Topološka obdelava slik

CERAR, MATJAŽ (author), Mramor Kosta, Nežka (mentor), Virk, Žiga (comentor)

V diplomskem delu si ogledamo implementacijo topološkega pristopa k analizi digitalnih 2-dimenzionalnih slik. Najprej predstavimo dva načina, kako sliko brez izgube informacij poenostavimo in pripravimo, da je primerna za nadaljnje algoritme. Obdelano sliko nato predstavimo kot topološko strukturo, ki jo imenujemo kubični kompleks. S pomočjo slednjega zgradimo vektorsko polje, ki odraža smeri, kamor funkcijske vrednosti padajo, ter pripadajoči seznam kritičnih celic. Iz obeh dobljenih struktur zgradimo Morsov kompleks, s katerim zajamemo bistvene informacije o posamezni sliki, in izračunamo Bettijeva števila, ki opisujejo ključne značilnosti slike. Za izračun Bettijevih števil prav tako predstavimo dva pristopa. Na koncu sledi še prikaz uporabe, kjer na izbranih primerih slik štejemo svetle objekte.

• Language: Slovenian
• Keywords: diskretna Morsova teorija, topološka analiza podatkov, kubični kompleks, Morsov kompleks, Bettijeva števila
• Work type: Undergraduate thesis
• Organization: FRI - Faculty of Computer and Information Science
• Year: 2015
• PID: 20.500.12556/RUL-42849
• Publication date in RUL: 10.07.2015

CERAR, MATJAŽ, 2015, Topološka obdelava slik [online]. Bachelor’s thesis. [Accessed 3 July 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=42849

Secondary language

• Language: English
• Title: Topological analysis of images

Abstract

In this thesis we present an implementation of a topological approach to 2-dimensional digital images. First, we present two methods for simplifying and preparing the image, without loss of information, for further algorithms. We represent the image as a topological structure called a cubical complex. On the cubical complex, a discrete vector field encoding the directions of descent of grey scale values is constructed, together with the corresponding list of critical cells. From these, the Morse complex, which captures the vital information about the image, is built. Using Betti numbers, important features in the image are described. We present two approaches to computing Betti numbers. The thesis concludes with a presentation of how the implemented algorithms can be used for counting bright objects on specific examples of images.

• Keywords: discrete Morse theory, topological data analysis, cubical complex, Morse complex, Betti number