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FilterReg: Robust and Efficient Probabilistic Point-Set Registration using Gaussian Filter and Twist Parameterization

Wei Gao, Russ Tedrake

arXiv:1811.10136v3cs.CV

TL;DR

Probabilistic point-set registration is more robust to noise, outliers, and occlusions than ICP-based approaches but is typically slower. The paper uses EM with Gaussian-filtered correspondence and twist parameterization, reporting robust, efficient registration across rigid, articulated, and deformable tasks.

  • Problem

    Probabilistic registration offers robustness to noise, outliers, and occlusions, but its computational cost limits practical use compared with ICP.

  • Method

    The method casts registration as maximum likelihood solved by EM, formulates correspondence as Gaussian filtering, and uses twist parameterization for articulated and deformable objects.

  • Results

    The method achieves state-of-the-art robustness and efficiency, including 3–7× faster CPU performance than modern ICP implementations and 7× faster GPU performance than its CPU implementation.

  • Takeaways & Limitations

    The combined formulation provides a robust, efficient, and general registration method that outperforms competitive baselines across varied registration tasks.

  • Takeaways & Limitations

    On a real-world dataset, point-to-point error is less accurate because it does not capture planar structure.

Abstract

from arXiv · show

Probabilistic point-set registration methods have been gaining more attention for their robustness to noise, outliers and occlusions. However, these methods tend to be much slower than the popular iterative closest point (ICP) algorithms, which severely limits their usability. In this paper, we contribute a novel probabilistic registration method that achieves state-of-the-art robustness as well as substantially faster computational performance than modern ICP implementations. This is achieved using a rigorous yet computationally-efficient probabilistic formulation. Point-set registration is cast as a maximum likelihood estimation and solved using the EM algorithm. We show that with a simple augmentation, the E step can be formulated as a filtering problem, allowing us to leverage advances in efficient Gaussian filtering methods. We also propose a customized permutohedral filter for improved efficiency while retaining sufficient accuracy for our task. Additionally, we present a simple and efficient twist parameterization that generalizes our method to the registration of articulated and deformable objects. For articulated objects, the complexity of our method is almost independent of the Degrees Of Freedom (DOFs), which makes it highly efficient even for high DOF systems. The results demonstrate the proposed method consistently outperforms many competitive baselines on a variety of registration tasks.

1. Introduction

Point-set registration must balance robustness to noise, outliers, and occlusions with the speed needed for practical vision systems. The paper proposes a probabilistic method targeting both robustness and substantially faster computation, while extending to articulated and deformable objects.

  • 1. Introduction: Point-set registration aligns two point clouds by estimating their relative transformation and supports systems including SLAM, pose estimation, reconstruction, and tracking.The paper positions registration as a component of several practical vision systems.
  • 1. Introduction: Probabilistic registration improves robustness to noise, outliers, and occlusions but is often too slow for large point clouds.ICP is fast but susceptible to these disturbances, motivating probabilistic alternatives.
  • 1. Introduction: 3–7 times faster than modern ICP implementations, while remaining as robust as state-of-the-art GMM-based methods.The CPU method is also orders of magnitude faster than typical robust GMM-based methods.
  • 1. Introduction: The method combines a computationally efficient probabilistic model, EM optimization, Gaussian filtering, and a customized permutohedral filter.The filter formulation targets improved efficiency while retaining sufficient accuracy.
  • 1. Introduction: Twist parameterization extends the method to articulated and node-graph deformable objects, with complexity almost independent of degrees of freedom.This makes the approach efficient for high-DOF systems.

2. Related Work

Prior work includes GMM-based probabilistic models and correlation-based methods for point-set registration. This paper assumes observations define a spatial probability distribution and combines that formulation with filter-based correspondence and twist parameterization.

  • 2. Related Work: GMM-based methods model observations probabilistically and have been extended to rigid, multi-rigid, articulated, and deformable registration.Several works derive EM procedures from this probabilistic formulation.
  • 2. Related Work: Correlation-based methods treat both point sets as probability distributions and minimize distances such as KL-divergence, using compact GMM representations for efficiency.Voxelization and support vector machines are cited as efficiency techniques.
  • 2. Related Work: The proposed formulation treats observations as a spatial probability distribution and moves model points toward positions with high posterior probability under kinematic constraints.The paper combines this formulation with filter-based correspondence and twist parameterization.

3. Probabilistic Model for Registration

The paper models registration by treating observation points as a probabilistic distribution and estimating motion through a constrained maximum-likelihood formulation solved with EM. This formulation supports robust correspondence, efficient computation, and extensions to feature- and plane-aware registration.

  • 3.1. Probabilistic Formulation: Registration maximizes the likelihood of the observation under geometric and kinematic potentials, with constraints such as X = X(θ) encoded in the kinematic model.The factorization assumes model-point conditional independence given θ and can include hard constraints or soft motion regularizers.
  • 3.1. Probabilistic Formulation: The EM E step computes weighted correspondences, and the M step optimizes motion using weight-averaged targets that replace ICP nearest neighbors.The formulation includes an outlier ratio parameter, and the averaged target is described as improving robustness to observation noise.
  • 3.1. Probabilistic Formulation: The proposed model treats observation points Y as inducing a GMM over space, while model points X are controlled by motion parameters θ.The formulation is described as an inverse of CPD: CPD models observations from a GMM introduced by model points, whereas this method models the observation-induced distribution directly.
  • 3.2. Discussion and Comparison: The formulation requires summation over Y only, allowing one spatial index to be reused when Gaussian variance remains fixed, unlike CPD’s updates over both Y and X.This reduces computation because CPD must rebuild its spatial index after model points change in each EM iteration.
  • 3.3. Several Extensions: The framework incorporates feature correspondence and point-to-plane objectives, while its Gaussian mixture includes a uniform outlier component and an explicit outlier-ratio parameter.Feature extensions can use normals, SHOT, learned features, or concatenated features; point-to-plane residuals use observation normals.

