Point-Set Registration: Coherent Point Drift (0905.2635v1)

Abstract: Point set registration is a key component in many computer vision tasks. The goal of point set registration is to assign correspondences between two sets of points and to recover the transformation that maps one point set to the other. Multiple factors, including an unknown non-rigid spatial transformation, large dimensionality of point set, noise and outliers, make the point set registration a challenging problem. We introduce a probabilistic method, called the Coherent Point Drift (CPD) algorithm, for both rigid and non-rigid point set registration. We consider the alignment of two point sets as a probability density estimation problem. We fit the GMM centroids (representing the first point set) to the data (the second point set) by maximizing the likelihood. We force the GMM centroids to move coherently as a group to preserve the topological structure of the point sets. In the rigid case, we impose the coherence constraint by re-parametrization of GMM centroid locations with rigid parameters and derive a closed form solution of the maximization step of the EM algorithm in arbitrary dimensions. In the non-rigid case, we impose the coherence constraint by regularizing the displacement field and using the variational calculus to derive the optimal transformation. We also introduce a fast algorithm that reduces the method computation complexity to linear. We test the CPD algorithm for both rigid and non-rigid transformations in the presence of noise, outliers and missing points, where CPD shows accurate results and outperforms current state-of-the-art methods.

• The paper presents a probabilistic framework using a Gaussian Mixture Model to perform robust, coherent point set registration.
• It introduces a motion coherence constraint that ensures centroids move collectively, enhancing both rigid and non-rigid registration.
• Closed-form solutions and fast implementations yield efficient performance on noisy, large-scale datasets for 2D and 3D applications.

Insightful Overview of "Point Set Registration: Coherent Point Drift"

The paper presents a sophisticated probabilistic methodology for both rigid and non-rigid point set registration, termed the Coherent Point Drift (CPD) algorithm. This work is crucial to the domain of computer vision, where point set registration is central to tasks such as stereo matching, shape recognition, and medical image registration. CPD frames the registration problem within a probabilistic context, leveraging a Gaussian Mixture Model (GMM) to represent one point set, and aligning it with another set of data points by maximizing likelihood through the Expectation-Maximization (EM) algorithm.

Problem Formulation and Contributions

The Coherent Point Drift algorithm introduces several innovative approaches to address the complexities inherent in point set registration:

1. Probabilistic Framework: CPD utilizes a GMM to represent the first point set (centroids) and aligns it with a second set (data points) under a likelihood maximization objective, distinguishing itself from conventional deterministic methods. This probabilistic setting inherently accounts for noise and outliers by incorporating a uniform distribution term within the GMM to model irrelevant points.
2. Motion Coherence: The core principle of CPD is the imposition of a coherence constraint on the GMM centroids, ensuring that they move collectively in a topologically consistent manner. This is achieved through explicit re-parameterization in rigid scenarios and regularization of the displacement field in non-rigid contexts.
3. Closed-form Solutions: For rigid transformations, CPD provides closed-form solutions for the maximization step of the EM algorithm that generalizes across arbitrary dimensions. This is a significant development over existing methods, such as Iterative Closest Point (ICP) and its variants, which typically require iterative and approximate solutions.
4. Efficient Non-rigid Registration: In the non-rigid case, CPD regularizes the displacement field using variational calculus, which leads to an optimal transformation characterized by a kernel form. The algorithm is designed to maintain computational feasibility through computational complexity reductions to linear scales via fast implementations like the fast Gauss transform (FGT) and low-rank matrix approximation.

Strong Numerical Results

The experimental section showcases CPD's efficacy across a variety of conditions:

• Robustness to Outliers and Noise: CPD outperforms traditional methods such as LM-ICP when dealing with both synthetic and real-world data contaminated with noise and outliers. For instance, in experiments involving the 2D and 3D data sets (e.g., fish and Stanford bunny point sets), CPD accurately aligns point sets despite substantial noise, missing data, and outliers.
• Speed and Accuracy: The paper demonstrates that CPD's fast implementation, equipped with FGT and low-rank approximations, processes large datasets expediently. It supports robust and efficient non-rigid registration, scaling to large point sets with minimal performance degradation.

Implications and Future Developments

From a theoretical perspective, CPD's introduction of coherence constraints and pure probabilistic framing could influence future research in point set registration and related areas. This includes potential applications in medical imaging for tracking anatomical changes over time, or in robotics for object recognition and manipulation.

Practically, the robust handling of noise and outliers makes CPD highly applicable in scenarios where precision is critical despite degraded data quality, such as remote sensing and autonomous navigation. The computational efficiencies gained through the fast implementations make CPD attractive for real-time applications.

Speculative Future Directions

Future developments might explore:

• Extension to Higher Dimensions: Expanding CPD's framework to seamlessly operate in higher-dimensional spaces could cater to complex volumetric data common in advanced medical imaging techniques.
• Hybrid Models: Integrating CPD with machine learning frameworks for improved initializations and adaptive parameter tuning might further enhance robustness and speed.
• Parallel Computation: Utilizing parallel processing and GPU acceleration could dramatically reduce computation times, making CPD viable for even larger datasets and real-time applications.

Conclusion

The Coherent Point Drift algorithm represents a significant advancement in the field of point set registration. By introducing a probabilistic framework and enforcing coherence constraints, it achieves robust and accurate registration under challenging conditions. Its computational strategies ensure feasibility for large datasets, contributing both theoretical innovation and practical utility to the field.