Mohammed Mostefa Mesmoudi

Morphology-driven simplification and multiresolution modeling of terrains

We propose a technique for simplification and multiresolution modeling of a terrain represented as a TIN. Our goal is to maintain the morphological structure of the terrain in the resulting multiresolution model. To this aim, we extend Morse theory, developed for continuous and differentiable functions, to the case of piecewise linear functions. We decompose a TIN into areas with uniform morphological properties (such as valleys, basins, etc.) separated by a network of critical lines and points. We describe an algorithm to compute the above decomposition and the critical net, and a TIN simplification algorithm that preserves them. On this basis, we build a multiresolution terrain model, which provides a representation of critical features at any level of detail.

Decomposing non-manifold objects in arbitrary dimensions

We address the problem of building valid representations of non-manifold d-dimensional objects through an approach based on decomposing a non-manifold d-dimensional object into an assembly of more regular componems. We first define a standard decomposition of d-dimensional non-manifold objects described by abstract simplicial complexes. This decomposition splits a non-manifold object into components that belong to a well-understood class of objects, that we call initial quasi-manifold. Initial quasi-manifolds cannot be decomposed without cutting them along manifold faces. They form a decidable superset of d-manifolds for d ≥ 3, and coincide with manifolds for d ≤ 2. We then present an algorithm that computes the standard decomposition of a general non-manifold complex. This decomposition is unique, and removes all singularities which can be removed without cutting the complex along its manifold faces.

Discrete distortion in triangulated 3-manifolds

We introduce a novel notion, that we call discrete distortion, for a triangulated 3‐manifold. Discrete distortion naturally generalizes the notion of concentrated curvature defined for triangulated surfaces and provides a powerful tool to understand the local geometry and topology of 3‐manifolds. Discrete distortion can be viewed as a discrete approach to Ricci curvature for singular flat manifolds. We distinguish between two kinds of distortion, namely, vertex distortion, which is associated with the vertices of the tetrahedral mesh decomposing the 3‐manifold, and bond distortion, which is associated with the edges of the tetrahedral mesh. We investigate properties of vertex and bond distortions. As an example, we visualize vertex distortion on manifold hypersurfaces in R4 defined by a scalar field on a 3D mesh. distance fields.

Discrete Curvature Estimation Methods for Triangulated Surfaces

We review some recent approaches to estimate discrete Gaussian and mean curvatures for triangulated surfaces, and discuss their characteristics. We focus our attention on concentrated curvature which is generally used to estimate Gaussian curvature. We present a result that shows that concentrated curvature can also be used to estimate mean curvature and hence principal curvatures. This makes concentrated curvature one of the fundamental notions in discrete computational geometry.

Surface Segmentation through Concentrated Curvature

Curvature is one of the most relevant notions that links the metric properties of a surface to its geometry and to its topology (Gauss-Bonnet theorem). In the literature, a variety of approaches exist to compute curvatures in the discrete case. Several techniques are computationally intensive or suffer from convergence problems. In this paper, we discuss the notion of concentrated curvature, introduced by Troyanov [24]. We discuss properties of this curvature and compare with a widely-used technique that estimates the Gaussian curvatures on a triangulated surface. We apply our STD method [13] for terrain segmentation to segment a surface by using different curvature approaches and we illustrate our comparisons through examples.

Morphological analysis of terrains based on discrete curvature and distortion

In order to characterize the morphology of a triangulated terrain, we define several discrete estimators that mimic mean and Gaussian curvatures in the discrete case. We start from concentrated curvature, a discrete notion of Gaussian curvature for polyhedral surfaces, defined by Troyanov [7]. Since concentrated curvature does not depend on the local geometric shape of the terrain, we introduce Ccurvature that allows us to obtain discrete counterparts of both Gaussian and mean curvature. Finally, we define distortion, which behaves as an approximation of mean curvature. We apply all such measures to the analysis of the morphology of triangulated terrains.

A Smale-like decomposition for discrete scalar fields

In this paper we address the problem of representing the structure of the topology of a d-dimensional scalar field as a basis for constructing a multiresolution representation of the structure of such afield. To this aim, we define a discrete decomposition of a triangulated d-dimensional domain, on whose vertices the values of the field are given. We extend a Smale decomposition, defined by Thom (1949) and Smale (1960) for differentiable functions, to the discrete case, to what we call a Smale-like decomposition. We introduce the notion of discrete gradient vector field, which indicates the growth of the scalar field and matches with our decomposition. We sketch an algorithm for building a Smale-like decomposition and a graph-based representation of this decomposition. We present results for the case of two-dimensional fields.

Smale-like decomposition and forman theory for discrete scalar fields

Forman theory, which is a discrete alternative for cell complexes to the well-known Morse theory, is currently finding several applications in areas where the data to be handled are discrete, such as image processing and computer graphics. Here, we show that a discrete scalar field f, defined on the vertices of a triangulated multidimensional domain Σ, and its gradient vector field Grad f through the Smale-like decomposition of f [6], are both the restriction of a Forman function F and its gradient field Grad F that extends f over all the simplexes of Σ. We present an algorithm that gives an explicit construction of such an extension. Hence, the scalar field f inherits the properties of Forman gradient vector fields and functions from field Grad F and function F.

Multiresolution Analysis of 3D Images Based on Discrete Distortion

We consider a model of a 3D image obtained by discretizing it into a multiresolution tetrahedral mesh known as a hierarchy of diamonds. This model enables us to extract crack-free approximations of the 3D image at any uniform or variable resolution, thus reducing the size of the data set without reducing the accuracy. A 3D intensity image is a scalar field (the intensity field) defined at the vertices of a 3D regular grid and thus the graph of the image is a hyper surface in

Discrete Distortion for Surface Meshes

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