The recently read article "A New Effective Parallel Plane Segmentation Method for Point Cloud Based on Dimension Reduction Correction" is translated into Chinese as: A new method for point cloud parallel plane segmentation based on dimensionality reduction correction.
The email addresses of two authors of this article: [email protected] and [email protected]. They were two scholars from Shanghai Jiao Tong University. I sent them emails when I was reading the literature, but unfortunately I didn't get a reply. Maybe they graduated.
As introduced in the abstract, plane segmentation plays a very important role in point cloud processing. In particular, dividing parallel planes and then calculating the distance between parallel planes also plays a great role in practical engineering applications. The algorithm proposed in the paper first estimates the normal vector of the parallel plane, and then corrects this normal vector in the two-dimensional part. Finally, the plane equation is obtained based on this normal vector.
The paper first assumes that the two planes are parallel to each other, or very close to being parallel to each other. The core idea of the algorithm is to first segment the first plane, and then segment the second plane parallel to the first plane. Then, the geometric equations of the first plane are updated based on the error in splitting the second plane.
The original words in the article are as follows:
Point cloud preprocessing
The point cloud preprocessing process includes downsampling operations, outlier removal and pre-segmentation operations.
The author believes that different 3D scanning equipment and different scanning methods will cause the obtained point cloud to have complex spatial distribution. Using downsampling operations can reduce the impact of equipment and scanning methods. At the same time, downsampling does not damage the spatial structure of the point cloud, but reduces the number of points, thus increasing the computing speed (with fewer points, the calculation must be faster). They used the voxel filtering method to achieve this operation, which is implemented in PCL.
Outliers will interfere with the accuracy and effectiveness of the algorithm, so they must be removed during preprocessing.
The last small step in point cloud preprocessing is point cloud pre-segmentation. The so-called pre-segmentation, from my understanding, is to roughly segment the area containing parallel planes from the point cloud to avoid interference from other points and difficult operation. . To put it simply, it is actually to select an ROI area first, or write an algorithm to automatically find the ROI area.
The original text is as follows:
The second step is to obtain the estimated normal direction
. The author believes that a plane should be segmented first, and then the normal direction of this plane should be used as a pre-estimated value. Even after preprocessing, there will still be some interference points, but the RANSAC algorithm can still segment the plane containing the most points in the space. The plane containing the most points is of course the largest plane. The RANSAC algorithm can extract this plane and obtain the normal vector N0 of the plane.
The third step is to divide the two planes. The
algorithm starts with the normal direction of one plane and finally obtains the equations of two parallel planes. First reduce the dimension of the point cloud to two dimensions, and then optimize the normal direction.
1. First select a plane Sk. The normal direction Nk of the plane is perpendicular to the normal direction N0 obtained in the second step. To put it bluntly, select a plane perpendicular to two parallel planes. In fact, the point cloud can be transformed into a rigid body.
2. According to N0, the point cloud can be rotated and translated to its new normal direction pointing to the x-axis or y-axis, then the plane perpendicular to the pair of parallel planes can be the XOY plane. Then the normal direction of the XOY plane is the unit vector (0, 0, 1) in the z-axis direction.
3. After rigid body transformation, the pair of parallel planes are perpendicular to the XOY plane, so the projection of the point cloud on the XOY plane is a pair of parallel lines. Since the points on the point cloud are projected onto the XOY plane, the Z values of these points are all 0, and the problem becomes simpler, and the purpose of dimensionality reduction is achieved.
4. As mentioned in the paper, the least squares method can then be used to directly fit the equations of the two parallel lines. I don’t quite understand how this step can be obtained directly using the least squares method.
5. The unit vector in the direction perpendicular to the obtained pair of parallel lines is the new normal vector N1. The author believes that this N1 is more accurate than N0. In fact, N1 and N0 are very close.
6. Based on this N1, the point cloud can be rigidly transformed to make it more perpendicular to the XOY plane, and then projected to the XOY plane again. According to the projected points, the least squares method mentioned in (4) can be used to fit the new equations of the two parallel lines. After several iterations, a very accurate normal vector can be obtained.
Note: As we all know, the normal direction and curvature of each point in the point cloud can be obtained using the PCA algorithm. Since the normal direction of a three-dimensional point can be obtained, naturally the normal direction of a two-dimensional point can also be obtained. Therefore, the equations of two parallel lines can be obtained by fitting the two parallel lines without the least squares method. The PCA algorithm can be used directly to solve the covariance matrix of the two-dimensional points projected onto XOY. The eigenvectors corresponding to the eigenvalues are the normal and tangential directions.
7. Based on this normal vector and the origin, the equations of two parallel planes can be calculated. I really don’t understand this point. What is the origin and how can we get the equation of a parallel plane?
(5) (6) (7) is very critical. The original text is as follows:
According to the obtained equations of the two parallel planes, calculate their distance.