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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. 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.

The cohomology of digraphs was introduced for the first time by Dimakis and Muller-Hoissen. Their algebraic definition is based on a differential calculus on an algebra of functions on the set of vertices with relations that follow naturally from the structure of the set of edges. A dual notion of homology of digraphs, based on the notion of path complex, was introduced by the authors, and the first methods for computing the (co)homology groups were developed. The interest in homology on digraphs is motivated by physical applications and relations between algebraic and geometrical properties of quivers. The digraph $G_B$ of the partially ordered set $B_{S}$ of simplexes of a simplicial complex $S$ has graph homology that is isomorphic to the simplicial homology of $S$. In this paper, we introduce the concept of cubical digraphs and describe their homology properties. In particular, we define a cubical subgraph $G_{S}$ of $G_B$, whose homologies are isomorphic to the simplicial homologies of $S$.