9789536
Robust global registration
We present an algorithm for the automatic alignment of two 3D shapes (data and model), without any assumptions about their initial positions. The algorithm computes for each surface point a descriptor based on local geometry that is robust to noise. A small number of feature points are automatically picked from the data shape according to the uniqueness of the descriptor value at the point. For each feature point on the data, we use the descriptor values of the model to find potential corresponding points. We then develop a fast branch-and-bound algorithm based on distance matrix comparisons to select the optimal correspondence set and bring the two shapes into a coarse alignment. The result of our alignment algorithm is used as the initialization to ICP (iterative closest point) and its variants for fine registration of the data to the model. Our algorithm can be used for matching shapes that overlap only over parts of their extent, for building models from partial range scans, as well as for simple symmetry detection, and for matching shapes undergoing articulated motion.
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.
364731
Example-based 3D scan completion
We present a novel approach for obtaining a complete and consistent 3D model representation from incomplete surface scans, using a database of 3D shapes to provide geometric priors for regions of missing data. Our method retrieves suitable context models from the database, warps the retrieved models to conform with the input data, and consistently blends the warped models to obtain the final consolidated 3D shape. We define a shape matching penalty function and corresponding optimization scheme for computing the non-rigid alignment of the context models with the input data. This allows a quantitative evaluation and comparison of the quality of the shape extrapolation provided by each model. Our algorithms are explicitly designed to accommodate uncertain data and can thus be applied directly to raw scanner output. We show on a variety of real data sets how consistent models can be obtained from highly incomplete input. The information gained during the shape completion process can be utilized for future scans, thus continuously simplifying the creation of complex 3D models.
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.
13195342
Geometric filtering of pairwise atomic interactions applied to the design of efficient statistical potentials
Distance-dependent, pairwise, statistical potentials are based on the concept that the packing observed in known protein structures can be used as a reference for comparing different 3D models for a protein. Here, packing refers to the set of all pairs of atoms in the molecule. Among all methods developed to assess three-dimensional models, statistical potentials are subject both to praise for their power of discrimination, and to criticism for the weaknesses of their theoretical foundations. Classical derivations of pairwise potentials assume statistical independence of all pairs of atoms. This assumption, however, is not valid in general. We show that we can filter the list of all interactions in a protein to generate a much smaller subset of pairs that retains most of the structural information contained in proteins. The filter is based on a geometric method called alpha shapes that captures the packing in a conformation. Statistical scoring functions derived from such subsets perform as well as scoring functions derived from the set of all pairwise interactions.
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.
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.
6460806
Distributed Proximity Maintenance in Ad Hoc Mobile Networks
We present an efficient distributed data structure, called the d-Spanner, for maintaining proximity information among communicating mobile nodes. The d-Spanner is a kinetic sparse graph spanner on the nodes that allows each node to quickly determine which other nodes are within a given distance of itself, to estimate an approximate nearest neighbor, and to perform a variety of other proximity related tasks. A lightweight and fully distributed implementation is possible, in that maintenance of the proximity information only requires each node to exchange a modest number of messages with a small number of mostly neighboring nodes. The structure is based on distance information between communicating nodes that can be derived using ranging or localization methods and requires no additional shared infrastructure other than an underlying communication network. Its modest requirements make it scalable to networks with large numbers of nodes.
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.