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Computing invariants of knotted graphs given by sequences of points in 3-dimensional space (2017)
Presentation / Conference Contribution
Kurlin, V., Carr, H., Garth, C., & Weinkauf, T. (2015, May). Computing invariants of knotted graphs given by sequences of points in 3-dimensional space. Presented at Topology-Based Methods in Visualization 2015., Annweiler, Germany

We design a fast algorithm for computing the fundamental group of the complement to any knotted polygonal graph in 3-space. A polygonal graph consists of straight segments and is given by sequences of vertices along edge-paths. This polygonal model i... Read More about Computing invariants of knotted graphs given by sequences of points in 3-dimensional space.

A fast persistence-based segmentation of noisy 2D clouds with provable guarantees (2015)
Journal Article
Kurlin, V. (2016). A fast persistence-based segmentation of noisy 2D clouds with provable guarantees. Pattern Recognition Letters, 83(Part 1), 3-12. https://doi.org/10.1016/j.patrec.2015.11.025

We design a new fast algorithm to automatically segment a 2D cloud of points into persistent regions. The only input is a dotted image without any extra parameters, say a scanned black-and-white map with almost closed curves or any image with detecte... Read More about A fast persistence-based segmentation of noisy 2D clouds with provable guarantees.

A one-dimensional Homologically Persistent Skeleton of an unstructured point cloud in any metric space (2015)
Journal Article
Kurlin, V. (2015). A one-dimensional Homologically Persistent Skeleton of an unstructured point cloud in any metric space. Computer Graphics Forum, 34(5), 253-262. https://doi.org/10.1111/cgf.12713

Real data are often given as a noisy unstructured point cloud, which is hard to visualize. The important problem is to represent topological structures hidden in a cloud by using skeletons with cycles. All past skeletonization methods require extra p... Read More about A one-dimensional Homologically Persistent Skeleton of an unstructured point cloud in any metric space.

Relaxed disk packing (2015)
Presentation / Conference Contribution
Edelsbrunner, E., Iglesias-Ham, M., Kurlin, V., Kouhestani, B., & Rappaport, D. (2015, August). Relaxed disk packing. Presented at 27th Canadian Conference on Computational Geometry., Queen's University, Kingston, Ontario, Canada

Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexago... Read More about Relaxed disk packing.

A Linear Time Algorithm for Visualizing Knotted Structures in 3 Pages (2015)
Presentation / Conference Contribution
Kurlin, V. (2015, March). A Linear Time Algorithm for Visualizing Knotted Structures in 3 Pages. Presented at IVAPP 2015 : 6th International Conference on Information Visualization Theory and Applications., Berlin, Germany

We introduce simple codes and fast visualization tools for knotted structures in molecules and neural networks. Knots, links and more general knotted graphs are studied up to an ambient isotopy in Euclidean 3-space. A knotted graph can be represented... Read More about A Linear Time Algorithm for Visualizing Knotted Structures in 3 Pages.

A fast and robust algorithm to count topologically persistent holes in noisy clouds (2014)
Presentation / Conference Contribution
Kurlin, V. (2023, June). A fast and robust algorithm to count topologically persistent holes in noisy clouds. Presented at CVPR : Computer Vision and Pattern Recognition, Columbus, Ohio, USA

Preprocessing a 2D image often produces a noisy cloud of interest points. We study the problem of counting holes in noisy clouds in the plane. The holes in a given cloud are quantified by the topological persistence of their boundary contours when th... Read More about A fast and robust algorithm to count topologically persistent holes in noisy clouds.

Computing a configuration skeleton for motion planning of two round robots on a metric graph (2014)
Presentation / Conference Contribution
Kurlin, V., & Safi-Samghabadi, M. (2014, October). Computing a configuration skeleton for motion planning of two round robots on a metric graph. Presented at 2014 Second RSI/ISM International Conference on Robotics and Mechatronics (ICRoM), Tehran

A connected metric graph G with n vertices and without loops and multiple edges is given as an n × n-matrix whose entry aij is the length of a single edge between vertices i ≠ j. A robot in the metric graph G is the metric ball with a center x ϵ G an... Read More about Computing a configuration skeleton for motion planning of two round robots on a metric graph.

Auto-completion of Contours in Sketches, Maps and Sparse 2D Images Based on Topological Persistence (2014)
Presentation / Conference Contribution
Kurlin, V., Winkler, F., Negru, V., Ida, T., Jebelean, T., Petcu, D., Watt, S., & Zaharie, D. (2014, December). Auto-completion of Contours in Sketches, Maps and Sparse 2D Images Based on Topological Persistence. Presented at Computational Topology in Image Context (workshop of SYNASC 2014: Symbolic and Numeric Algorithms for Scientific Computing, http://synasc.ro/2014), Timisoara, Romania

We design a new fast algorithm to automatically complete closed contours in a finite point cloud on the plane. The only input can be a scanned map with almost closed curves, a hand-drawn artistic sketch or any sparse dotted image in 2D without any ex... Read More about Auto-completion of Contours in Sketches, Maps and Sparse 2D Images Based on Topological Persistence.

