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Correspondence-Free Non-Rigid Point Set Registration Using Unsupervised Clustering Analysis

Mingyang Zhao, Jingen Jiang, Lei Ma, Shiqing Xin, Gaofeng Meng, Dong-Ming Yan

TL;DR

This paper develops a holistic framework where the source and target point sets are formulated as clustering centroids and clustering members, separately, and adopts Tikhonov regularization with an $\ell_{1}$-induced Laplacian kernel instead of the commonly used Gaussian kernel to ensure smooth and more robust displacement fields.

Abstract

This paper presents a novel non-rigid point set registration method that is inspired by unsupervised clustering analysis. Unlike previous approaches that treat the source and target point sets as separate entities, we develop a holistic framework where they are formulated as clustering centroids and clustering members, separately. We then adopt Tikhonov regularization with an $\ell_1$-induced Laplacian kernel instead of the commonly used Gaussian kernel to ensure smooth and more robust displacement fields. Our formulation delivers closed-form solutions, theoretical guarantees, independence from dimensions, and the ability to handle large deformations. Subsequently, we introduce a clustering-improved Nyström method to effectively reduce the computational complexity and storage of the Gram matrix to linear, while providing a rigorous bound for the low-rank approximation. Our method achieves high accuracy results across various scenarios and surpasses competitors by a significant margin, particularly on shapes with substantial deformations. Additionally, we demonstrate the versatility of our method in challenging tasks such as shape transfer and medical registration.

Correspondence-Free Non-Rigid Point Set Registration Using Unsupervised Clustering Analysis

TL;DR

This paper develops a holistic framework where the source and target point sets are formulated as clustering centroids and clustering members, separately, and adopts Tikhonov regularization with an -induced Laplacian kernel instead of the commonly used Gaussian kernel to ensure smooth and more robust displacement fields.

Abstract

This paper presents a novel non-rigid point set registration method that is inspired by unsupervised clustering analysis. Unlike previous approaches that treat the source and target point sets as separate entities, we develop a holistic framework where they are formulated as clustering centroids and clustering members, separately. We then adopt Tikhonov regularization with an -induced Laplacian kernel instead of the commonly used Gaussian kernel to ensure smooth and more robust displacement fields. Our formulation delivers closed-form solutions, theoretical guarantees, independence from dimensions, and the ability to handle large deformations. Subsequently, we introduce a clustering-improved Nyström method to effectively reduce the computational complexity and storage of the Gram matrix to linear, while providing a rigorous bound for the low-rank approximation. Our method achieves high accuracy results across various scenarios and surpasses competitors by a significant margin, particularly on shapes with substantial deformations. Additionally, we demonstrate the versatility of our method in challenging tasks such as shape transfer and medical registration.

Paper Structure

This paper contains 36 sections, 1 theorem, 15 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Non-rigid registration.
  4. Deformation representation.
  5. Preliminaries on Clustering Analysis
  6. Fuzzy Clustering Analysis
  7. Elkan $k$-Means Clustering
  8. Proposed Method
  9. Problem formulation.
  10. Clustering-Induced Non-Rigid Registration
  11. Observations.
  12. Regularization.
  13. Virtues of the Newly-Defined Function
  14. Information theory view.
  15. Optimization view.
  16. ...and 21 more sections

Key Result

Proposition 1

The low-rank approximation error $\epsilon=\|\mathbf{L}-\mathbf{E}\mathbf{W}^{-1}\mathbf{E}^{T}\|_{F}$ in terms of the Laplacian kernel is bounded by where $\|\cdot\|_{F}$ is the matrix Frobenious norm, $T=\max_i|\mathbf{P}_i|$, $q=\sum_{j=1}^{C}\|\bm{y}_j-\bm{z}_{c'(j)}\|_2^2$ is the clustering quantization error with $c'(j)={\rm argmin}_{i=1, \cdots, C'}\|\bm{y}_j-\bm{z}_i\|_2$, and $\gamma$ is

Figures (11)

  • Figure 1: Non-rigid registration on 3D point sets. The blue and gray models represent the source and target point clouds, respectively, while the yellow models are our registration results. Our method achieves successful registrations even for shapes with challenging deformations.
  • Figure 2: Left: Comparison between the Gaussian kernel and Laplacian kernel with the same bandwidth $\gamma=3$, where the latter delivers considerably thicker lower and upper tails, indicating higher robustness. Right: Laplacian kernel with different $\gamma$.
  • Figure 3: Test samples from the 2D hand pose dataset with different subjects. The red $\ast$ and the blue $\circ$ indicate the source and the target point sets, respectively.
  • Figure 4: Qualitative comparisons on the 2D hand pose dataset.
  • Figure 5: Robustness comparisons against external disturbances.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof