Emil Saucan

Sampling and Reconstruction of Surfaces and Higher Dimensional Manifolds

Abstract We present new sampling theorems for surfaces and higher dimensional manifolds. The core of the proofs resides in triangulation results for manifolds with boundary, not necessarily bounded. The method is based upon geometric considerations that are further augmented for 2-dimensional manifolds (i.e surfaces). In addition, we show how to apply the main results to obtain a new, geometric proof of the classical Shannon sampling theorem, and also to image analysis.

Forman curvature for complex networks

We adapt Formans discretization of Ricci curvature to the case of undirected networks, both weighted and unweighted, and investigate the measure in a variety of model and real-world networks. We find that most nodes and edges in model and real networks have a negative curvature. Furthermore, the distribution of Forman curvature of nodes and edges is narrow in random and small-world networks, while the distribution is broad in scale-free and real-world networks. In most networks, Forman curvature is found to display significant negative correlation with degree and centrality measures. However, Forman curvature is uncorrelated with clustering coefficient in most networks. Importantly, we find that both model and real networks are vulnerable to targeted deletion of nodes with highly negative Forman curvature. Our results suggest that Forman curvature can be employed to gain novel insights on the organization of complex networks.

Combinatorial Ricci Curvature and Laplacians for Image Processing

A new Combinatorial Ricci curvature and Laplacian oper- ators for grayscale images are introduced and tested on 2D synthetic, natural and medical images. Analogue formulae for voxels are also ob- tained. These notions are based upon more general concepts developed by R. Forman. Further applications, in particular a fltting Ricci ∞ow, are discussed.

Ricci curvature of the Internet topology

Analysis of Internet topologies has shown that the Internet topology has negative curvature, measured by Gromovs “thin triangle condition”, which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci curvature of the Internet, defined by Ollivier [1], Lin et al. [2], etc. Ricci curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.

Curvature based clustering for DNA microarray data analysis

Clustering is a technique extensively employed for the analysis, classification and annotation of DNA microarrays. In particular clustering based upon the classical combinatorial curvature is widely applied. We introduce a new clustering method for vertex-weighted networks, method which is based upon a generalization of the combinatorial curvature. The new measure is of a geometric nature and represents the metric curvature of the network, perceived as a finite metric space. The metric in question is natural one, being induced by the weights. We apply our method to publicly available yeast and human lymphoma data. We believe this method provides a much more delicate, graduate method of clustering then the other methods which do not undertake to ascertain all the relevant data. We compare our results with other works. Our implementation is based upon Trixy (as available at http://tagc.univ-mrs.fr/bioinformatics/trixy.html), with some appropriate modifications to befit the new method.

The existence of quasimeromorphic mappings in dimension 3

We prove that a Kleinian group G acting on H3 admits a nonconstant G-automorphic function, even if it has torsion elements, provided that the orders of the elliptic elements are uniformly bounded. This is accomplished by developing a method for meshing distinct fat triangulations which is fatness preserving. We further show how to adapt the proof to higher dimensions.

Isometric Embeddings in Imaging and Vision: Facts and Fiction

We explore the practicability of Nash’s Embedding Theorem in vision and imaging sciences. In particular, we investigate the relevance of a result of Burago and Zalgaller regarding the existence of PL isometric embeddings of polyhedral surfaces in ℝ3 and we show that their proof does not extended directly to higher dimensions.

Characterizing Complex Networks with Forman-Ricci Curvature and Associated Geometric Flows

We introduce Forman-Ricci curvature and its corresponding flow as characteristics for complex networks attempting to extend the common approach of node-based network analysis by edge-based characteristics. Following a theoretical introduction and mathematical motivation, we apply the proposed network-analytic methods to static and dynamic complex networks and compare the results with established node-based characteristics. Our work suggests a number of applications for data mining, including denoising and clustering of experimental data, as well as extrapolation of network evolution.

Metric Ricci Curvature for Manifolds

We introduce a metric notion of Ricci curvature for manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers theorem, for surfaces as well as for a large class of higher dimensional manifolds.

Curvature estimation over smooth polygonal meshes using the half tube formula

The interest, in recent years, in the geometric processing of polygonal meshes, has spawned a whole range of algorithms to estimate curvature properties over smooth polygonal meshes. Being a discrete approximation of a C2 continuous surface, these methods attempt to estimate the curvature properties of the original surface. The best known methods are quite effective in estimating the total or Gaussian curvature but less so in estimating the mean curvature. In this work, we present a scheme to accurately estimate the mean curvature of smooth polygonal meshes using a one sided tube formula for the volume above the surface. In the presented comparison, the proposed scheme yielded results whose accuracy is amongst the highest compared to similar techniques for estimating the mean curvature.