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Koliko točk določa katero geometrijsko ploskev?
POLANC, MIHA (Author), Fijavž, Gašper (Mentor) More about this mentor... This link opens in a new window

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Abstract
Geometrijsko ploskev lahko približno opišemo s končnim vzorcem njenih točk. V delu se ukvarjamo z vprašanjem, s koliko točkami, glede na način vzorčenja in rod ploskve, lahko zanesljivo rekonstruiramo originalno geometrijsko ploskev. Najprej opišemo različne načine vzorčenja točk s ploskve, kaj je enakomerni in kaj je slučajni vzorec točk z izbrane ploskve. Vzorce lahko obravnavamo s topološkimi metodami, natančneje metodami vztrajne homologije. Iz vzorca točk s programskim paketom Javaplex konstruiramo filtracijo Vietoris-Ripsovih simplicialnih kompleksov in opazujemo črtni diagram Bettijevih števil. V zaključku predstavimo računske rezultate za sfero in torus glede na različna modela vzorčenja, ter nekaj možnosti za nadaljnje izboljšave.

Language:Slovenian
Keywords:Vietoris-Ripsov kompleks, Bettijeva števila, Javaplex, vztrajna homologija, sfera, torus.
Work type:Undergraduate thesis (m5)
Organization:FRI - Faculty of computer and information science
Year:2016
Views:535
Downloads:500
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Secondary language

Language:English
Title:How many points are needed to determine a geometric surface?
Abstract:
A geometric surface can be approximately described using a finite point-sample. The main question of this thesis is the following: how many points, depending on the sampling model and surface genus, are needed to confidently reconstruct the original geometric surface. First we present different sampling models of surface points — the uniform and the random sample. We use topological methods, in particular persistent homology, to process our data. Using Javaplex software package we construct a filtration with Vietoris-Rips simplicial complexes and consider the bar-code diagram of its Betti numbers. Finally we present our computational results for both the sphere and the geometric torus with respect to the two sampling models , and several options for further improvements.

Keywords:Vietoris-Rips complex, Betti numbers, Javaplex, persistent homology, sphere, torus.

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