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points whose distance is less than a given threshold. In order to reduce the computational cost of this process, more complex metrics have been used, which include statistical information associated to the underlying point distributions. The main disadvantage of this kind of approach is its strong dependence on the number of distributions associated with the map. In this work, we propose to use point clouds to build a density function over a 3D bounding box by fitting a local density function to each point and summing these functions into an accumulated global density function. This global density function is used to directly extract a boundary surface that summarizes the data within a density interval. This leads to a clustering strategy that has several important advantages: (1) It is robust to noise and outliers, (2) provides a high semantic level model which allows for Boolean set operations over different point clouds, and (3) it is efficient as far as time complexity is concerned. Fig. 1 illustrates the proposed approach, where the red dashed boxes, on the left, represent a prior data; the acquired data is shown in continuous blue boxes, in the middle; and the input-output is shown in black boxes, on the right. Fig. 1. Problem statement: Given 3D data acquired by a robot and a known map on the environment, the robot detects and segments changes in the scene using implicit clusters and Boolean operations. The remainder of the paper is organized as follows: Sec. II briefly reviews related works. Sec. III describes our change detection approach. Experimental results are shown in Sec. IV and, finally, Sec. V presents the conclusions and future work directions. II. RELATED WORK In the last decade, the behavior of an autonomous mobile robot working in dynamic environments has been intensively studied. In order to tame the complexity of the problem, several approaches have typically used as the main strategy to ignore dynamic objects from the modeling process in order to improve navigation and localization tasks [10]. However, these changes in the robot’s surroundings may actually be relevant depending on the application. Following this idea, Andreasson et al. [1] presented a system for autonomous change detection with a security patrol robot using 3D laser range data and, unlike our approach, they also used images from a color camera. Change detection, in the context of surveillance robots, was also addressed in the work of Vieira Neto and Nehmzow [12], where they compare the use of self-organizing maps with incremental PCA in order to model the world and identify changes therein. Differently from the work that we present in this paper, they applied their method to visual colored data, where visual attention was employed based on salience maps. Kaestner et al. [4] presented a method based on “difference image” where a probabilistic approach is used for alignment and change detection using range sensor data. Their alignment method provides a non-rigid point cloud registration to deal with estimation errors and changes between datasets are detected using a probabilistic threshold to recognize changes. However, robustness and performance are not discussed. A combination of Gaussian Mixture Model (GMM) and the Earth Mover’s Distance(EMD) algorithms was proposed by Drews et al. [3] in order to detect changes in raw 3D point clouds. Furthermore, they retrieved superquadric shapes from changes, in order to classify the encountered changes. In spite of the impressive results attained, the computation time of the proposed techniques were not suitable for large datasets. This was mainly due to the use of the Expectation Maximization algorithm to retrieve GMMs. That work was extended in Núñez et al. [6], where the structural matching algorithm was used instead of EMD to determine changes in GMM space. In spite of the limited number of tests they conducted, the results showed that the new approach represented an improvement in terms of computational cost and sensitivity to the number of Gaussians needed for the representation. The main limitation of using 3D raw point clouds to detect changes is related to the clustering of data without the knowledge of surface normals and neighborhood information. The notion of neighborhood of a point in a point cloud subjectively suggests the idea of connectivity. In this work, we introduce a method based on implicit functions [2] to extract a boundary surface that clusters the point cloud, allowing efficient change detection, Boolean operations and segmentation. Differently from [13], where a signed distance field is used to obtain an implicit surface fitting the data points at zero level, our goal is to define a level surface to cluster data points. III. METHODOLOGY In this section, we detail how to build our global density function from a point cloud set and how to use the implicit function theorem to define a surface that clusters points according to spatial density values. Finally, Boolean operations over two point clouds is presented using its corresponding density functions. A. Density Function from Point Cloud The main idea of our method is to estimate an unknown global density function G : R 3 → R from a given point cloud P = {(xi, yi, zi) , 1 ≤ i ≤ n} such that larger values occurs in points close to the surface where P was sampled. This global density function should define a continuous smooth
