Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes

A quadrangulation is a graph embedded on the sphere such that each face is bounded by a walk of length 4, parallel edges allowed. All quadrangulations can be generated by a sequence of graph operations called vertex splitting, starting from the path P_2 of length 2. We define the degree D of a splitting S and consider restricted splittings S_{i,j} with i <= D <= j. It is known that S_{2,3} generate all simple quadrangulations. Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}. First we show that the splittings S_{1,2} are exactly the monotone ones in the sense that the resulting graph contains the original as a subgraph. Then we show that they define a set of nontrivial ancestors beyond P_2 and each quadrangulation has a unique ancestor. Our results have a direct geometric interpretation in the context of mechanical equilibria of convex bodies. The topology of the equilibria corresponds to a 2-coloured quadrangulation with independent set sizes s, u. The numbers s, u identify the primary equilibrium class associated with the body by V\'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate all primary classes from a finite set of ancestors which is closely related to their geometric results. If, beyond s and u, the full topology of the quadrangulation is considered, we arrive at the more refined secondary equilibrium classes. As Domokos, L\'angi and Szab\'o showed recently, one can create the geometric counterparts of unrestricted splittings to generate all secondary classes. Our results show that S_{1,2} can only generate a limited range of secondary classes from the same ancestor. The geometric interpretation of the additional ancestors defined by monotone splittings shows that minimal polyhedra play a key role in this process. We also present computational results on the number of secondary classes and multiquadrangulations.Comment: 21 pages, 11 figures and 3 table

Digital homotopy with obstacles

AbstractIn (Ayala et al. (Discrete Appl. Math. 125 (1) (2003) 3) it was introduced the notion of a digital fundamental group π1d(O/S;σ) for a set of pixels O in relation to another set S which plays the role of an “obstacle”. This notion intends to be a generalization of the digital fundamental groups of both digital objects and their complements in a digital space. However, the suitability of this group was only checked for digital objects in that paper. As a sequel, we extend here the results in Ayala et al. (2003) for complements of objects. More precisely, we prove that for arbitrary digital spaces the group π1d(O/S;σ) maps onto the usual fundamental group of the difference of continuous analogues |AO∪S|−|AS|. Moreover, this epimorphism turns to be an isomorphism for a large class of digital spaces including most of the examples in digital topology

BPM: Blended Piecewise Moebius Maps

We propose a novel Moebius interpolator that takes as an input a discrete map between the vertices of two planar triangle meshes, and outputs a smooth map on the input domain. The output map interpolates the discrete map, is continuous between triangles, and has low quasi-conformal distortion when the input map is discrete conformal. Our map leads to considerably smoother texture transfer compared to the alternatives, even on very coarse triangulations. Furthermore, our approach has a closed-form expression, is local, applicable to any discrete map, and leads to smooth results even for extreme deformations. Finally, by working with local intrinsic coordinates, our approach is easily generalizable to discrete maps between a surface triangle mesh and a planar mesh, i.e., a planar parameterization. We compare our method with existing approaches, and demonstrate better texture transfer results, and lower quasi-conformal errors

Large-scale Geometric Data Decomposition, Processing and Structured Mesh Generation

Mesh generation is a fundamental and critical problem in geometric data modeling and processing. In most scientific and engineering tasks that involve numerical computations and simulations on 2D/3D regions or on curved geometric objects, discretizing or approximating the geometric data using a polygonal or polyhedral meshes is always the first step of the procedure. The quality of this tessellation often dictates the subsequent computation accuracy, efficiency, and numerical stability. When compared with unstructured meshes, the structured meshes are favored in many scientific/engineering tasks due to their good properties. However, generating high-quality structured mesh remains challenging, especially for complex or large-scale geometric data. In industrial Computer-aided Design/Engineering (CAD/CAE) pipelines, the geometry processing to create a desirable structural mesh of the complex model is the most costly step. This step is semi-manual, and often takes up to several weeks to finish. Several technical challenges remains unsolved in existing structured mesh generation techniques. This dissertation studies the effective generation of structural mesh on large and complex geometric data. We study a general geometric computation paradigm to solve this problem via model partitioning and divide-and-conquer. To apply effective divide-and-conquer, we study two key technical components: the shape decomposition in the divide stage, and the structured meshing in the conquer stage. We test our algorithm on vairous data set, the results demonstrate the efficiency and effectiveness of our framework. The comparisons also show our algorithm outperforms existing partitioning methods in final meshing quality. We also show our pipeline scales up efficiently on HPC environment

An Architecture for Online Affordance-based Perception and Whole-body Planning

The DARPA Robotics Challenge Trials held in December 2013 provided a landmark demonstration of dexterous mobile robots executing a variety of tasks aided by a remote human operator using only data from the robot's sensor suite transmitted over a constrained, field-realistic communications link. We describe the design considerations, architecture, implementation and performance of the software that Team MIT developed to command and control an Atlas humanoid robot. Our design emphasized human interaction with an efficient motion planner, where operators expressed desired robot actions in terms of affordances fit using perception and manipulated in a custom user interface. We highlight several important lessons we learned while developing our system on a highly compressed schedule

A system that learns to recognize 3-D objects

A system that learns to recognize 3-D objects from single and multiple views is presented. It consists of three parts: a simulator of 3-D figures, a Learner, and a recognizer. The 3-D figure simulator generates and plots line drawings of certain 3-D objects. A series of transformations leads to a number of 2-D images of a 3-D object, which are considered as different views and are the basic input to the next two parts. The learner works in three stages using the method of Learning from examples. In the first stage an elementary-concept learner learns the basic entities that make up a line drawing. In the second stage a multiple-view learner learns the definitions of 3-D objects that are to be recognized from multiple views. In the third stage a single-view learner learns how to recognize the same objects from single views. The recognizer is presented with line drawings representing 3-D scenes. A single-view recognizer segments the input into faces of possible 3-D objects, and attempts to match the segmented scene with a set of single-view definitions of 3-D objects. The result of the recognition may include several alternative answers, corresponding to different 3-D objects. A unique answer can be obtained by making assumptions about hidden elements (e. g. faces) of an object and using a multiple-view recognizer. Both single-view and multiple-view recognition are based on the structural relations of the elements that make up a 3-D object. Some analytical elements (e. g. angles) of the objects are also calculated, in order to determine point containment and conveziti. The system performs well on polyhedra with triangular and quadrilateral faces. A discussion of the system's performance and suggestions for further development is given at the end. The simulator and the part of the recognizer that makes the analytical calculations are written in C. The learner and the rest of the recognizer are written in PROLOG

Finsler bordifications of symmetric and certain locally symmetric spaces

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces $X/\Gamma$ for arbitrary discrete subgroups $\Gamma< G$. These bordifications result from attaching $\Gamma$-quotients of suitable domains of proper discontinuity at infinity. We further prove that such bordifications are compactifications in the case of Anosov subgroups. We show, conversely, that Anosov subgroups are characterized by the existence of such compactifications among uniformly regular subgroups. Along the way, we give a positive answer, in the torsion free case, to a question of Ha\"issinsky and Tukia on convergence groups regarding the cocompactness of their actions on the domains of discontinuity.Comment: 88 page