1. Region segmentation of images based on a raster-scan paradigmLuka Lukač, Andrej Nerat, Damjan Strnad, Štefan Horvat, Borut Žalik, 2024, izvirni znanstveni članek Opis: This paper introduces a new method for the region segmentation of images. The approach is based on the raster-scan paradigm and builds the segments incrementally. The pixels are processed in the raster-scan order, while the construction of the segments is based on a distance metric in regard to the already segmented pixels in the neighbourhood. The segmentation procedure operates in linear time according to the total number of pixels. The proposed method, named the RSM (raster-scan segmentation method), was tested on selected images from the popular benchmark datasets MS COCO and DIV2K. The experimental results indicate that our method successfully extracts regions with similar pixel values. Furthermore, a comparison with two of the well-known segmentation methods—Watershed and DBSCAN—demonstrates that the proposed approach is superior in regard to efficiency while yielding visually similar results.
Ključne besede: segment, image analysis, distance metric, Watershed, DBSCAN
Objavljeno v DKUM: 05.12.2024; Ogledov: 0; Prenosov: 2
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2. On metric dimensions of hypercubesAleksander Kelenc, Aoden Teo Masa Toshi, Riste Škrekovski, Ismael G. Yero, 2023, izvirni znanstveni članek Opis: In this note we show two unexpected results concerning the metric, the edge metric and the mixed metric dimensions of hypercube graphs. First, we show that the metric and the edge metric dimensions of ▫$Q_d$▫ differ by at most one for every integer ▫$d$▫. In particular, if ▫$d$▫ is odd, then the metric and the edge metric dimensions of ▫$Q_d$▫ are equal. Second, we prove that the metric and the mixed metric dimensions of the hypercube ▫$Q_d$▫ are equal for every ▫$d \ge 3$▫. We conclude the paper by conjecturing that all these three types of metric dimensions of ▫$Q_d$▫ are equal when d is large enough.
Ključne besede: edge metric dimension, mixed metric dimension, metric dimension, hypercubes
Objavljeno v DKUM: 21.05.2024; Ogledov: 123; Prenosov: 11
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3. Distance-based Invariants and Measures in GraphsAleksander Kelenc, 2019, doktorska disertacija Opis: This doctoral dissertation is concerned with aspects on distance related topics in graphs. We study three main topics, namely a recently introduced measure called the Hausdorff distance of graphs and two new graph invariants - the edge metric dimension and the mixed metric dimension of graphs. All three topics are part of the metric graph theory since they are tightly connected with the basic concept of distance between two vertices of a graph.
The Hausdorff distance is a relatively new measure of the similarity of graphs.
The notion of the Hausdorff distance considers a special kind of common subgraph of the compared graphs and depends on the structural properties outside of the common subgraph. We study the Hausdorff distance between certain families of graphs that often appear in chemical graph theory. Next to a few results for general graphs, we determine formulae for the distance between paths and cycles.
Previously, there was no known efficient algorithm for the problem of determining the Hausdorff distance between two trees, and in this dissertation we present a polynomial-time algorithm for it. The algorithm is recursive and it utilizes the divide and conquer technique. As a subtask it also uses a procedure that is based on the well-known graph algorithm for finding a maximum bipartite matching.
The edge metric dimension is a graph invariant that deals with distinguishing the edges of a graph. Let $G=(V(G),E(G))$ be a connected graph, let $w \in V(G)$ be a vertex, and let $e=uv \in E(G)$ be an edge. The distance between the vertex $w$ and the edge $e$ is given by $d_G(e,w)=\min\{d_G(u,w),d_G(v,w)\}$. A vertex $w \in V(G)$ distinguishes two edges $e_1,e_2 \in E(G)$ if $d_G(w,e_1) \ne d_G(w,e_2)$. A set $S$ of vertices in a connected graph $G$ is an edge metric generator of $G$ if every two distinct edges of $G$ are distinguished by some vertex of $S$. The smallest cardinality of an edge metric generator of $G$ is called the edge metric dimension and is denoted by $dim_e(G)$. The concept of the edge metric dimension is new. We study its mathematical properties. We make a comparison between the edge metric dimension and the standard metric dimension of graphs while presenting some realization results concerning the two. We prove that computing the edge metric dimension of connected graphs is NP-hard and give some approximation results. Moreover, we present bounds and closed formulae for the edge metric dimension of several classes of graphs.
The mixed metric dimension is a graph invariant similar to the edge metric dimension that deals with distinguishing the elements (vertices and edges) of a graph. A vertex $w \in V(G)$ distinguishes two elements of a graph $x,y \in E(G)\cup V(G)$ if $d_G(w,x) \ne d_G(w,y)$. A set $S$ of vertices in a connected graph $G$ is a mixed metric generator of $G$ if every two elements $x,y \in E(G) \cup V(G)$ of $G$, where $x \neq y$, are distinguished by some vertex of $S$. The smallest cardinality of a mixed metric generator of $G$ is called the mixed metric dimension and is denoted by $dim_m(G)$. In this dissertation, we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds. We also give results on the mixed metric dimension of certain families of graphs and present an upper bound with respect to the girth of a graph. Finally, we prove that the problem of determining the mixed metric dimension of a graph is NP-hard in the general case.
