Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Registration of algebraic varieties using Riemannian optimization

Florentin Goyens, Coralia Cartis, Stéphane Chrétien

TL;DR

The paper addresses point cloud registration by leveraging a low-dimensional nonlinear structure: each cloud is first approximated by an algebraic variety using polynomial features on the Grassmann manifold, then a rigid transform $(Q,a)$ is found to align the resulting varieties without requiring point correspondences. The denoising and registration problems are cast as smooth nonconvex optimizations on Riemannian manifolds, solved with second-order methods, and accompanied by Stein’s unbiased risk estimates to bound denoising error. Empirical results on synthetic data and dental scans show accurate denoising and robust registration even with partial or no overlap, highlighting the approach’s potential when surfaces admit polynomial approximations. The method, while effective for low-dimensional data, faces scaling challenges that the authors propose addressing via patch-based strategies, orthogonal polynomial bases, and stochastic optimization in future work.

Abstract

We consider the point cloud registration problem, the task of finding a transformation between two point clouds that represent the same object but are expressed in different coordinate systems. Our approach is not based on a point-to-point correspondence, matching every point in the source point cloud to a point in the target point cloud. Instead, we assume and leverage a low-dimensional nonlinear geometric structure of the data. Firstly, we approximate each point cloud by an algebraic variety (a set defined by finitely many polynomial equations). This is done by solving an optimization problem on the Grassmann manifold, using a connection between algebraic varieties and polynomial bases. Secondly, we solve an optimization problem on the orthogonal group to find the transformation (rotation $+$ translation) which makes the two algebraic varieties overlap. We use second-order Riemannian optimization methods for the solution of both steps. Numerical experiments on real and synthetic data are provided, with encouraging results. Our approach is particularly useful when the two point clouds describe different parts of an objects (which may not even be overlapping), on the condition that the surface of the object may be well approximated by a set of polynomial equations. The first procedure -- the approximation -- is of independent interest, as it can be used for denoising data that belongs to an algebraic variety. We provide statistical guarantees for the estimation error of the denoising using Stein's unbiased estimator.

Registration of algebraic varieties using Riemannian optimization

TL;DR

The paper addresses point cloud registration by leveraging a low-dimensional nonlinear structure: each cloud is first approximated by an algebraic variety using polynomial features on the Grassmann manifold, then a rigid transform is found to align the resulting varieties without requiring point correspondences. The denoising and registration problems are cast as smooth nonconvex optimizations on Riemannian manifolds, solved with second-order methods, and accompanied by Stein’s unbiased risk estimates to bound denoising error. Empirical results on synthetic data and dental scans show accurate denoising and robust registration even with partial or no overlap, highlighting the approach’s potential when surfaces admit polynomial approximations. The method, while effective for low-dimensional data, faces scaling challenges that the authors propose addressing via patch-based strategies, orthogonal polynomial bases, and stochastic optimization in future work.

Abstract

We consider the point cloud registration problem, the task of finding a transformation between two point clouds that represent the same object but are expressed in different coordinate systems. Our approach is not based on a point-to-point correspondence, matching every point in the source point cloud to a point in the target point cloud. Instead, we assume and leverage a low-dimensional nonlinear geometric structure of the data. Firstly, we approximate each point cloud by an algebraic variety (a set defined by finitely many polynomial equations). This is done by solving an optimization problem on the Grassmann manifold, using a connection between algebraic varieties and polynomial bases. Secondly, we solve an optimization problem on the orthogonal group to find the transformation (rotation translation) which makes the two algebraic varieties overlap. We use second-order Riemannian optimization methods for the solution of both steps. Numerical experiments on real and synthetic data are provided, with encouraging results. Our approach is particularly useful when the two point clouds describe different parts of an objects (which may not even be overlapping), on the condition that the surface of the object may be well approximated by a set of polynomial equations. The first procedure -- the approximation -- is of independent interest, as it can be used for denoising data that belongs to an algebraic variety. We provide statistical guarantees for the estimation error of the denoising using Stein's unbiased estimator.
Paper Structure (13 sections, 2 theorems, 38 equations, 14 figures, 2 tables)

This paper contains 13 sections, 2 theorems, 38 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions and outline of the paper
  4. Preliminaries: algebraic varieties and polynomial features
  5. Approximation by an algebraic variety (denoising)
  6. Formulation as an optimization problem
  7. Statistical error estimation
  8. Numerical results for denoising of algebraic varieties
  9. Registration of algebraic varieties
  10. Formulation as an optimization problem
  11. Numerical results for registration
  12. Conclusions
  13. Acknowledgement

Key Result

Theorem 3.1

Let $\mathcal{M}_1, \mathcal{M}_2, \mathcal{M}_3$ be manifolds. Let $F\colon \mathcal{M}_1 \times \mathcal{M}_2 \to \mathcal{M}_3$ be smooth and let $(x_0, y_0) \in \mathcal{M}_1 \times \mathcal{M}_2$ with $F(x_0, y_0) =0$. If $\mathrm{D}_2 F(x_0, y_0)$ is invertible, there exists open neighbourhood

Figures (14)

  • Figure 1: Point-to-point ICP.
  • Figure 2: Point-to-plane ICP bellekens2014survey.
  • Figure 3: Denoising of an algebraic variety
  • Figure 4: Denoising a circle.
  • Figure 5: Denoising a nonsmooth algebraic variety.
  • ...and 9 more figures

Theorems & Definitions (6)

  • Definition 2.1: Algebraic variety model cox1994ideals
  • Theorem 3.1: abraham2012manifolds, Prop. 3.3.13
  • proposition 1
  • proof
  • Remark 4.1
  • Remark 4.2