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.
1. Introduction. Point set registration is a fundamental problem in computer vision and geometric data analysis, with wide-ranging applications in 3D reconstruction [28], sensor alignment [11], motion tracking [10], and robotic perception [36]. While rigid and affine methods have been extensively studied [4,42,48], registering point sets under smooth non-rigid deformations remains a significant challenge due to the need for expressive mappings in high-dimensional spaces. Such techniques are crucial in modern systems ranging from visual metrology [28] to augmented reality [50] and autonomous navigation [14].
Most existing non-rigid registration methods adopt non-parametric models, such as radial basis functions [17], Gaussian mixtures [33], or statistical estimation frameworks [25,19]. While flexible, these models typically require large numbers of basis functions or parameters to express complex deformations, resulting in high computational complexity and increased risk of overfitting [31,35,43]. To ensure numerical stability, additional regularization is often introduced, which may obscure geometric structure and hinder interpretability.
From a mathematical perspective, multivariate Taylor expansions offer a classical framework for approximating smooth functions via structured polynomial bases. In particular, Vetter’s work [45] provides a general formulation for matrix-and vector-valued Taylor expansions, enabling analytic mappings to be expressed with controllable complexity and explicit algebraic structure. Despite its theoretical appeal, this formulation remains underexplored in computer vision and registration tasks.
In this paper, we present the first application of multivariate Taylor expansion of vector-valued functions to non-rigid point set registration. By constructing a low-dimensional admissible function space spanned by truncated Taylor basis terms, we provide a compact and expressive framework capable of representing smooth deformations without relying on kernel-based models or dense parameter sets.
To illustrate the expressive capacity of low-order analytic expansions, we show in Fig. 1 a series of 2D deformations generated from a second-order Taylor model with only three parameters. Despite its simplicity, the model can produce a rich variety of nonlinear and smooth transformations.
Based on this representation, we design a progressive fitting algorithm that lifts the identity map to higher-order analytic forms via structured quasi-Newton optimization. This method is integrated into a standard ICP loop-with (by default) nearest-neighbor correspondences-to construct Analytic-ICP, a fast and robust registration algorithm that supports hierarchical refinement of rigid, affine, and nonlinear mappings.
Our main contributions are summarized as follows:
• We present the first structured approximation framework based on multivariate Taylor expansions of vector-valued functions in computer vision. This formulation enables interpretable and hierarchical modeling of smooth deformations, and provides a principled foundation for registration and beyond. (0, 0, 0) (0.5, 0, 0) (0, 0.9, 0) (0, 0, 0.7) (-0.5, -1.2, 0) (0, 0.9, -0.5) (0.5, 0, -0.7) (-0.2, 1.2, 0.8)
Fig. 1: Illustrative effect of second-order analytic deformation. Even with only three tunable coefficients (a 1 , a 2 , a 3 ), the second-order Taylor expansion model produces a diverse range of smooth, nonlinear transformations.
• We formalize a matrix-vector representation of analytic mappings using generalized derivative matrices and generalized monomial vectors, and prove a Structured Approximation Theorem establishing the approximation capability of truncated Taylor models over smooth mappings. • We construct a low-dimensional function space spanned by truncated Taylor basis terms, unifying rigid, affine, and nonlinear transformations within a single closed-form expression. • We design a progressive quasi-Newton fitting algorithm that incrementally lifts the identity map into higher-order approximants, offering controllable model complexity and efficient convergence. • We integrate the proposed framework into an ICP-style optimization routine. The resulting Analytic-ICP algorithm achieves quasi-linear complexity and outperforms classical non-rigid methods (e.g., CPD, TPS-RPM) in both accuracy and efficiency on 2D and 3D registration tasks. To facilitate reproducibility, source code and data are available; see Section 7.1 for details.
- Related Work. We propose a novel analytic mapping model that can be integrated into many unsupervised registration frameworks to improve their performance. In this section, we briefly review the development of rigid, affine, and non-rigid registration techniques that are relevant to our method.
2.1. Rigid and Affine Registration. The Iterative Closest Point (ICP) algorithm [17] is one of the most fundamental and widely used geometric registration methods. It alternates between estimating correspond
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