4. E Step: Filter-based Correspondence

The E step is transformed into a generalized Gaussian filtering computation through feature and value augmentation. A customized permutohedral filter then accelerates this computation while addressing small-variance indexing and repeated index construction.

  • 4.1. General Formulation: IFGT is considered but is reported as insufficiently faster for registration because its internal k-means tree would require too many centroids for typical parameters.The paper therefore uses Gaussian filtering algorithms as the computational alternative.
  • 4.1. General Formulation: The E-step computations are expressed in a generalized Gaussian-transform form and evaluated with Gaussian filtering algorithms instead of direct summation.The paper notes that the proposed filtering operation is a subset of the general Gaussian transform and uses Gaussian filtering for improved efficiency.
  • 4.1. General Formulation: Augmenting the input with features and values from both point clouds lets the filter retrieve G(f_xi) for model features not contained in observation set Y.The augmented inputs use F_X, F_Y, and V_Y to make the cross-cloud query compatible with the filtering operation.
  • 4.1. General Formulation: The permutohedral filter performs splatting onto a lattice, blurring among nearby lattice points, and slicing the filtered values back to input positions.Barycentric weights are used for interpolation during both splat and slice operations.
  • 4.2. Permutohedral Lattice Filter: The authors customize the permutohedral filter because small Gaussian variance makes index construction inefficient and naive application rebuilds the index every EM iteration.The customization is intended to retain sufficient accuracy while improving efficiency for this registration task.

5. M Step: Efficient Twist Parameterization

The M step uses twist coordinates to linearize rigid transformations and solve the resulting least-squares problems efficiently with Gauss–Newton. This parameterization extends to articulated and node-graph deformable models, with articulated complexity nearly independent of joint count.

  • General Kinematic Model: Twist coordinates represent the locally linearized change of an SE(3) transformation as a six-vector combining rotation and translation components.The associated 3×6 matrix uses skew(x_i) and the 3×3 identity to map twist changes to point displacement.
  • General Kinematic Model: The motion-parameter optimizations become least-squares problems solved with Gauss–Newton, whose main computational bottleneck is assembling matrices A and b.Each iteration updates the motion parameters using Δθ = −A^-1b.
  • 5.1. Articulated Model: For articulated models, rigid bodies are connected in a kinematic tree, and the point transformations depend on joint angles through body-specific rigid transforms.The spatial velocity Jacobian can be obtained from off-the-shelf rigid-body simulators.
  • 5.1. Articulated Model: O(6^2M) complexity is almost independent of N_joint, versus O(N_joint^2M) for previous articulated registration methods.The dominant algorithm lines are simple to implement and can be GPU-parallelized, while prior methods require customized kinematic-tree implementations for real-time performance.
  • 5.2. Deformable Model: Node graphs model large deformations such as rope or cloth through spatial motion fields, with point transformations interpolated from nearby nodes using fixed skinning weights.Dual quaternions interpolate the neighboring SE(3) transformations.

6. Results

Experiments evaluate robustness, accuracy, efficiency, and generality across synthetic, real-world, feature-based, articulated, and deformable registration tasks. The proposed method is consistently robust and efficient, outperforming competitive baselines across these settings.

  • 6.1. Robustness Test on Synthetic Data: The proposed method and CPD significantly outperform robust ICP under increasing outlier levels.The experiment adds randomly placed points to both model and observation clouds, averaging results over 30 independent runs.
  • 6.1. Robustness Test on Synthetic Data: Under increasing Gaussian noise, the proposed method and CPD are more accurate than robust ICP.Noise is scaled by the Bunny diameter, and clean clouds are shown only for visualization.
  • 6.1. Robustness Test on Synthetic Data: The proposed method matches CPD's robustness while running about 7 times faster than TrICP and two orders of magnitude faster than CPD.The comparison uses clean point clouds; the fixed-σ variant is faster per iteration but requires more iterations overall.
  • 6.2. Rigid Registration on Real-World Data: On the Stanford Lounge dataset, plane-error evaluation gives the proposed method state-of-the-art accuracy and significantly better computation than all baselines.Point-to-point error is less accurate because it does not capture planar structure; the plane-error variant addresses this limitation.
  • 6.3. Global Pose Estimation using Learned Features: For feature-based global registration with severe occlusion and ambiguous outliers, the proposed method reaches the correct pose while the baseline remains in bad alignment.The method also has a higher success rate and is more efficient than the baseline.
  • 6.4. Articulated Tracking: In articulated tracking, twist parameterization is an order of magnitude faster than direct parameterization, enabling real-time performance and improvement over articulated ICP and.For node-graph deformable objects, GPU filtering gives 8 times CPU speedup and twist parameterization is about 20 times faster than [8].

7. Conclusion

The paper concludes that its probabilistic registration method achieves robustness, accuracy, and efficiency by reformulating correspondence search as Gaussian filtering and extending registration to articulated and deformable objects.

  • The method achieves state-of-the-art robustness, accuracy, and efficiency across the evaluated registration tasks.
  • Formulating correspondence search as a filtering problem enables the use of efficient Gaussian filtering methods.
  • A simple twist parameterization generalizes the method to articulated and deformable objects.
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