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CLIPPER+: A Fast Maximal Clique Algorithm for Robust Global Registration

Kaveh Fathian, Tyler Summers

TL;DR

This work tackles robust global registration by casting putative data associations as a consistency graph and solving for the largest jointly consistent set, i.e., the maximum clique. It introduces CLIPPER+, a hybrid solver that combines a fast degeneracy-ordered greedy algorithm with a continuous-relaxation optimization, augmented by core-number pruning to reduce search space. A homotopy-based relaxation with a penalty parameter $d$ guides the solution toward a binary state, yielding a maximal clique when $d$ is sufficiently large relative to $n$. Empirical results on DIMACS and real-world point-cloud benchmarks show that CLIPPER+ achieves state-of-the-art accuracy while offering strong runtime performance, enabling robust registration under extreme outlier ratios (e.g., over $99\%$ outliers in some tests).

Abstract

We present CLIPPER+, an algorithm for finding maximal cliques in unweighted graphs for outlier-robust global registration. The registration problem can be formulated as a graph and solved by finding its maximum clique. This formulation leads to extreme robustness to outliers; however, finding the maximum clique is an NP-hard problem, and therefore approximation is required in practice for large-size problems. The performance of an approximation algorithm is evaluated by its computational complexity (the lower the runtime, the better) and solution accuracy (how close the solution is to the maximum clique). Accordingly, the main contribution of CLIPPER+ is outperforming the state-of-the-art in accuracy while maintaining a relatively low runtime. CLIPPER+ builds on prior work (CLIPPER [1] and PMC [2]) and prunes the graph by removing vertices that have a small core number and cannot be a part of the maximum clique. This will result in a smaller graph, on which the maximum clique can be estimated considerably faster. We evaluate the performance of CLIPPER+ on standard graph benchmarks, as well as synthetic and real-world point cloud registration problems. These evaluations demonstrate that CLIPPER+ has the highest accuracy and can register point clouds in scenarios where over $99\%$ of associations are outliers. Our code and evaluation benchmarks are released at https://github.com/ariarobotics/clipperp.

CLIPPER+: A Fast Maximal Clique Algorithm for Robust Global Registration

TL;DR

This work tackles robust global registration by casting putative data associations as a consistency graph and solving for the largest jointly consistent set, i.e., the maximum clique. It introduces CLIPPER+, a hybrid solver that combines a fast degeneracy-ordered greedy algorithm with a continuous-relaxation optimization, augmented by core-number pruning to reduce search space. A homotopy-based relaxation with a penalty parameter guides the solution toward a binary state, yielding a maximal clique when is sufficiently large relative to . Empirical results on DIMACS and real-world point-cloud benchmarks show that CLIPPER+ achieves state-of-the-art accuracy while offering strong runtime performance, enabling robust registration under extreme outlier ratios (e.g., over outliers in some tests).

Abstract

We present CLIPPER+, an algorithm for finding maximal cliques in unweighted graphs for outlier-robust global registration. The registration problem can be formulated as a graph and solved by finding its maximum clique. This formulation leads to extreme robustness to outliers; however, finding the maximum clique is an NP-hard problem, and therefore approximation is required in practice for large-size problems. The performance of an approximation algorithm is evaluated by its computational complexity (the lower the runtime, the better) and solution accuracy (how close the solution is to the maximum clique). Accordingly, the main contribution of CLIPPER+ is outperforming the state-of-the-art in accuracy while maintaining a relatively low runtime. CLIPPER+ builds on prior work (CLIPPER [1] and PMC [2]) and prunes the graph by removing vertices that have a small core number and cannot be a part of the maximum clique. This will result in a smaller graph, on which the maximum clique can be estimated considerably faster. We evaluate the performance of CLIPPER+ on standard graph benchmarks, as well as synthetic and real-world point cloud registration problems. These evaluations demonstrate that CLIPPER+ has the highest accuracy and can register point clouds in scenarios where over of associations are outliers. Our code and evaluation benchmarks are released at https://github.com/ariarobotics/clipperp.
Paper Structure (12 sections, 6 equations, 6 figures, 2 tables, 3 algorithms)

This paper contains 12 sections, 6 equations, 6 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Graph-Theoretic Robust Registration
  4. Maximal Clique Algorithms
  5. Degeneracy-Ordered Greedy Maximal Clique Algorithm
  6. Continuous-Relaxation Maximal Clique Algorithm
  7. CLIPPER+ Maximal Clique Algorithm
  8. Experimental Evaluations
  9. Maximum Clique Benchmark---DIMACS
  10. Registration Benchmark---Stanford Bunny
  11. Registration Benchmark---Real-World Point Clouds
  12. Conclusion and Future Directions

Figures (6)

  • Figure 1: An example of maximum clique formulation for robust global registration. (Left) Putative associations (red: outliers, green: inliers). (Right) Graph formulation with maximum clique indicating inlier associations.
  • Figure 2: The effect of outliers on the consistency graph. (Left) Putative associations (red: outliers; green: inliers). (Right) Resulting consistency graph. The higher the outlier percentage, the sparser the graph.
  • Figure 3: The maximum clique estimation accuracy ratio on the Stanford Bunny benchmark (the closer to $1$, the better). Each point corresponds to the accuracy ratio from a single trial. CLIPPER+ outperforms all algorithms.
  • Figure 4: Mean maximum clique estimation accuracy across different outlier percentages on the Stanford Bunny benchmark (the closer to $1$, the better).
  • Figure 5: Mean runtime across different outlier percentages on the Stanford Bunny benchmark (the lower, the better).
  • ...and 1 more figures