Combinatorial models for topology-based geometric modeling

Many combinatorial (topological) models have been proposed in geometric modeling, computational geometry, image processing or analysis, for representing subdivided geometric objects, i.e. partitionned into cells of different dimensions: vertices, edges, faces, volumes, etc. We can distinguish among models according to the type of cells (regular or not regular ones), the type of assembly ("manifold" or "non manifold"), the type of representation (incidence graphs or ordered models), etc

Volumetric data analysis using Morse-Smale complexes

The 3D Morse-Smale complex is a fundamental topological construct that partitions the domain of a real-valued function into regions having uniform gradient flow behavior. In this paper, we consider the construction and selective presentation of cells of the Morse-Smale complex and their use in the analysis and visualization of scientific datasets. We take advantage of the fact that cells of different dimension often characterize different types of features present in the data. For example, critical points pinpoint changes in topology by showing where components of the level sets are created, destroyed or modified in genus. Edges of the Morse-Smale complex extract filament-like features that are not explicitly modeled in the original data. Interactive selection and rendering of portions of the Morse-Smale complex introduces fundamental data management challenges due to the unstructured nature of the complex even for structured inputs. We describe a data structure that stores the Morse-Smale complex and allows efficient selective traversal of regions of interest. Finally, we illustrate the practical use of this approach by applying it to cryo-electron microscopy data of protein molecules

Conversion between chains of maps and chains of surfaces; application to the computation of incidence graphs homology

Many combinatorial cellular structures have been defined in order to represent the topology of subdivided geometric objects. Two main classes can be distinguished. According to the terminology of [8], one is related to incidence graphs and the other to ordered models. Both classes have their own specificities and their use is relevant in different contexts. It is thus important to create bridges between them. So we define here chains of surfaces (a subclass of incidence graphs) and chains of maps without multi-incidence (a subclass of ordered models), which are able to represent the topology of subdivided objects, whose cells have " manifold-like " properties. We show their equivalence by providing conversion operations. As a consequence, it is hence possible to directly apply on each model results obtained on the other. We extend here classical results related to homology computation obtained for incidence graphs corresponding to regular CW −complexes and recent results about combinatorial cell complexes where cells are not necessarily homeomorphic to balls

TetGen, towards a quality tetrahedral mesh generator

TetGen is a C++ program for generating quality tetrahedral meshes aimed to support numerical methods and scientific computing. It is also a research project for studying the underlying mathematical problems and evaluating algorithms. This paper presents the essential meshing components developed in TetGen for robust and efficient software implementation. And it highlights the state-of-the-art algorithms and technologies currently implemented and developed in TetGen for automatic quality tetrahedral mesh generation

Representations for Cognitive Vision : a Review of Appearance-Based, Spatio-Temporal, and Graph-Based Approaches

The emerging discipline of cognitive vision requires a proper representation of visual information including spatial and temporal relationships, scenes, events, semantics and context. This review article summarizes existing representational schemes in computer vision which might be useful for cognitive vision, a and discusses promising future research directions. The various approaches are categorized according to appearance-based, spatio-temporal, and graph-based representations for cognitive vision. While the representation of objects has been covered extensively in computer vision research, both from a reconstruction as well as from a recognition point of view, cognitive vision will also require new ideas how to represent scenes. We introduce new concepts for scene representations and discuss how these might be efficiently implemented in future cognitive vision systems

The dual half-edge-a topological primal/dual data structure and construction operators for modelling and manipulating cell complexes

© 2016 by the authors. There is an increasing need for building models that permit interior navigation, e.g., for escape route analysis. This paper presents a non-manifold Computer-Aided Design (CAD) data structure, the dual half-edge based on the Poincaré duality that expresses both the geometric representations of individual rooms and their topological relationships. Volumes and faces are expressed as vertices and edges respectively in the dual space, permitting a model just based on the storage of primal and dual vertices and edges. Attributes may be attached to all of these entities permitting, for example, shortest path queries between specified rooms, or to the exterior. Storage costs are shown to be comparable to other non-manifold models, and construction with local Euler-type operators is demonstrated with two large university buildings. This is intended to enhance current developments in 3D Geographic Information Systems for interior and exterior city modelling

The Surface Evolver

The Surface Evolver is a computer program that minimizes the energy of a surface subject to constraints. The surface is represented as a simplicial complex. The energy can include surface tension, gravity, and other forms. Constraints can be geometrical constraints on vertex positions or constraints on integrated quantities such as body volumes. The minimization is done by evolving the surface down the energy gradient. This paper describes the mathematical model used and the operations available to interactively modify the surface

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology