Keynote speakers

Gilles Bertrand
ESIEE Paris, France

Tat Yung Kong
City University of New York, USA

Philippe Salembier
Universitat Politècnica de Catalunya, Spain

 

Gilles Bertrand
ESIEE Paris, France

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"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 field. 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 define 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 unified 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,
he was with the Thomson-CSF company where he designed image processing systems for aeronautical applications. He received his Ph.D from the Ecole Centrale in 1986. Until 2018, he was teaching and doing research with the Computer Science and Telecommunication Department of ESIEE-Paris and with the Laboratoire d’Informatique Gaspard-Monge of Université Paris-Est. His research interests are digital and combinatorial topology, mathematical morphology, image analysis.

Tat Yung Kong
City University of New York, USA

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"Hereditarily homology-simple sets and homology critical kernels of binary images on sets of convex polytopes"

Abstract

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 I1[1] is finite: We say I is a binary image on X and call each element of I1[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(I1[1] \ D) U I1[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:

  • A set D ⊆ I1[1] is hereditarily homology-simple in I if and only if every D-intersection in I’s critical kernel is a subset of a 1 of I that is not in D.

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 I1[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
Universitat Politècnica de Catalunya, Spain

"Processing Radar Images with Hierarchical Region-Based Representations and Graph Signal Processing Tools"

Abstract

This talk will discuss the interest of hierarchical region-based representations of images such as maxtree, mintree and Binary Partition Trees for radar images. These representations can be considered as an initial abstraction from the signal in which raw pixels are grouped to form regions which are hierarchically structured by inclusion in a tree. They provide multiple resolutions of description and easy access to subsets of regions. This approach and the associated notions will be discussed for both maxtree description of Synthetic Aperture Radar (SAR) image and for Binary Partition Tree for Polarimetric SAR images.

Once constructed, these hierarchical representations can be used for many applications including filtering, segmentation, classification and object detection. Many processing strategies consist in populating the tree with features of interest for the application and in applying a specific graph-cut called pruning. These pruning ideas will be illustrated in particular for polarimetric SAR image segmentation and speckle reduction.

The tree representation itself is a specific graph structure. As a result, an alternative processing strategy consists in populating the tree with attributes but considering the resulting data as graph attribute signals which can be processed with graph filters. The goal of this filtering step is to exploit the correlation existing between attribute values on neighboring tree nodes. Considering that trees are specific graphs where the connectivity towards ancestors and descendants may have a different meaning, several filtering strategies can be defined. Beside classical Graph filters, two new filtering notions can be used: Tree and Branch filters. These ideas will be illustrated in the context of ship detection in SAR images.

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.

His research interests include image and video sequence processing, compression and indexing, mathematical morphology, level sets and nonlinear filtering, as well as remote sensing image processing and signal processing tools for genomics. In terms of standardization activities, he has been involved in the definition of the MPEG-7 standard ("Multimedia Content Description Interface") as chair of the "Multimedia Description Scheme" group between 1999 and 2001.

He served as a Associate Editor of various journals including the Journal of Visual Communication and Image Representation (Academic Press), Signal Processing (Elsevier), Signal Processing: Image Communication(Elsevier), the Eurasip Journal on Image and Video Processing, the IEEE Transactions on Image Processing, the IEEE Transactions on Circuits and Systems for Video Technology and the IEEE Signal Processing Letters.He was member of the Image and Multidimensional Signal Processing Technical Committee of the IEEE Signal Processing Society between 2000-2006, an AdCom officer of the European Association for Signal Processing (EURASIP) between 1994-1999 and was technical chair (with Prof. Ed. Delp) of the IEEE Int. Conf. on Image Processing, ICIP'2003 organized in Barcelona. Philippe Salembier is a Fellow of the IEEE.

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