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"Lienhardt, Pascal"
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Combinatorial maps : efficient data structures for computer graphics and image processing
\"Although they are less widely known than other models, combinatorial maps are very powerful data structures and can be useful in many applications, including computer graphics and image processing. The book introduces these data structures, describes algorithms and data structures associated with them, makes connections to other common structures, and demonstrates how to use these structures in geometric modeling and image processing. The data structures and algorithms introduced in the book will be available in a C++ library on the authors' website\"-- Provided by publisher.
Book
Combinatorial Maps
This book gathers important ideas related to combinatorial maps and explains how the maps are applied in geometric modeling and image processing. It focuses on two subclasses of combinatorial maps: n-Gmaps and n-maps. The book presents the data structures, operations, and algorithms that are useful in handling subdivided geometric objects. It shows how to study data structures for the explicit representation of subdivided geometric objects and describes operations for handling the structures. The book also illustrates results of the design of data structures and operations.
eBook
Combinatorial Maps
A Versatile Framework for Handling Subdivided Geometric ObjectsCombinatorial Maps: Efficient Data Structures for Computer Graphics and Image Processing gathers important ideas related to combinatorial maps and explains how the maps are applied in geometric modeling and image processing. It focuses on two subclasses of combinatorial maps: n-Gmaps and n-maps.Suitable for researchers and graduate students in geometric modeling, computational and discrete geometry, computer graphics, and image processing and analysis, the book presents the data structures, operations, and algorithms that are useful in handling subdivided geometric objects. It shows how to study data structures for the explicit representation of subdivided geometric objects and describes operations for handling the structures. The book also illustrates results of the design of data structures and operations.
eBook
Homology of Cellular Structures Allowing Multi-incidence
This paper focuses on homology computation over ‘cellular’ structures that allow multi-incidence between cells. We deal here with combinatorial maps, more precisely chains of maps and subclasses such as maps and generalized maps. Homology computation on such structures is usually achieved by computing simplicial homology on a simplicial analog. But such an approach is computationally expensive because it requires computing this simplicial analog and performing the homology computation on a structure containing many more cells (simplices) than the initial one. Our work aims at providing a way to compute homologies directly on a cellular structure. This is done through the computation of incidence numbers. Roughly speaking, if two cells are incident, then their incidence number characterizes how they are attached. Having these numbers naturally leads to the definition of a boundary operator, which induces a homology. Hence, we propose a boundary operator for chains of maps and provide optimization for maps and generalized maps. It is proved that, under specific conditions, the homology of a combinatorial map as defined in the paper is equivalent to the homology of its simplicial analogue.
Journal Article
Equivalence between Closed Connected n-G-Maps without Multi-Incidence and n-Surfaces
Many combinatorial structures have been designed to represent the topology of space subdivisions and images. We focus here on two particular models, namely the
n
-
G
-maps used in geometric modeling and computational geometry and the
n
-surfaces used in discrete imagery. We show that a subclass of
n
-
G
-maps is equivalent to
n
-surfaces. To achieve this, we provide several characterizations of
n
-surfaces. Finally, the proofs being constructive, we show how to switch from one representation to another effectively.
Journal Article
Concluding Remarks
Numerous data structures and operations have been proposed in order to
represent and handle subdivided geometric objects, for various applications
within different fields, e.g. geometric modeling, computational geometry,
discrete geometry, computer graphics, image processing and analysis, etc.
[17, 8, 167, 214, 127, 107, 215, 93, 102, 87, 38, 163, 162, 94, 10, 195, 128,
170, 181, 203, 108, 57, 39, 132, 183, 153, 165, 104, 184, 154, 110, 49, 19, 159,
31, 45, 90, 84, 88, 66, 175, 185, 157, 60, 50, 158].
Book Chapter
Preliminary Notions
Two sets of basic notions are studied in this chapter, which is mainly based
upon [1, 125]:• the first ones are related to the objects we are interested in: subdivisions of geometric objects. They are illustrated by following Griffith’s
approach about surface classification [125], and then extended for higher
dimensions;• the second ones are related to the representations of these subdivisions.
Here we are interested in defining data structures which can be handled
in geometric softwares: such representations are algebraic ones, based
upon well-known discrete structures equivalent to graphs.
Book Chapter
n-maps
An n-map is a combinatorial data structure allowing to describe an ndimensional oriented quasi-manifold with or without boundary. The main
difference with n-Gmaps introduced in the previous chapter is the fact that
n-maps cannot describe nonorientable quasi-manifolds. The main interest of
n-maps comparing to n-Gmaps is to use twice less darts for representing orientable quasi-manifolds1. The main drawback is the “inhomogeneity” of the
definition, which often involves more complex algorithms. We structure this
chapter as for n-Gmaps, in order to emphasize the similarities and the differences between the two data structures. n-maps are defined in Section 5.1,
as the basic notions of cells, incidence and adjacency relations between the
cells. Some basic operations allowing to modify existing n-maps are presented
in Section 5.2. These operations allow to add/remove darts, increase/decrease
the dimension of an n-map, merge/split n-maps; the sew/unsew operations
allow to identify cells. We show in Section 5.3 that these operations make a
small basis of operations allowing to build any n-map. Moreover, it is pointed
out that it is possible to take multi-incidence between cells into account, and
some specific configurations are illustrated, related to multi-incidence such as
dangling cells and folded cells. A possible data structure for implementing
n-maps is proposed in Section 5.4, and also some algorithms allowing to develop a computer software handling n-maps. Some additional notions related
to n-maps are presented in Section 5.5. The relations between n-maps and
n-Gmaps are studied in Section 5.5.2.
Book Chapter
Introduction
Now, everyone knows several applications for which it is necessary to represent
geometric objects in a computer, for instance in the fields of architecture,
computer-aided design and manufacturing, geology, medical simulation, image
synthesis, video games, medical or biological image processing and analysis,
etc.
Book Chapter
Cellular Structures as Structured Simplicial Structures
So far we have been concerned with the definitions of n-Gmaps and n-maps,
construction operations and embedding definitions for applications in geometric modeling, computational geometry, discrete geometry, computer graphics,
image processing and analysis. But note that only examples of geometric objects which can be associated with n-Gmaps and n-maps have been provided.
An important question remains: what are the geometric objects which correspond to n-Gmaps and n-maps, and more generally to cellular structures,
including incidence graphs?
Book Chapter