Abstract. We present a programming framework for discrete mathematical morphology centered on the concept of genericity. We show that formal definitions of morphological algorithms can be translated into actual code, usable on virtually any kind of compatible images, provided a general definition of the concept of image is given. This work is implemented in Milena, a generic, efficient, and user-friendly image processing library 3 1

This paper describes an efficient and robust algorithm for computing a common refinement of two meshes modeling a common surface of arbitrary shape by overlaying them on top of each other. A common refinement is an important data structure for transferring data between meshes that have different topological structures. Our algorithm is optimal in time and space, with linear complexity. Special treatments are introduced to handle discretization and rounding errors and to ensure robustness with imprecise computations. It also addresses the additional complexities caused by degeneracies, sharp edges, sharp corners, and nonmatching boundaries. The algorithm has been implemented and demonstrated to be robust for complex geometries from real-world applications.

Cell complexes are potentially very useful in many fields, including image segmentation, numerical analysis, and computer graphics. However, in practice they are not used as widely as they could be. This is partly due to the difficulties in actually implementing algoritStms on top of cell complexes. We propose to use generic programming to design cell complex data stxuctures that are easy to use, efficient, and flexible. The implementation of the new design is demonstrated for a number of common cell complex types and an example algorithm.

We propose a method to model and control topological changes by a smooth deformation of a polyhedral mesh using curves and loops. As changing the genus of a surface is not a continuous transformation, the topological change is made when an intermediate shape between the two topologies has been obtained. The creation and the deletion of holes are studied. The deletion of a hole uses non nullhomotopic loops to designate the hole to be deleted. A method computing two independent loops associated to a hole is presented.

In this report, a set of C++ classes is presented for representing unstructured triangular meshes of intrinsic dimension two; i.e. oriented 2-manifolds. Simple classes for the basic mesh objects, i.e. vertices, triangles, and line segments, are described. They define abstractions based on their incidence relations and a few geometric primitives for a mesh class, which is an intelligent container class of three lists of these simple mesh objects. The classes are intended to be components in an object oriented approach to software for meshing applications described in the report. This context differentiates the roles of the mesh class and the simple mesh object classes; these latter can be extended as the carriers of the applications data. The capability of the classes of this report to simultaneously simplify the coding of mesh methods and facilitate generalization of the code is discussed with examples. The report provides an overview of the class design and use, tutorial e...

The main question this dissertation addresses is the following: Is it possible to design a computer graphics API such that modeling primitives, computing visibility, and generating shadows from point, linear, and area light sources can be conveniently and concisely expressed? The thesis answers this question in the affirmative by describing a framework for geometric computing in computer graphics. The classes in the layered system constituting the framework are described using the UML notation and each algorithm presented is encapsulated in a member method of a class in the hierarchy. We identify a number of abstractions for object–space graphics such as transparent visibility and opaque visibility. These abstractions are somewhat harder to implement than standard rasterized abstractions as they rely on graphs and planar maps. Nevertheless, these notions prove to be fundamental in this work on object–space graphics and also appear to be fundamental for computer graphics in general. We propose that clients of a graphics API such as the one presented here should be relieved from the onus of computing shadows and we show that the computation of shadows can be automated and encapsulated in the framework. We address illumination under a point, a linear,

This paper describes the design and the implementation of the three-dimensional triangulation package 1 of the Computational Geometric Algorithms Library Cgal 2 . We focus on representation issues and especially insist on how the cases of degenerate dimensions are treated. The algorithmic issues are not examined in this short paper. 1 Introduction A three-dimensional triangulation is a three-dimensional simplicial complex, pure connected and without singularities [BY98]. It is a set of cells (tetrahedra) such that two cells either do not intersect or share a common facet, edge or vertex. Generalizing the storage of 2D triangulations [tri99] to the 3D case, we choose to explicitly represent only cells and vertices, together with adjacency and incidence relations: a cell has pointers to its four vertices and to its four neighbors, a vertex has a pointer to one of the cells having this vertex. Design Overview We follow the design in three layers proposed for polyhedral surfaces by L...

This chapter provides important background material that will be needed for Part II. Formulating and solving motion planning problems require defining and manipulating complicated geometric models of a system of bodies in space. Section 3.1 introduces geometric modeling, which focuses mainly on semi-algebraic modeling because it is an important part of Chapter 6. If your interest is mainly in Chapter 5, then understanding semi-algebraic models is not critical. Sections 3.2 and 3.3 describe how to transform a single body and a chain of bodies, respectively. This will enable the robot to “move. ” These sections are essential for understanding all of Part II and many sections beyond. It is expected that many readers will already have some or all of this background (especially Section 3.2, but it is included for completeness). Section 3.4 extends the framework for transforming chains of bodies to transforming trees of bodies, which allows modeling of complicated systems, such as humanoid robots and flexible organic molecules. Finally, Section 3.5 briefly covers transformations that do not assume each body is rigid.

Cgal est une biblioth#que d'algorithmes g#om#triques, d#velopp#e dans le cadre d'un projet Esprit LTR. Le but de cette biblioth#que est de rendre les algorithmes g#om#triques d#veloppes par la communaut# de g#om#trie algorithmique disponibles pour une utilisation dans l'industrie, ainsi que par les chercheurs et les praticiens du domaines et des domaines connexes. Les buts en sont AEexibilit#, eOEcacit#, correction, et robustesse. Le design de la biblioth#que est pr#sent# de fa#on synth#tique, tout en montrant les techniques qui permettent de concilier au mieux ces exigences apparemment contradictoires. Ces techniques sont pour une large part inspir#es de la programmation g#n#rique et de la Standard Template Library (STL) de C++.