Support Vector Machines for Classification of Geometric Primitives in Point Clouds

Abstract

Classification of point clouds by different types of geometric primitives is an essential part in the reconstruction process of CAD geometry. We use support vector machines (SVM) to label patches in point clouds with the class labels tori, ellipsoids, spheres, cones, cylinders or planes. For the classification features based on different geometric properties like point normals, angles, and principal curvatures are used. These geometric features are estimated in the local neighborhood of a point of the point cloud. Computing these geometric features for a random subset of the point cloud yields a feature distribution. Different features are combined for achieving best classification results. To minimize the time consuming training phase of SVMs, the geometric features are first evaluated using linear discriminant analysis (LDA).

LDA and SVM are machine learning approaches that require an initial training phase to allow for a subsequent automatic classification of a new data set. For the training phase point clouds are generated using a simulation of a laser scanning device. Additional noise based on an laser scanner error model is added to the point clouds. The resulting LDA and SVM classifiers are then used to classify geometric primitives in simulated and real laser scanned point clouds.

Compared to other approaches, where all known features are used for classification, we explicitly compare novel against known geometric features to prove their effectiveness.

A Additional Geometric Features and Model Selection

For comparison additional geometrical features were tested, see Table 4. For optimization results of the SVM model selection refer to Table 5.

Table 4. True positive rate of additional geometric features.

Table 5. SVM parameters C and $\gamma$ for best classification results for each feature.

• A.1 Centroid distances are computed as distance of random points to the bounding box centroid, [OFCD02].

• A.2 Cube cell count is computed by subdividing the point cloud’s bounding box into equally sized cells and counting the points in each cell. This feature is not invariant to rotation.

• A.3 K-Median points are computed by clustering the points into k clusters, such that the sum of distances of points in the cluster to their median is minimized. The coordinates of the resulting medians are concatenated to one feature vector of size 3k.

• A.4 The moduli of principal curvatures $\kappa _1$, $\kappa _2$ as in Sect. 2.1 yield two concatenated histograms.

• A.5 , A.6, and A.7 mean curvatures $H ={(\kappa _1 + \kappa _2)}/{2}$, Gauss curvatures $K = \kappa _1 \kappa _2$, and curvature ratios $|{\kappa _1}/{\kappa _2}|$.

• A.8 Curvature changes are computed as the absolute difference between a random point’s principal curvatures and those of its nearest neighbor and yield two concatenated histograms.

• A.9 Shape indices as in [KvD92].

The histograms of the geometric features A.5, A.6, A.7, and A.8 are cropped to range between the 0.05 and 0.95 percentiles. All these features are less effective in the experiments with a true-positive-rate below 0.5.