7378273
Learning smooth objects by probing
This video considers the problem of discovering the boundary S of an unknown smooth object O. The discovery process consists of moving a point probing device in the free space around O so that it repeatedly comes in contact with S. We present a probing strategy for generating a sequence of sample points of S, from which a PL-approximation of S can be constructed, within any desired accuracy. This strategy can be applied in any dimension, although its output is guaranteed only for objects embedded in the plane or in 3-space. For pedagogical purpose, the video focuses on the planar case.
3023549
Manifold reconstruction in arbitrary dimensions using witness complexes
It is a well-established fact that the witness complex is closelyrelated to the restricted Delaunay triangulation in lowdimensions. Specifically, it has been proved that the witness complexcoincides with the restricted Delaunay triangulation on curves, and isstill a subset of it on surfaces, under mild samplingassumptions. Unfortunately, these results do not extend tohigher-dimensional manifolds, even under stronger samplingconditions. In this paper, we show how the sets of witnesses andlandmarks can be enriched, so that the nice relations that existbetween both complexes still hold on higher-dimensional manifolds. Wealso use our structural results to devise an algorithm thatreconstructs manifolds of any arbitrary dimension or co-dimension atdifferent scales. The algorithm combines a farthest-point refinementscheme with a vertex pumping strategy. It is very simple conceptually,and it does not require the input point sample W to be sparse. Itstime complexity is bounded by c(d) |W|2, where c(d) is a constantdepending solely on the dimension d of the ambient space.
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.
2060671
Landmark Selection and Greedy Landmark-Descent Routing for Sensor Networks
We study the problem of landmark selection for landmark-based routing in a network of fixed wireless communication nodes. We present a distributed landmark selection algorithm that does not rely on global clock synchronization, and a companion local greedy landmark-based routing scheme. We assume no node location information, and that each node can communicate with some of its geographic neighbors. Each node is named by its hop count distances to a small number of nearby landmarks. Greedy routing at a node is performed to equalize its vector of landmark distances to that of the destination. This is done by following the shortest path to the landmark that maximizes the ratio of its distances to the source and the destination. In addition, we propose a method to alleviate the difficulty in routing to destinations near the boundaries by virtually expanding the network boundaries. The greedy routing, when combined with our landmark selection scheme, has a provable bounded path stretch relative to the best path possible, and guarantees packet delivery in the continuous domain. In the discrete domain, our simulations show that the landmark selection scheme is effective, and the companion routing scheme performs well under realistic settings. Both the landmark selection and greedy routing assumes no specific communication model and works with asymmetric links. Although some of the analysis are non-trivial, the algorithms are simple, flexible and cost-effective enough to warrant a real-world deployment.
10145935
Partial and approximate symmetry detection for 3D geometry
"Symmetry is a complexity-reducing concept [...]; seek it every-where." - Alan J. PerlisMany natural and man-made objects exhibit significant symmetries or contain repeated substructures. This paper presents a new algorithm that processes geometric models and efficiently discovers and extracts a compact representation of their Euclidean symmetries. These symmetries can be partial, approximate, or both. The method is based on matching simple local shape signatures in pairs and using these matches to accumulate evidence for symmetries in an appropriate transformation space. A clustering stage extracts potential significant symmetries of the object, followed by a verification step. Based on a statistical sampling analysis, we provide theoretical guarantees on the success rate of our algorithm. The extracted symmetry graph representation captures important high-level information about the structure of a geometric model which in turn enables a large set of further processing operations, including shape compression, segmentation, consistent editing, symmetrization, indexing for retrieval, etc.
14829043
The Robot Localization Problem
We consider the following problem: given a simple polygon ${\cal P}$ and a star-shaped polygon ${\cal V}$, find a point (or the set of points) in ${\cal P}$ from which the portion of ${\cal P}$ that is visible is translation-congruent to ${\cal V}$. The problem arises in the localization of robots equipped with a range finder and a compass---${\cal P}$ is a map of a known environment, ${\cal V}$ is the portion visible from the robot's position, and the robot must use this information to determine its position in the map. We give a scheme that preprocesses ${\cal P}$ so that any subsequent query ${\cal V}$ is answered in optimal time O(m + log n + A), where m and n are the number of vertices in ${\cal V}$ and ${\cal P}$ and A is the number of points in ${\cal P}$ that are valid answers (the output size). Our technique uses O(n5) space and preprocessing in the worst case; within certain limits, we can trade off smoothly between the query time and the preprocessing time and space. In the process of solving this problem, we also devise a data structure for output-sensitive determination of the visibility polygon of a query point inside a polygon ${\cal P}$. We then consider a variant of the localization problem in which there is a maximum distance to which the robot can "see"---this is motivated by practical considerations, and we outline a similar solution for this case. We finally show that a single localization query ${\cal V}$ can be answered in time O(mn) with no preprocessing.
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.
14624187
Locating and Bypassing Holes in Sensor Networks
In real sensor network deployments, spatial distributions of sensors are usually far from being uniform. Such networks often contain regions without enough sensor nodes, which we call holes. In this paper, we show that holes are important topological features that need to be studied. In routing, holes are communication voids that cause greedy forwarding to fail. Holes can also be defined to denote regions of interest, such as the “hot spots” created by traffic congestion or sensor power shortage. In this paper, we define holes to be the regions enclosed by a polygonal cycle which contains all the nodes where local minima can appear. We also propose simple and distributed algorithms, the Tent rule and BoundHole, to identify and build routes around holes. We show that the boundaries of holes marked using BoundHole can be used in many applications such as geographic routing, path migration, information storage mechanisms and identification of regions of interest.