD的有效性及优越的精度-效率性能。