An efficient initial guess for the ICP method
Introduction
Registration of point clouds obtained from different perspective is a challenging problem in computer vision and computer graphics. Currently, the widely accepted solution is the Iterative Closest Point (ICP) method [1], but it requires that a point cloud is the subset of another, and the initial pose is close enough. Many variations of the ICP method has been proposed to increase the robustness in [2], [3], [4], [5].
Let {Mi} and {Nk} be the point clouds to be registered. Here, and are the three-dimensional coordinates of the points. If Mi and Nk are a pair of corresponding points, which satisfies , then the transformation between {Mi} and {Nk} is called the rigid transformation. Here, R is a 3 × 3 rotation matrix and T is a 3 × 1 translation vector. Let and let be the transformation matrix, then the rigid transformation is . If there are at least 3 pair of corresponding points, then the rigid transformation is found. However, the sequence of the three-dimensional points in {Mi} and {Nk} may be totally different due to the reason that they are sorted, or they are just scanned in a different view. If the correspondence is unknown, then the rigid transformation is not easy to solve.
The ICP method uses a heuristic strategy to find the point to point correspondence. In the heuristic strategy, each point finds its closest target in another point cloud and then treats it as the corresponding point. With this strategy, the point to point correspondence problem is solved and the translation matrix is found iteratively. But the problem is that this strategy works only if the initial pose of the point clouds is close enough. If the initial pose of the point clouds is not properly set, then the ICP method will dramatically fail.
To find the initial position efficiently, the feature-based methods [2], [6], [8], [9], [10], [14] and the 2D/3D based methods [11], [12], [13] are proposed. In the feature-based methods, the corresponding points are estimated by artificial statistics. With the artificial statistics, a certain amount of the real corresponding points is found, and the partially overlapped problem is solved too. Since not all the corresponding points are the correct pairs, the RANSAC method [7] is employed to find the correct matches. After the correspondence problem is solved, the rigid transformation is found. In the 2D/3D based methods, the image points are employed. With the additional information, additional topology and additional features are found. Since additional information is obtained, the corresponding points are found more easily. Although the point to point problem is partly solved by the 2D/3D based methods, the image information is not always available.
In this paper, a pseudo closed form solution is proposed for the rigid point cloud registration problem. The proposed solution is based on the covariance matrix. The advantage of the proposed solution is that the point to point correspondence is not needed, so that the rigid transformation is found with linear time. If the point cloud is transformed and sorted, then the proposed solution is the best solution.
For the partially overlapped point clouds, the proposed method finds the solution based on the covariance method and the statistic-based method. Firstly, a statistic (the local feature) is proposed to describe the local structure of each point. Secondly, the potential overlapped regions are found based on the similarity between the local features. Thirdly, the transformations between the overlapped regions are found based on the covariance matrix. Fourthly, the interior regions are found. Finally, the transformation is found according to the interior regions.
This paper is organized as follows. Section 2 defines the scope of the problem. Section 3 introduces the proposed theory. Section 4 shows the experimental result and compares the proposed method with other works. Section 5 gives the conclusion.
Section snippets
Motivation
Previous point-set registration methods treat the point set registration problem as a combination of two intertwined sub problems, which are the pose estimation problem and the point correspondence problem. However, if the point cloud is fully overlapped, then the transformation is determined by the covariance matrix. In this case, the point to point correspondence in not needed.
The advantage of finding the transformation using the covariance matrix is that the pseudo closed form solution is
Find the rigid transformation from the covariance matrix
Suppose {Mi} and {Nk} are two fully overlapped point clouds, which satisfies (one-to-one mapping). Let be the covariance matrix of {Nk}, and let be the covariance matrix of {Mi}. Then, CN and CM satisfies (1), here, ‖CN‖2 is the 2-norm of CN. Because the correspondence is not involved in (1), the closed form solution is found without the point to point correspondence. And because the covariance matrix is a statistical method, there is no
Experiments and comparison
In the experiments, 131 non-asymmetric point cloud models (found in the threepark.net, referred as the downloaded models hereafter) and some scanned models (obtained from home-made scanners) are employed to test the effectiveness of the proposed method. The downloaded models are interpolated so that the sparse point clouds are changed to dense point cloud. If the size of a mesh is larger than 1.5 times of the average size, then a new point is interpolated in the center of that mesh. This
Conclusion and discussion
In this paper, a covariance-based pseudo closed form solution is proposed for the point match problem. The rotation and the translation are linearly found in the proposed method. Experiments demonstrate that the proposed method is robust and accurate. If the point cloud is transformed and sorted, then the solution found by (4) and (5) solve the rigid transformation with linear time. In our experiment, the proposed statistics is faster than the FPFH method [14], this is probably because their
Declaration of Competing Interest
The authors have declared that no conflict of interest exists.
Acknowledgments
The authors would like to thank Professor Zhan.Song for his advices.
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