Efficient and Deterministic Search Strategy Based on Residual Projections for Point Cloud Registration with Correspondences

TL;DR

The paper tackles robust rigid point cloud registration under high outlier rates by introducing a novel pose decoupling strategy that uses $L_ty$ residual projections to split the 6-DOF problem into three axis-wise sub-problems. Each sub-problem is solved deterministically via a branch-and-bound search conducted on a reduced 2D domain, with translations estimated implicitly through interval stabbing, avoiding translation-domain initialization. The method refines the coarse solution with SVD on the consensus set and extends naturally to simultaneous pose and correspondence registration (SPCR) through interval merging-based bounds. Comprehensive experiments on synthetic and real-world datasets show superior efficiency and competitive robustness relative to state-of-the-art deterministic and learning-based approaches, with practical applicability to challenging outdoor LiDAR data such as Bremen, ETH, and KITTI.

Abstract

Estimating the rigid transformation between two LiDAR scans through putative 3D correspondences is a typical point cloud registration paradigm. Current 3D feature matching approaches commonly lead to numerous outlier correspondences, making outlier-robust registration techniques indispensable. Many recent studies have adopted the branch and bound (BnB) optimization framework to solve the correspondence-based point cloud registration problem globally and deterministically. Nonetheless, BnB-based methods are time-consuming to search the entire 6-dimensional parameter space, since their computational complexity is exponential to the solution domain dimension in the worst-case. To enhance algorithm efficiency, existing works attempt to decouple the 6 degrees of freedom (DOF) original problem into two 3-DOF sub-problems, thereby reducing the search space. In contrast, our approach introduces a novel pose decoupling strategy based on residual projections, decomposing the raw registration problem into three sub-problems. Subsequently, we embed interval stabbing into BnB to solve these sub-problems within a lower two-dimensional domain, resulting in efficient and deterministic registration. Moreover, our method can be adapted to address the challenging problem of simultaneous pose and registration. Through comprehensive experiments conducted on challenging synthetic and real-world datasets, we demonstrate that the proposed method outperforms state-of-the-art methods in terms of efficiency while maintaining comparable robustness.

Figures (12)

• Figure 1: The proposed method can efficiently address the rigid registration problem in different scenarios with high outlier rates or low overlap. The input point clouds are selected from (a) Bremen datasetborrmann2013thermal, (b) ETH datasettheiler2014keypoint, (c) KITTI datasetgeiger2012we, and (d) Bunny datasetcurless1996volumetric, respectively. The source point cloud is green, the target point cloud is yellow, and the aligned point cloud is blue. Compared with state-of-the-art (SOTA) correspondence-based methods, the proposed method achieves significant performance in terms of robustness and efficiency. Besides, the proposed method also can solve the SPCR problem efficiently and robustly.
• Figure 2: A toy 2D registration example to demonstrate $L_\infty$ residual projection. Specifically, $\left\{(\bm{p}_i,\bm{q}_i)\right\}_{i=1}^3$ is the set of input correspondences, $\bm{r}_j$, $j=X, Y$, is the transpose of each row of the rotation matrix, and $t_j$, $j=X, Y$, is the component of the translation vector. The red line segments represent the projections of the residual on the coordinate axes $X$ and $Y$, i.e., $\left| \bm{r}_j^\mathrm{T}\bm{p}_1+t_j-q_1^j \right|$, $j=X, Y$. The inlier constraint for $L_\infty$ residual indicates that $(\bm{p}_1,\bm{q}_1)$ is an inlier only if both residual projections on the coordinate axes are not larger than the inlier threshold.
• Figure 3: The solution domain before and after exponential mapping, and the pipeline of our proposed BnB algorithm. The original solution domain of the vector $\bm{r}$ is a unit sphere in the 3D Euclidean space. The exponential mapping method maps the unit sphere to two identical 2D disks, representing the solution domains of $\bm{r}$ and $-\bm{r}$, respectively. We can only branch one 2D-disk domain during each iteration, followed by the calculation of upper and lower bounds for each sub-branch. The proposed BnB algorithm converges until the optimal solution $\bm{r}^*$ is found, and the optimal $t^*$ is found by interval stabbing simultaneously. In the visualization results of interval stabbing, the black line segments are the candidate intervals of each correspondence, and the red line segments are the intervals crossed by the blue probe with the max-stabbing number. The probe position is the max-stabbing position.
• Figure 4: The visualization results for the 24-th iteration of a representative SPCR test on synthetic data. The black line segments are the intervals after interval merging, and the red line segments are the intervals crossed by the blue probe with the max-stabbing number. After interval merging, interval stabbing is utilized to calculate the upper bounds. Notably, the blue probe resulting from interval stabbing can only penetrate at most one interval for each point $\bm{p}_i$. The final SPCR upper bound for $\mathbb{S}^{2}$ is 38.
• Figure 5: Controlled experiments with $N=\{1000,2000,\dots,5000\}$. The results include average rotation errors, average translation errors, and average running times.

Theorems & Definitions (20)

• Lemma 1
• Proposition 1
• Lemma 2
• proof
• Proposition 2: Upper bound for $\mathbb{S}^{2+}$
• proof
• Proposition 3: Upper bound for $\mathbb{S}^{2-}$
• proof
• Proposition 4: Upper bound for $\mathbb{S}^{2}$
• proof