34799453
Estimating surface normals in noisy point cloud data
In this paper we describe and analyze a method based on local least square fitting for estimating the normals at all sample points of a point cloud data (PCD) set, in the presence of noise. We study the effects of neighborhood size, curvature, sampling density, and noise on the normal estimation when the PCD is sampled from a smooth curve in ℝ2 or a smooth surface in ℝ3, and noise is added. The analysis allows us to find the optimal neighborhood size using other local information from the PCD. Experimental results are also provided.
207156516
Registration of point cloud data from a geometric optimization perspective
We propose a framework for pairwise registration of shapes represented by point cloud data (PCD). We assume that the points are sampled from a surface and formulate the problem of aligning two PCDs as a minimization of the squared distance between the underlying surfaces. Local quadratic approximants of the squared distance function are used to develop a linear system whose solution gives the best aligning rigid transform for the given pair of point clouds. The rigid transform is applied and the linear system corresponding to the new orientation is build. This process is iterated until it converges. The point-to-point and the point-to-plane Iterated Closest Point (ICP) algorithms can be treated as special cases in this framework. Our algorithm can align PCDs even when they are placed far apart, and is experimentally found to be more stable than point-to-plane ICP. We analyze the convergence behavior of our algorithm and of point-to-point and point-to-plane ICP under our proposed framework, and derive bounds on their rate of convergence. We compare the stability and convergence properties of our algorithm with other registration algorithms on a variety of scanned data.
34184916
Deformable spanners and applications
For a set S of points in ℝ(d), an s-spanner is a subgraph of the complete graph with node set S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + ε)-spanner with O(n/ε(d)) edges, where ε is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decompositions, and approximate k-centers.
9104347
Reconstruction Using Witness Complexes
We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions.
1839876
Discrete mobile centers
\emph{We propose a new randomized algorithm for maintaining a set of c lusters among moving nodes in the plane. Given a specified cluster radius, our algorithm selects and maintains a variable subset of the nodes as cluster centers. This subset has the property that (1) balls of the given radius centered at the chosen nodes cover all the others and (2) the number of centers selected is a constant-factor approximation of the minimum possible. As the nodes move, an event-based kinetic data structure updates the clustering as necessary. This kinetic data structure is shown to be responsive, efficient, local, and compact. The produced cover is also smooth, in the sense that wholesale cluster re-arrangements are avoided. The algorithm can be implemented without exact knowledge of the node positions, if each node is able to sense its distance to other nodes up to the cluster radius. Such a kinetic clustering can be used in numerous applications where mobile devices must be interconnected into an ad-hoc network to collaboratively perform some task.}
42656979
Geometric spanner for routing in mobile networks
5682640
A dual-space approach to tracking and sensor management in wireless sensor networks
Wireless ad hoc sensor networks have the advantage of spanning a large geographical region and being able to collaboratively detect and track non-local spatio-temporal events. This paper presents a dual-space approach to event tracking and sensor resource management in sensor networks. The dual-space transformation maps a non-local phenomenon, e.g., the edge of a half-plane shadow, to a single point in the dual space, and maps locations of distributed sensor nodes to a set of lines that partitions the dual space. The detection problem becomes finding and tracking the cell that contains the point in the arrangement defined by these lines. This mechanism can be effectively used for power management of the sensor network - nodes that will not be immediately visited by an event can be turned off to save energy required for sensing, processing, and communication. The approach has been successfully demonstrated on a laboratory testbed built using the UC Berkeley motes sensors. An implemented application of detecting and tracking light shadow edges moving over a sensor field is described.