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SPARE: Symmetrized Point-to-Plane Distance for Robust Non-Rigid 3D Registration

Yuxin Yao, Bailin Deng, Junhui Hou, Juyong Zhang

TL;DR

SPARE tackles robust non-rigid 3D registration by combining a robust symmetrized point-to-plane distance with as-rigid-as-possible regularization to better estimate deformed normals. The approach uses a majorization-minimization solver that yields closed-form updates and incorporates a deformation-graph–based coarse alignment to initialize and constrain the solution, improving convergence and efficiency. Adaptive weights mitigate unreliable correspondences, enabling accurate alignment under noise, partial overlap, and large deformations. Extensive experiments across synthetic and real datasets demonstrate state-of-the-art accuracy and competitive speed versus both optimization-based and learning-based methods, indicating strong practical impact for 3D reconstruction, tracking, and motion capture. SPARE thus provides a robust, scalable framework for precise non-rigid surface registration with principled geometric regularization.

Abstract

Existing optimization-based methods for non-rigid registration typically minimize an alignment error metric based on the point-to-point or point-to-plane distance between corresponding point pairs on the source surface and target surface. However, these metrics can result in slow convergence or a loss of detail. In this paper, we propose SPARE, a novel formulation that utilizes a symmetrized point-to-plane distance for robust non-rigid registration. The symmetrized point-to-plane distance relies on both the positions and normals of the corresponding points, resulting in a more accurate approximation of the underlying geometry and can achieve higher accuracy than existing methods. To solve this optimization problem efficiently, we introduce an as-rigid-as-possible regulation term to estimate the deformed normals and propose an alternating minimization solver using a majorization-minimization strategy. Moreover, for effective initialization of the solver, we incorporate a deformation graph-based coarse alignment that improves registration quality and efficiency. Extensive experiments show that the proposed method greatly improves the accuracy of non-rigid registration problems and maintains relatively high solution efficiency. The code is publicly available at https://github.com/yaoyx689/spare.

SPARE: Symmetrized Point-to-Plane Distance for Robust Non-Rigid 3D Registration

TL;DR

SPARE tackles robust non-rigid 3D registration by combining a robust symmetrized point-to-plane distance with as-rigid-as-possible regularization to better estimate deformed normals. The approach uses a majorization-minimization solver that yields closed-form updates and incorporates a deformation-graph–based coarse alignment to initialize and constrain the solution, improving convergence and efficiency. Adaptive weights mitigate unreliable correspondences, enabling accurate alignment under noise, partial overlap, and large deformations. Extensive experiments across synthetic and real datasets demonstrate state-of-the-art accuracy and competitive speed versus both optimization-based and learning-based methods, indicating strong practical impact for 3D reconstruction, tracking, and motion capture. SPARE thus provides a robust, scalable framework for precise non-rigid surface registration with principled geometric regularization.

Abstract

Existing optimization-based methods for non-rigid registration typically minimize an alignment error metric based on the point-to-point or point-to-plane distance between corresponding point pairs on the source surface and target surface. However, these metrics can result in slow convergence or a loss of detail. In this paper, we propose SPARE, a novel formulation that utilizes a symmetrized point-to-plane distance for robust non-rigid registration. The symmetrized point-to-plane distance relies on both the positions and normals of the corresponding points, resulting in a more accurate approximation of the underlying geometry and can achieve higher accuracy than existing methods. To solve this optimization problem efficiently, we introduce an as-rigid-as-possible regulation term to estimate the deformed normals and propose an alternating minimization solver using a majorization-minimization strategy. Moreover, for effective initialization of the solver, we incorporate a deformation graph-based coarse alignment that improves registration quality and efficiency. Extensive experiments show that the proposed method greatly improves the accuracy of non-rigid registration problems and maintains relatively high solution efficiency. The code is publicly available at https://github.com/yaoyx689/spare.
Paper Structure (25 sections, 1 theorem, 45 equations, 15 figures, 8 tables, 2 algorithms)

This paper contains 25 sections, 1 theorem, 45 equations, 15 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Alignment Error Metric
  4. Deformation Field
  5. Correspondences
  6. Proposed Method
  7. Preliminary: Symmetrized Point-To-Plane Distance
  8. Non-rigid Registration with Robust SP2P Distance
  9. Numerical Solver
  10. Coarse Alignment Using a Deformation Graph
  11. Results
  12. Experiment Settings
  13. Comparison with State-of-the-Art Methods
  14. Effectiveness of Components
  15. Limitations and Future Work
  16. ...and 10 more sections

Key Result

Proposition 1

If $\mathbf{d}\neq \mathbf{0}$, then $f(\mathbf{R}_i)$ in Eq. eq:f-Ri satisfies where $\mathcal{P}$ is a plane containing all vectors $\mathbf{h} \in \mathbb{R}^3$ that satisfy i.e., $f(\mathbf{R}_i)$ is the squared distance from $\mathbf{R}_i\mathbf{n}_i$ to the plane $\mathcal{P}$, scaled by a factor $\|\mathbf{d}\|^2$.

Figures (15)

  • Figure 1: The target positions of the points on the source surface when minimizing different distance metrics. Here, we only show the position and the normal vector of a potential minimum point. When considering the point-to-plane distance, the position of the minimum point may slide along the tangent plane of the target point. For the symmetrized point-to-plane distance, the minimum point may slide along a symmetric surface that passes through the source point. Additionally, for the point-to-point distance and the point-to-plane distance, the normal vector of the minimum point may rotate around the minimum point.
  • Figure 2: Results from different methods on two problem instances from the "handstand"(top) and "march1"(bottom) sequences from vlasic2008articulated. For each method, we show the deformed mesh (left), alignment result (right-bottom), and an error map (right-top) that visualizes the distance between each point and its ground-truth corresponding points, as well as the RMSE and the computational time.
  • Figure 3: The results obtained from different methods on problem instances with partial overlaps from vlasic2008articulated. For each method, we show the deformed mesh (left), the alignment result (right-bottom), and an error map (right-top) that visualizes the distance between the ground-truth correspondences in the overlapping area defined by Eq. \ref{['eq:overlap_rate']} (Points with no correspondence are marked in gray), and label the value of $\text{Corr}_\text{err}$ and the computational time.
  • Figure 4: The results obtained from different methods on two problem instances from the SHREC'20 non-rigid correspondence dataset Dyke2020tracka. For each method, we show the deformed mesh (left), the alignment result (right-bottom), and an error map (right-top) that visualizes the distance between the ground-truth correspondences (Points with no correspondences are marked in gray). We also label the value of $\text{Corr}_\text{err}$ and the computational time for each method.
  • Figure 5: The results obtained from different methods on two problem instances from the BEHAVE dataset bhatnagar22behave. For each method, we show the deformed mesh (left), the alignment results (right-bottom), and an error map (right-top) that visualizes the distance between each point and the ground-truth positions (right-top). We also label the value of RMSE and the computational time. The red dots on the source surface in the top-left corner mark the locations of the landmark points.
  • ...and 10 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof