Abstract. We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allows us to address the various interactions between forms and vector fields (such as Lie derivatives) which are important in applications. Previous attempts at discrete exterior calculus have addressed only differential forms. We also introduce the notion of a circumcentric dual of a simplicial complex. The importance of dual complexes in this field has been well understood, but previous researchers have used barycentric subdivision or barycentric duals. We show that the use of circumcentric duals is crucial in arriving at a theory of discrete

Abstract—Aspects of the geometric discretization of electromagnetic fields on simplicial lattices are considered. First, the convenience of the use of exterior differential forms to represent the field quantities through their natural role as duals (cochains) of the geometric constituents of the lattice (chains = nodes, edges, faces, volumes) is briefly reviewed. Then, the use of the barycentric subdivision to decompose the (ordinary) simplicial primal lattice together with the (twisted) non-simplicial barycentric dual lattice into simplicial elements is considered. Finally, the construction of lattice Hodge operators by using Whitney maps on the first barycentric subdivision is described. The objective is to arrive at a discrete formulation of electromagnetic fields on general lattices which better adheres to the

A new method for the deformation of curves is presented. It is based upon a decomposition of the linear elasticity problem into basic physical laws.

The space-time geometric structure of Maxwell's equations is examined and a subset of them is found to define a pair of exact discrete time-stepping relations. The desirability of adopting an approach to the discretization of electromagnetic problems which exploits this fact is advocated, and the name topological time-stepping for numerical schemes complying with it is suggested. The analysis of the equations leading to this kind of time-stepping reveals that these equations are naturally written in terms of integrated field quantities associated with space-time domains. It is therefore suggested that these quantities be adopted as state variables within numerical methods. A list of supplementary prescriptions for a discretization of electromagnetic problems suiting this philosophy is given, with particular emphasis on the necessity to adopt a space-time approach in each discretization step. It is shown that some existing methods already comply with this tenets, but that this fact is not explicitly recognized and exploited. The role of the constitutive equations in this discretization philosophy is briefly analyzed. The extension of this approach to more general kinds of space-time meshes, to other sets of basic time-stepping equations and to other field theories is finally considered.

One physical process involved in many computer vision problems is the heat diffusion process. Such Partial differential equations are continuous and have to be discretized by some techniques, mostly mathematical processes like finite differences or finite elements. The continuous domain is subdivided into sub-domains in which there is only one value. The diffusion equation comes from the energy conservation then it is valid on a whole domain. We use the global equation instead of discretize the PDE obtained by a limit process on this global equation. To encode these physical global values over pixels of different dimensions, we use a computational algebraic topology (CAT)-based image model. This model has been proposed by Ziou and Allili and used for the deformation of curves and optical flow. It introduces the image support as a decomposition in terms of points, edges, surfaces, volumes, etc. Images of any dimensions can then be handled. After decomposing the physical principles of the heat transfer into basic laws, we recall the CAT-based image model and use it to encode the basic laws. We then present experimental results for nonlinear graylevel diffusion for denoising, ensuring thin features preservation.

Abstract—We identify primal and dual formulations in the finite element method (FEM) solution of the vector wave equation using a geometric discretization based on differential forms. These two formulations entail a mathematical duality denoted as Galerkin duality. Galerkin-dual FEM formulations yield identical nonzero (dynamical) eigenvalues (up to machine precision), but have static (zero eigenvalue) solution spaces of different dimensions. Algebraic relationships among the degrees of freedom of primal and dual formulations are explained using a deep-rooted connection between the Hodge–Helmholtz decomposition of differential forms and Descartes–Euler polyhedral formula, and verified numerically. In order to tackle the fullness of dual formulation, algebraic and topological thresholdings are proposed to approximate inverse mass matrices by sparse matrices. Index Terms—Differential forms, finite element methods (FEMs), Maxwell equations, sparse matrices. I.

Abstract—This paper demonstrates how discrete exterior calculus tools may be useful in computer vision and graphics. Avariational approach provides a link with mechanics. I.

Abstract. We study hierachical segmentation in the framework of edgeweighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical edgesegmentations. We end this paper by showing how the proposed framework allows to see constrained connectivity as a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.

This paper proposes an alternative to partial differential equations (PDEs) for the solution of diffusion (Perona and Malik scheme), using the heat transfer problem. Traditionally, the method for solving such physics-based problems is to discretize and solve a PDE by a mathematical process. We propose to use the global heat equation and decompose it into simpler laws. Some of these laws admit an exact global version since they arise from conservation principles while the assumptions on the others can be made wisely, taking into account knowledge about the problem. A computational algebraic topology-based image model allows us to write directly discrete equations. The numerical scheme is derived in a straightforward way from the problem modeled. It thus provides a physical explanation of each solving step in the solution. Finally, we present results for non linear diffusion.

et de nationalité italienne acceptée sur proposition du jury: Prof. D. Floreano, directeur de thèse Dr. I. Harvey, rapporteur