用泰勒展开构建低复杂度点云配准模型，实现高效高精度非刚性匹配。
Structured Analytic Mappings for Point Set Registration
- 基于多变量函数泰勒展开构造结构化映射空间，支持平滑变形的显式表达
- 在2D/3D数据上比CPD、TPS-RPM更准确且收敛更快，尤其适合小而平滑形变
- 无需核函数或高维参数化，嵌入ICP框架，计算复杂度接近线性
我们提出一种基于向量值函数多重泰勒展开的非刚性点集配准解析近似模型。通过利用泰勒展开的代数结构，构建由截断基项张成的结构化函数空间，使平滑变形可低复杂度、显式表示。为估计该空间内的映射，我们设计了一种拟牛顿优化算法，逐步将恒等映射提升至更高阶解析形式。该结构化框架在单一闭式公式下统一了刚性、仿射与非线性变形，无需依赖核函数或高维参数化。所提模型嵌入标准ICP循环中（默认使用最近邻对应），形成Analytic-ICP，一种具有准线性时间复杂度的高效配准算法。在2D和3D数据集上的实验表明，Analytic-ICP在小而平滑形变场景下，相比经典方法如CPD和TPS-RPM，具有更高的精度和更快的收敛速度。
原文摘要 · Abstract (English)
We present an analytic approximation model for non-rigid point set registration, grounded in the multivariate Taylor expansion of vector-valued functions. By exploiting the algebraic structure of Taylor expansions, we construct a structured function space spanned by truncated basis terms, allowing smooth deformations to be represented with low complexity and explicit form. To estimate mappings within this space, we develop a quasi-Newton optimization algorithm that progressively lifts the identity map into higher-order analytic forms. This structured framework unifies rigid, affine, and nonlinear deformations under a single closed-form formulation, without relying on kernel functions or high-dimensional parameterizations. The proposed model is embedded into a standard ICP loop -- using (by default) nearest-neighbor correspondences -- resulting in Analytic-ICP, an efficient registration algorithm with quasi-linear time complexity. Experiments on 2D and 3D datasets demonstrate that Analytic-ICP achieves higher accuracy and faster convergence than classical methods such as CPD and TPS-RPM, particularly for small and smooth deformations.
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