45865813
Persistence Barcodes for Shapes
In this paper, we initiate a study of shape description and classification via the application of persistent homology to tangential constructions on geometric objects. Our techniques combine the differentiating power of geometry with the classifying power of topology. The homology of our first construction, the tangent complex, can distinguish between topologically identical shapes with different "sharp" features, such as corners. To capture "soft" curvature-dependent features, we define a second complex, the filtered tangent complex, obtained by parameterizing a family of increasing subcomplexes of the tangent complex. Applying persistent homology, we obtain a shape descriptor, called a barcode, that is a finite union of intervals. We define a metric over the space of such intervals, arriving at a continuous invariant that reflects the geometric properties of shapes. We illustrate the power of our methods through a number of detailed studies of parameterized families of mathematical shapes.
31949610
A barcode shape descriptor for curve point cloud data
In this paper, we present a complete computational pipeline for extracting a compact shape descriptor for curve point cloud data (PCD). Our shape descriptor, called a barcode, is based on a blend of techniques from differential geometry and algebraic topology. We also provide a metric over the space of barcodes, enabling fast comparison of PCDs for shape recognition and clustering. To demonstrate the feasibility of our approach, we implement our pipeline and provide experimental evidence in shape classification and parametrization.
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.
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.
147952
Locating and bypassing routing holes in sensor networks
Many algorithms for routing in sensor networks exploit greedy forwarding strategies to get packets to their destinations. We study a fundamental difficulty such strategies face: the "local minimum phenomena" that can cause packets to get stuck. We give a definition of stuck nodes where packets may get stuck in greedy multi-hop forwarding, and develop a local rule, the TENT rule, for each node in the network to test whether a packet can get stuck at that node. To help the packets get out of stuck nodes, we describe a distributed algorithm, BOUNDHOLE, to build routes around holes, which are connected regions of the network with boundaries consisting of all the stuck nodes. We show that these hole-surrounding routes can be used in many applications such as geographic routing, path migration, information storage mechanisms and identification of regions of interest.
11907465
Deformable Free-Space Tilings for Kinetic Collision Detection†
We present kinetic data structures for detecting collisions between a set of polygons that are moving continuously. Unlike classical collision detection methods that rely on bounding volume hierarchies, our method is based on deformable tilings of the free space surrounding the polygons. The basic shape of our tiles is that of a pseudo-triangle, a shape sufficiently flexible to allow extensive deformation, yet structured enough to make detection of self-collisions easy. We show different schemes for maintaining pseudo-triangulations as a kinetic data structure, and we analyze their performance. Specifically, we first describe an algorithm for maintaining a pseudo-triangulation of a point set, and show that the pseudo-triangulation changes only quadratically many times if points move along algebraic arcs of constant degree. In addition, by refining the pseudo-triangulation, we show triangulations of points that only change about O(n 7 / 3 ) times for linear motion. We then describe an algorithm for maintaining a pseudo-triangulation of a set of convex polygons. Finally, we extend our algorithm to the general case of maintaining a pseudo-triangulation of a set of moving or deforming simple polygons.
8275001
Zonotopes as bounding volumes
Zonotopes are centrally symmetric polytopes with a very special structure: they are Minkowski sums of line segments. In this paper we propose to use zonotopes as bounding volumes for geometry in collision detection and other applications where the spatial relationship between two pieces of geometry is important. We show how to construct optimal, or approximately optimal zonotopes enclosing given set of points or other geometry. We also show how zonotopes can be used for efficient collision testing, based on their representation via their defining line segments --- without ever building their explicit description as polytopes. This implicit representation adds flexibility, power, and economy to the use of zonotopes as bounding volumes.
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.