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Reconstruction from point cloud by using discrete functions Morse

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Author(s):
Helton Hideraldo Biscaro
Total Authors: 1
Document type: Doctoral Thesis
Press: São Carlos.
Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
Defense date:
Examining board members:
Antonio Castelo Filho; Renato Cardoso Mesquita; Luis Gustavo Nonato; Manuel Menezes de Oliveira Neto; Luiz Carlos Pacheco Rodrigues Velho
Advisor: Antonio Castelo Filho; Luis Gustavo Nonato
Abstract

Performing extensive geometrical computaiions is a major source of problems in reconstruction from sample points. Such operations are computationally intensive, and introduce numerical instabilities that may be difficult to handle. We introduce an innovative approach for generating a piecewise linear approximations from sample points that reduces geometrical calculations drastically by converting geometrical information into topological attributes. Geometrical tests are employed only at an initial step, to compute a Delaunay triangulation of the points. Discrete Morse theory is used to derive a topological framework towards reconstruction. The result is a robust reconstruction algorithm that handles multiple components and has reduced computational costs. In this work we introduce an innovative approach for generating 3D piecewise linear approximations from sample points that is strongly based on topological information, thus reducing the computational cost and numerical instabilities typically associated with geometric computations. Discrete Morse theory provides the basis of a topological framework for reconstruction that results in a robust reconstruction algorithm capable of handling multiple components and with low computational cost. We describe the proposed approach and introduce the reconstruction algorithm, called Tor - Topological Object Reconstructor. Reconstruction results are presented and the performance of Tor is compared with that of other reconstruction approaches for some standard point sets. Theoretical guarantees of reconstruction are also provided. (AU)