Manifold reconstruction from discrete point sets

PhD Thesis Proposal 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 dimension. This motivates manifold reconstruction to 
study the geometrical and topological properties of the manifold. Given a set 
of point samples drawn 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 four tasks: detect the manifold 
dimension, estimate the tangent spaces of the manifold, construct an implicit 
function whose zero-set is homeomorphic to the manifold and produce a 
simplicial complex homeomorphic to the manifold. In this proposal, we present a 
method to estimate the tangent space with provably small angular error. We also 
prove an implicit function whose zero-set is geometrically close to and 
homeomorphic to the manifold with arbitrary dimension. Preliminary results and 
experimental results are presented.

In the future, we will design an algorithm to produce a simplicial complex from 
the zero-set of the implicit function and implement the algorithm to show that 
it is practical. Also, we will experimentally show that the zero-set allows the 
synthesis of new data points on the manifold.


Date:                   Friday, 25 January 2013

Time:                   2:00pm - 4:00pm

Venue:                  Room 3501
                         lifts 25/26

Committee Members:      Prof. Siu-Wing Cheng (Supervisor)
                         Dr. Sunil Arya (Chairperson)
 			Prof. Mordecai Golin
 			Dr. Ke Yi


**** ALL are Welcome ****