scalar field over which an isovalue density threshold h can be used to cluster points in the implicit volume defined by the set S = {p ∈ R 3 ; G (p) ≥ h}. Given a point cloud set P with n points, we define, for each point pi ∈ P , a simple local density gi : R 3 → R which is a kernel function chosen to be a Gaussian: gi (p) = exp . − pi−p2 2σ 2 The parameter σ smooths the influence of pi over its neighborhood. We set σ with the value of the maximum Euclidean distance from two points pi, pj ∈ P to be considered neighbors. Hence, if pi and pj are neighbors, which means pi − pj ≤ σ, the density contribution δ between pi and pj is given by δ = gi (pj) = e − pi−pj 2 2σ2 1 − ≥ e 2 . In order to define the global density function G, we accumulate the local density functions gi contributed by all points pi, 1 ≤ i ≤ n, n G (p) = gi (p). i=1 Since the local density function gi (pi) = 1 for all pi ∈ P , if pi has k neighbors, then G (pi) ≥ 1 + kδ. Our method clusters only points with k or more neighbors by setting a threshold value h = 1 + kδ. Fig. 2 and 3 illustrates this clustering strategy for 1D and 2D point cloud, respectively. Fig. 2. Example of the point clustering in 1D. Local density functions are accumulated to define a global density function. Notice that accumulated density for p5 is insufficient to cluster it, given the threshold. (a) (b) (c) Fig. 3. Example of the point clustering in 2D. Increasing values of smoothing parameter σ, lead to increasing cluster volume and more points are clustered. In (a), two points are considered outliers, in (b) only one point is considered outliers and, in (c), no outliers is considered. In order to speed up computation, we obtain a discrete sampling for the global density function by computing its values for a 3D grid defined for the bounding box of the point cloud P . For each point pi ∈ P , its local density contribution is calculated and summed only for a constant number of grid points on its neighborhood according to the kernel mask size s. So, the global density function G is sampled, for all points on the grid, in time O (n) for a point cloud with n points. The grid values is used as a discrete approximation for G. B. Clustering by Implicit Surface From the theorem of implicit functions, we have that if f : R 3 → R is smooth and h ∈ R is a regular value, that is, ∇f (p) = 0 for all p ∈ f −1 (h), then f −1 (h) is a surface that partitions the domain R 3 into three sets: S1 = {p ∈ R 3 ; f(p) < h}, S2 = {p ∈ R 3 ; f(p) = h}, S3 = {p ∈ R 3 ; f(p) > h}. The global density function G constructed as described in the previous section is a smooth scalar function in R 3 and setting a constant threshold h > 1, we define a surface S2 ⊂ R 3 as the boundary of a continuous volume region that clusters the point set P . Setting h = 1 + kδ we define a set to cluster points that have at least k neighbors whose distance is less or equal than σ. This set is implicitly defined as S = {p ∈ R 3 ; G (p) > h}. In order represent and visualize the set S, we extract a piecewise linear boundary surface for this cluster using the algorithm Marching Cubes [5]. The method Marching Cubes produces a mesh of triangles approximating the preimage G −1 (h) from a grid of cubes where the values of the function G are known at its vertices. Each cube vertex on the grid is classified either as positive or as negative, according to its relative position with respect to the set S. The resulting mesh represents a boundary surface that clusters the raw point cloud. Different level sets can be extracted for the same point cloud as long as we use a different isovalue h. Figure 4-a shows an example of a point cloud with some outliers; its level set surface extracted for a global density value h = 1 + 2δ is depicted in Figure 4-b, and another level set surface extracted for density level h = 1 + 3δ is shown in Figure 4-c. Notice that multiple components are well clustered. Notice, also, that in Figure 4-c the higher density level clustering leads to a disconnected component (a) (b) (c) Fig. 4. Example of the point cloud clustering in 3D. Raw point cloud (a), the clustering set for density level h = 1 + 2δ (b) and the clustering set for density level h = 1 + 3δ (c).
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