Registration of 3D Point Sets Using Exponential-based Similarity Matrix

TL;DR

The paper addresses robust registration of 3D point sets under large rotational differences and non-Gaussian noise by introducing Exponential Similarity Matrix ICP (ESM-ICP). ESM-ICP builds a symmetric similarity matrix M with weights w_i = exp(-||s_i - t_{c(i)}||^2/(2 sigma^2)) and computes a weighted cross-covariance H = sum (s_i - s_bar) M (t_i - t_bar) to estimate the rigid transformation via SVD; the optimization minimizes E_k = sum w_i^(k) ||R_k s_i + tau_k - t_i||^2 with weights in (0,1], and as iterations progress w_i -> 1, causing M to approach the identity and effectively recovering standard ICP. The authors demonstrate that the weights drive robust alignment in the presence of large rotations and outliers, outperforming both traditional geometry-based methods and several learning-based approaches on datasets like ModelNet40 and the Stanford Bunny, with the code released on GitHub. This work advances practical 3D mapping and SLAM applications by providing a robust, open-source registration tool suited to noisy and challenging sensor data.

Abstract

Point cloud registration is a fundamental problem in computer vision and robotics, involving the alignment of 3D point sets captured from varying viewpoints using depth sensors such as LiDAR or structured light. In modern robotic systems, especially those focused on mapping, it is essential to merge multiple views of the same environment accurately. However, state-of-the-art registration techniques often struggle when large rotational differences exist between point sets or when the data is significantly corrupted by sensor noise. These challenges can lead to misalignments and, consequently, to inaccurate or distorted 3D reconstructions. In this work, we address both these limitations by proposing a robust modification to the classic Iterative Closest Point (ICP) algorithm. Our method, termed Exponential Similarity Matrix ICP (ESM-ICP), integrates a Gaussian-inspired exponential weighting scheme to construct a similarity matrix that dynamically adapts across iterations. This matrix facilitates improved estimation of both rotational and translational components during alignment. We demonstrate the robustness of ESM-ICP in two challenging scenarios: (i) large rotational discrepancies between the source and target point clouds, and (ii) data corrupted by non-Gaussian noise. Our results show that ESM-ICP outperforms traditional geometric registration techniques as well as several recent learning-based methods. To encourage reproducibility and community engagement, our full implementation is made publicly available on GitHub. https://github.com/aralab-unr/ESM_ICP

Figures (6)

• Figure 1: Registration results using ESM-ICP. Top row: input source and target pairs. Bottom row: aligned outputs after applying ESM-ICP.
• Figure 2: (a) shows the 2 bunny rabbit point cloud (Red: Source and Green: Target) down-sampled using voxel grid filtering of leaf size $0.01$. The transformation between them is $(r,p,y)=(0,0,4.53)$ and $({\tau}_x,{\tau}_y,{\tau}_z) = (0,0,0.3)$ (b) shows the alignment after 14 iterations. (c) shows the alignment after 15 iterations.
• Figure 3: (a) Evolution of the similarity matrix $\mathbf{M}$ ($21 \times 21$) across iterations. (Note: the bunny rabbit point cloud dataset was sampled with a leaf size of $0.06$, reducing the matrix size for better visualization purposes.) Yellow entries denote high-weight correspondences; blue entries indicate near-zero weights.
• Figure 4: Comparison of ESM-ICP alignment to other geometric-based approaches.
• Figure 5: Comparison to deep learning methods.

Theorems & Definitions (4)

• Theorem 1
• proof
• Corollary 1
• proof