Manifold Reconstruction from Discrete Point Sets

The Hong Kong University of Science and Technology
Department of Computer Science and Engineering


PhD Thesis Defence


Title: "Manifold Reconstruction from Discrete Point Sets"

By

Mr. Man Kwun CHIU


Abstract

In many applications involving datasets in high dimensional spaces, it is 
often postulated that the data points lie on an unknown manifold of much 
lower dimension than the ambient space dimension. This motivates manifold 
reconstruction to study the geometrical and topological properties of the 
manifold. Given a set of points sampled from an unknown manifold, manifold 
reconstruction is to produce a representation with the same topology as 
the manifold and geometrically close to it.

We divide the reconstruction problem into three tasks: detecting the 
manifold dimension, estimating the tangent spaces of the manifold and 
constructing an implicit function whose zero-set is homeomorphic to the 
manifold. In this thesis, we address the second and third tasks assuming 
that the manifold dimension has been determined. We present a method to 
estimate the tangent space with provably small angular error. We also show 
how to construct an implicit function whose zero-set contains a 
homeomorphic approximation of the manifold and is geometrically close to 
the manifold. Some experimental results are presented in both tasks.


Date:			Friday, 6 December 2013

Time:			2:00pm – 4:00pm

Venue:			Room 3494
 			Lifts 25/26

Chairman:		Prof. Daniel Palomar (ECE)

Committee Members:	Prof. Siu-Wing Cheng (Supervisor)
 			Prof. Sunil Arya
 			Prof. Mordecai Golin
 			Prof. Beifang Chen (MATH)
                        Prof. Lusheng Wang (Comp. Sci., CityU.)


**** ALL are Welcome ****