Ključne besede: Hausdorff distance, distance between graphs, graph algorithms, trees, graph similarity, edge metric dimension, edge metric generator, mixed metric dimension, metric dimension
Objavljeno v DKUM: 03.08.2020; Ogledov: 1587; Prenosov: 124
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5. Closed embeddings into Lipscomb's universal spaceUroš Milutinović, 2007, objavljeni povzetek znanstvenega prispevka na konferenci Ključne besede: matematika, topologija, dimenzija pokrivanja, posplošena krivulja Sierpińskega, univerzalni prostor, Lipscombov univerzalni prostor, vložitev, razširitev, poln metrični prostor, zaprta vložitev, mathematics, topology, covering dimension, embedding, closed embedding, generalized Sierpiński curve, universal space, Lipscomb universal space, complete metric space, extension
Objavljeno v DKUM: 10.07.2015; Ogledov: 1401; Prenosov: 28
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6. Nonexistence of face-to face four-dimensional tilings in the Lee metricSimon Špacapan, 2007, izvirni znanstveni članek Opis: A family of ▫$n$▫-dimensional Lee spheres ▫$mathcal{L}$▫ is a tiling of ▫${mathbb{R}}^n$▫ if ▫$cupmathcal{L} = {mathbb{R}}^n$▫ and for every ▫$L_u, L_v in mathcal{L}$▫, the intersection ▫$L_u cap L_v$▫ is contained in the boundary of ▫$L_u$▫. If neighboring Lee spheres meet along entire ▫$(n-1)$▫-dimensional faces, then ▫$mathcal{L}$▫ is called a face-to-face tiling. We prove nonexistence of a face-to-face tiling of ▫${mathbb{R}}^4$▫, with Lee spheres of different radii.
Ključne besede: delitev, Leejeva metrika, popolne kode, tiling, Lee metric, perfect codes
Objavljeno v DKUM: 10.07.2015; Ogledov: 1003; Prenosov: 84
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7. On the canonical metric representation, average distance, and partial Hamming graphsSandi Klavžar, 2006, izvirni znanstveni članek Opis: Average distance of a graph is expressed in terms of its canonical metric representation. The equality can be modified to an inequality in such a way that it characterizes isometric subgraphs of Hamming graphs. This approach simplifies recognition of these graphs and computation of their average distance.
Povprečna razdalja grafa je izražena s pomočjo kanonične metrične reprezentacije. Enakost lahko preoblikujemo v neenakost tako, da karakterizira izometrične podgrafe Hammingovih grafov. Ta pristop poenostavlja prepoznavanje teh grafov ter izračun povprečne razdalje.
Ključne besede: matematika, teorija grafov, kanonična metrična reprezentacija, Hammingovi grafi, delni Hammingovi grafi, Wienerjev indeks, algoritem prepoznavanja, mathematics, graph theory, cononical metric representation, Hamming graphs, partial Hamming graphs, Wiener index, recognition algorithm
Objavljeno v DKUM: 10.07.2015; Ogledov: 1357; Prenosov: 135
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8. Closed embeddings into Lipscomb's universal spaceIvan Ivanšić, Uroš Milutinović, 2006 Opis: Naj bo ▫${mathcal{J}}(tau)$▫ Lipscombov enodimenzionalni prostor in ▫$L_n(tau) = {x in {mathcal{J}}(tau)^{n+1}|$▫ vsaj ena koordinata od ▫{sl x}▫ je iracionalna ▫$} subseteq {mathcal{J}}(tau)^{n+1}$▫ Lipscombov ▫$n$▫-dimenzionalni univerzalni prostor s težo ▫$tau ge aleph_0$▫. V tem članku dokazujemo, da če je ▫$X$▫ poln metrizabilni prostor in velja ▫$dim X le n$▫, ▫$wX le tau$▫, tedaj obstaja zaprta vložitev prostora ▫$X$▫ v ▫$L_n(tau)$▫. Še več, vsako zvezno funkcijo ▫$f: X to {mathcal{J}}(tau)^{n+1}$▫ lahko poljubno natančno aproksimiramo z zaprto vložitvijo ▫$psi: X to L_n(tau)$▫. Razen tega sta dokazani relativna verzija in punktirana verzija. V primeru separabilnosti je dokazan analogni rezultat, v katerem je klasična trikotna krivulja Sierpińskega (ki je homeomorfna ▫${mathcal{J}}(3)$▫) nadomestila ▫${mathcal{J}(aleph_0)}$▫.
Ključne besede: matematika, topologija, dimenzija pokrivanja, posplošena krivulja Sierpińskega, univerzalni prostor, Lipscombov univerzalni prostor, vložitev, razširitev, poln metrični prostor, zaprta vložitev, mathematics, topology, covering dimension, embedding, closed embedding, generalized Sierpiński curve, universal space, Lipscomb universal space, complete metric space, extension
Objavljeno v DKUM: 10.07.2015; Ogledov: 1098; Prenosov: 90
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9. Dress Andreas, Huber Katharina T., Lesser Alice, Moulton Vincent: Hereditarily optimal realizations of consistent metrics. Ann. Comb. 10 (2006), no. 1, 63-76Boštjan Brešar, 2006, recenzija, prikaz knjige, kritika Ključne besede: tight span, finite metric space, optimal realization, weakly compatible
Objavljeno v DKUM: 31.05.2012; Ogledov: 1405; Prenosov: 38
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