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25 Mar 2019 ESIEE Paris (France)
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Keynote speakersGilles Bertrand Tat Yung Kong Philippe Salembier
Gilles Bertrand "An axiomatic approach to combinatorial topology" Abstract We present an attempt to build an axiomatic approach to combinatorial topology. The purpose is to introduce a ground set of definitions which allow us to describe structures and to derive basic constructions relative to that fi eld. A main ingredient of our project is the notion of completion. Completions are inductive properties which may be expressed in a declarative way and which may be combined. Intuitively, a completion may be seen as a rewriting rule acting on sets. In this talk, we first introduce two completions in order to defi ne a remarkable collection of acyclic simplicial complexes, namely the collection of dendrites. We give few basic properties of this collection. Also, we present a theorem which shows the equivalence between this collection and the one made of all simplicial complexes which are acyclic in the sense of homology. Then, we introduce several completions for describing dyads. A dyad is a pair of complexes which are, in a certain sense, linked by a "relative topology". We give some basic properties of dyads, and introduce a second set of axioms for relative dendrites. We establish a theorem which provides a link between dyads, relative dendrites, and dendrites. At last, using again the framework of completions, we propose an extension of simple homotopy by considering homotopic pairs. Intuitively, a homotopic pair is a couple of objects (X,Y) such that X is included in Y and (X,Y) may be transformed to a trivial couple by simple homotopic deformations that keep X inside Y. Thus, these objects are linked by a "relative homotopy relation". Our main result is a theorem that makes clear the link between homotopic pairs and dyads. Thus, we prove that, in the unifi ed framework of completions, it is possible to handle notions relative to both homotopy and homology. Biography Gilles Bertrand received his Ingénieur’s degree from the Ecole Centrale des Arts et Manufactures in 1976. Until 1983, Tat Yung Kong
"Hereditarily homology-simple sets and homology critical kernels of binary images on sets of convex polytopes" Abstract (The pdf file is available here.) We define a binary image to be a mapping I : X → {0, 1} in which X is a set of nonempty sets (e.g., a set of cubical voxels) in a Euclidean space and I−1[1] is finite: We say I is a binary image on X and call each element of I−1[1] a 1 of I. For any set S of 1s of I we use the term S-intersection to mean a nonempty set that is the intersection of a nonempty subset of S. Thus an S-intersection is either an element of S or a nonempty intersection of two or more elements of S. Let D be any set of 1s of a binary image I. If the inclusion U(I−1[1] \ D) → U I−1[1] induces homology isomorphisms in all dimensions, then we say D is homology-simple in I. If every subset of D is homology-simple in I, then we say D is hereditarily homology-simple in I. A local characterization of hereditarily homology-simple sets can be useful for designing parallel thinning algorithms or for checking the topological soundness of proposed parallel thinning algorithms. When I is a binary image on the grid cells of a Cartesian grid of dimension ≤ 4, it can be deduced from results of Bertrand and Couprie that the sets D of 1s that are hereditarily homology-simple in I can be locally characterized as follows in terms of Bertrand's concept of critical kernel:
After discussing this characterization and some of its consequences, we will explain how we can generalize it to a local characterization of hereditarily homology-simple sets of 1s in any binary image I on an arbitrary set of convex polytopes of any dimension. To do this, we need only replace ``I's critical kernel'' in the above characterization with ``I's homology critical kernel''. We define the latter to be the set of all I−1[1]-intersections c for which the intersection of c with the union of the 1s of I that do not contain c either is empty or is disconnected or has non-trivial homology in some positive dimension. Biography T. Yung Kong is Professor of Computer Science at Queens College of the City University of New York. Much of his published work has dealt with topological problems relating to binary images, especially problems of digital topology. He has also worked on some topics in other research areas (such as the theory of fuzzy segmentation and algorithms for convex feasibility in recent years). A member of the Computer Science faculty at Queens College since 1989, he previously held faculty positions in computer science at Ohio University and in mathematics at City College of the City University of New York. In 2000-01 he was a Visiting Professor in the Computer and Information Sciences Department at Temple University. He has a B.A. in mathematics and an M.Math. from the University of Cambridge (earned in 1981 and 1982). In 1982-85 he was a doctoral student at the Oxford University Computing Laboratory, where he pursued research on digital topology. He received a D.Phil. from the University of Oxford in 1986.
Philippe Salembier
"Processing Radar Images with Hierarchical Region-Based Representations and Graph Signal Processing Tools" Abstract Biography Philippe Salembier received an engineering degree from the Ecole Polytechnique, Paris, France, in 1983 and an engineering degree from the Ecole Nationale Supérieure des Télécommunications, Paris, France, in 1985. From 1985 to 1989, he worked at Laboratoires d'Electronique Philips, Limeil-Brevannes, France, in the fields of digital communications and signal processing for HDTV. In 1989, he joined the Signal Processing Laboratory of the Swiss Federal Institute of Technology in Lausanne, Switzerland, to work on image processing and received a PhD in 1991. At the beginning of 1992, after a stay at the Harvard Robotics Laboratory as a Postdoctoral Fellow, he joined the Technical University of Catalonia, Barcelona, Spain, where he is currently professor lecturing on the area of digital signal and image processing. |
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