Structured Analytic Mappings for Point Set Registration

TL;DR

An analytic approximation model for non-rigid point set registration, grounded in the multivariate Taylor expansion of vector-valued functions, which is embedded into a standard ICP loop, resulting in Analytic-ICP, an efficient registration algorithm with quasi-linear time complexity.

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.

Figures (16)

• Figure 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.
• Figure 1: Pipeline of the structured analytic mapping approach. At iteration $t$, we form correspondences $C^{(t)}$ and fit in stages: rigid $\mathrm{SE}(n,\mathbb{R}) \rightarrow$ affine (residual) $\mathrm{AGL}(n,\mathbb{R})/\mathrm{SE}(n,\mathbb{R})$, then take a mutually exclusive branch: restricted projective $\mathrm{PGL}(n{+}1,\mathbb{R})/\mathrm{AGL}(n,\mathbb{R})$ for planar/homography–dominant cases, or structured Taylor lifting of order $m_t$ for smooth non-rigid deformation (if the stop test fails, increase $m_{t+1}\!\leftarrow m_t{+}1$). The diamond marks the stopping rule ($\Delta\mathrm{RMSE}$ / $\|\Delta\theta\|$ / order / iteration). A negative test returns to correspondence (outer ICP-style loop); a positive test outputs the moved point set and the final mapping $\tau$. When the correspondence step uses nearest-neighbor updates (ICP-style), we refer to this instantiation as Analytic-ICP; the coefficient fitting routine is detailed later (AMVFF).
• Figure 1: Hierarchical decomposition of linear components. The registration pipeline follows the subgroup chain $\mathrm{SE}(n,\mathbb{R})\subset \mathrm{AGL}(n,\mathbb{R})\subset \mathrm{PGL}(n{+}1,\mathbb{R})$. Each stage fits residual degrees of freedom along the corresponding homogeneous direction: first rigid motions $\mathrm{SE}(n,\mathbb{R})$, then affine modes modulo rigid $\mathrm{AGL}(n,\mathbb{R})/\mathrm{SE}(n,\mathbb{R})$ (scales, shears), and finally projective modes modulo affine $\mathrm{PGL}(n{+}1,\mathbb{R})/\mathrm{AGL}(n,\mathbb{R})$.
• Figure 1: Performance of 2D stepwise Analytic-ICP. The registration is decomposed into three stages: orthogonal (rigid), affine, and a structured analytic refinement. The analytic stage is implemented as a composition of quadratic (second-order) Taylor lifts, yielding an overall low-order but expressive mapping. In each row, the right panel shows the fixed (red) and moving (green) point sets; the left panels visualize the intermediate results after each stage.
• Figure 2: Residual visualization under stagewise order lifting. Each row shows a 2D point set (fish, numeral 8, curve). Left to right: moving set, affine result, and compositions of low-order stages with increasing stage orders (labels indicate a possible order sequence and the corresponding effective degree upper bound$\le\!\prod_s o_s$). Residuals decrease as effective order increases.

Theorems & Definitions (18)

• Definition 4.1: Generalized Derivative Matrix
• Definition 4.2: Generalized Monomial Vector
• Definition 4.3: Structured Taylor Expansion
• Theorem 4.4: Structured Taylor Approximation with Remainder
• Proof 1: Proof Sketch
• Lemma 4.5: Nontriviality of the Structured Remainder
• Proof 2
• Theorem 4.6: Structured Approximation of Smooth Registration Mappings
• Proof 3: Proof Sketch
• Lemma 4.7: Density of Structured Taylor Approximants in $C^\infty$