Certified geometric computation: radial basis function based isosurfaces and Morse-Smale complexes

PhD ceremony: Mr. A. Chattopadhyay, 13.15 uur, Academiegebouw, Broerstraat 5, Groningen

Title: Certified geometric computation: radial basis function based isosurfaces and Morse-Smale complexes

Promotor(s): prof. G. Vegter

Faculty: Mathematics and Natural Sciences

Certified geometric computation, a newly emerging branch of computing science, is a computation paradigm where the main goal is not only numerical accuracy, but above all the geometric and topological correctness of the output. More precisely, in certified geometric computation, the challenge is to develop algorithms for computing topologically correct and geometrically close approximations of implicitly or explictly given input shapes.

The current thesis focuses on an important class of problems in computational geometry and topology. The first part of the thesis deals with certified surface reconstruction using the radial basis function (RBF) method, which is being used in Computer Aided Geometric Design (CAGD), in visualization, and in medical applications. The surface-reconstruction method using radial basis functions consists of two steps: (i) computing an interpolating implicit function, the zero set of which contains the points in the data set, followed by (ii) extraction of isocurves or isosurfaces. The second step of the method, that is the extraction of certified isosurfaces, has been developed in this research project.

In the second part of the thesis we consider the problem of certified computation of Morse-Smale complexes corresponding to Morse-Smale gradient vector fields defined on bounded planar domains or on implicit surfaces. The Morse-Smale complex is an important tool for the global topological analysis of complex geometrical shapes or data. Here the problem is to compute certified separatrices of a Morse-Smale system connecting a saddle to a source or a sink, and separating attracting regions of sinks, and repelling regions of sources of the gradient field. We propose new techniques for certified computation of the Morse-Smale complex. Computing the Morse-Smale complex, i.e., the configurations of singular points and separatrices of a Morse-Smale gradient field, can be extremely challenging because of the arbitrarily complex nature of the input data, whereas available computational resources are limited.