"Shape and Image Cognition, Construction and Compression via Tools from Differential Geometry"
Professor Franz-Erich Wolter, Welfenlab, Institute of Man-Machine-Communication
Leibniz University Hannover, Germany, http://welfenlab.de/en/index.html
Thursday, September 11, 2014, 12:00pm to 1:00pm | Room 33-116
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We shall describe how concepts from differential geometry have been providing powerful tools creating major advances in geometric modeling, geometry processing and image analysis dealing with the topics presented in the title of this address. This talk includes a retrospective compiling contributions of the author's works showing how concepts from local and global differential geometry have introduced new methods into geometric modeling and shape interrogation and classification finally ending with modern state of the art research on geometry processing and image processing. A major part of this seminar starting with works in the late nineties at the author's laboratory is dedicated to discussing how "efficient finger prints" useful for indexing and clustering digital data collections can be derived from spectra of Laplace operators being naturally associated with geometric objects such as surfaces and solids as well as (colored) images including medical 2d- and 3d-images. Recently the latter works obtained particular attention in the area of medical imaging. Geometric aspects of the Laplacian operator lead to generalizations of the Laplacian operating on line bundles allowing to compute eigen-functions whose iso-surfaces yield smooth non-orientable Seifert surfaces spanning knot complements in three-dimensional manifolds. Next we focus on cut loci, the medial axis and its inverse in Euclidean and Riemannian worlds.
This work starts with basic medial axis results presented by the the author in the early nineties. Those results state: The Medial Axis Transform can be used to reconstruct, modify and design a given shape ("Shape Reconstruction Theorem"). Under some weak assumptions the medial axis contains the essence of the topological shape of the geometric object as it is a deformation retract of the given shape ("Topological Shape Theorem"). Therefore the medial axis contains the homotopy type of the given shape. We present recent results showing how geodesic Voronoi diagrams, geodesic medial axis and its inverse can be computed in 3d- or higher-dimensional Riemannian spaces. The "medial axis inverse" allows to construct a medial modeler providing efficient features for shape optimization with respect to shape dependent mechanical properties.
Dr. F.-E. Wolter has been a full professor of computer science at Leibniz Universität Hannover (LUH) since the winter term of academic year 1994-1995, where he heads the Institute of Man-Machine-Communication and directs the Division of Computer Graphics and Geometric Modeling called Welfenlab. Before coming to Hannover, Dr. Wolter held faculty positions at the University of Hamburg (in 1994), MIT (1989-1993) and Purdue University (1987-1989). Prior to this he developed industrial expertise as a software and development engineer with AEG in Germany (1986-1987). Dr. Wolter obtained his Ph.D. in 1985 from the department of mathematics at the Technical University of Berlin in the area of Riemannian manifolds. In 1980 he graduated in mathematics and theoretical physics from the Free University of Berlin. At MIT Dr. Wolter co-developed the geometric modeling system Praxiteles for the US Navy. Since then he has been publishing various papers that broke new ground applying concepts from differential geometry and topology on problems and design of new methods used in geometric modeling and CAD systems as well as shape and image analysis. These works include pioneering contributions on medial axis theory, the computation of medial axes and Voronoi diagrams and geodesics in Riemannian space as well as pioneering works on Laplace spectra as finger prints for multi dimensional geometric objects and images with applications in biomedical imaging. The works on laplacian operators recently lead to computational studies on generalizations of Laplacians operating on line bundles allowing to compute eigen-functions whose iso-surfaces yield smooth orientable and non-orientable Seifert surfaces in three-dimensional manifolds. During the last decade research on Virtual Reality systems with an emphasis on haptic and tactile perception has been subject of Dr. Wolter's research in Hannover. More recently his research includes the development of medical imaging systems and bio-mechanical simulation systems including hearing mechanics of the cochlear.