Structured Analytic Mappings for Point Set Registration

TL;DR

This paper introduces a structured analytic model for non-rigid point set registration using Taylor expansions, enabling efficient, unified deformation estimation with higher accuracy and faster convergence.

Contribution

It develops a novel analytic approximation framework that unifies various deformation types under a single explicit form, avoiding kernel functions and high-dimensional parameters.

Findings

Achieves higher registration accuracy than classical methods.

Demonstrates faster convergence on 2D and 3D datasets.

Handles small and smooth deformations effectively.

Abstract

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.