Koliko točk določa katero geometrijsko ploskev?

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 Vietoris-Ripsov kompleks, Bettijeva števila, Javaplex, vztrajna homologija, sfera, torus. Undergraduate thesis (m5) FRI - Faculty of computer and information science 2016 535 500 (0 votes) Voting is allowed only to logged in users. AddThis uses cookies that require your consent. Edit consent...

## Secondary language

Language: English How many points are needed to determine a geometric surface? 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. Vietoris-Rips complex, Betti numbers, Javaplex, persistent homology, sphere, torus.