How Many Wireless Sensors are Needed to Guarantee Connectivity of a One-Dimensional Network with Random Inter-Node Spacing? (2013)
Journal Article
Kurlin, V., & Mihaylova, L. (2013). How Many Wireless Sensors are Needed to Guarantee Connectivity of a One-Dimensional Network with Random Inter-Node Spacing?. Journal of applied probability and statistics, 8(2), 27-50

An important problem in wireless sensor networks is to nd an optimal number of randomly deployed sensors to guarantee connectivity of the resulting network with a given probability. The authors describe a general method to compute the probabilities o... Read More about How Many Wireless Sensors are Needed to Guarantee Connectivity of a One-Dimensional Network with Random Inter-Node Spacing?.

Reconstructing persistent graph structures from noisy images (2013)
Journal Article
Chernov, A., & Kurlin, V. (2013). Reconstructing persistent graph structures from noisy images. Imagen-a, 3(5), 19-22

Let a point cloud be a noisy dotted image of a graph on the plane. We present a new fast algorithm for reconstructing the original graph from the given point cloud. Degrees of vertices in the graph are found by methods of persistent topology. Necessa... Read More about Reconstructing persistent graph structures from noisy images.

Computing braid groups of graphs with applications to robot motion planning (2012)
Journal Article
Kurlin, V. (2012). Computing braid groups of graphs with applications to robot motion planning. Homology, Homotopy and Applications, 14(1), 159-180. https://doi.org/10.4310/hha.2012.v14.n1.a8

An algorithm is designed to write down presentations of graph braid groups. Generators are represented in terms of actual motions of robots moving without collisions on a given connected graph. A key ingredient is a new motion planning algorithm whos... Read More about Computing braid groups of graphs with applications to robot motion planning.

Recognizing trace graphs of closed braids (2010)
Journal Article
Fiedler, T., & Kurlin, V. (2010). Recognizing trace graphs of closed braids. Osaka Journal of Mathematics, 47(4), 885-909

To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetrahedral moves. For closed braids with a fixed numbe... Read More about Recognizing trace graphs of closed braids.

A 1-parameter approach to links in a solid torus (2010)
Journal Article
Fiedler, T., & Kurlin, V. (2010). A 1-parameter approach to links in a solid torus. Journal of the Mathematical Society of Japan, 62(1), 167-211. https://doi.org/10.2969/jmsj/06210167

To an oriented link in a solid torus we associate a trace graph in a thickened torus in such a way that links are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. The key ingredient is a study of cod... Read More about A 1-parameter approach to links in a solid torus.

Fiber quadrisecants in knot isotopies (2008)
Journal Article
Fiedler, T., & Kurlin, V. (2008). Fiber quadrisecants in knot isotopies. Journal of Knot Theory and Its Ramifications, 17(11), 1415-1428. https://doi.org/10.1142/s0218216508006695

Fix a straight line L in Euclidean 3-space and consider the fibration of the complement of L by half-planes. A generic knot K in the complement of L has neither fiber quadrisecants nor fiber extreme secants such that K touches the corresponding half-... Read More about Fiber quadrisecants in knot isotopies.

All 2–dimensional links in 4–space live inside a universal 3–dimensional polyhedron (2008)
Journal Article
Kearton, C., & Kurlin, V. (2008). All 2–dimensional links in 4–space live inside a universal 3–dimensional polyhedron. Algebraic & geometric topology, 8(3), 1223-1247. https://doi.org/10.2140/agt.2008.8.1223

The hexabasic book is the cone of the 1–dimensional skeleton of the union of two tetrahedra glued along a common face. The universal 3–dimensional polyhedron UP is the product of a segment and the hexabasic book. We show that any closed 2–dimensional... Read More about All 2–dimensional links in 4–space live inside a universal 3–dimensional polyhedron.

The Baker-Campbell-Hausdorff Formula in the Free Metabelian Lie Algebra (2007)
Journal Article
Kurlin, V. (2007). The Baker-Campbell-Hausdorff Formula in the Free Metabelian Lie Algebra. Journal of Lie theory, 17(3), 525-538

The classical Baker-Campbell-Hausdorff formula gives a recursive way to compute the Hausdorff series $H=\ln(e^Xe^Y)$ for non-commuting $X,Y$. Formally $H$ lives in the graded completion of the free Lie algebra $L$ generated by $X,Y$. We present a clo... Read More about The Baker-Campbell-Hausdorff Formula in the Free Metabelian Lie Algebra.

Peripherally Specified Homomorphs of Link Groups (2007)
Journal Article
Kurlin, V., & Lines, D. (2007). Peripherally Specified Homomorphs of Link Groups. Journal of Knot Theory and Its Ramifications, 16(6), 719-740. https://doi.org/10.1142/s0218216507005440

Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic... Read More about Peripherally Specified Homomorphs of Link Groups.

Three-page encoding and complexity theory for spatial graphs (2007)
Journal Article
Kurlin, V. (2007). Three-page encoding and complexity theory for spatial graphs. Journal of Knot Theory and Its Ramifications, 16(01), 59-102. https://doi.org/10.1142/s021821650700521x

A finitely presented semigroup RSGn is constructed for n ≥ 2. The centre of RSGn encodes uniquely up to rigid ambient isotopy in 3-space all nonoriented spatial graphs with vertices of degree ≤ n. This encoding is obtained by using three-page embeddi... Read More about Three-page encoding and complexity theory for spatial graphs.

A Homologically Persistent Skeleton is a fast and robust descriptor of interest points in 2D images (1999)
Book Chapter
Kurlin, V. (1999). A Homologically Persistent Skeleton is a fast and robust descriptor of interest points in 2D images. In G. Azzopardi, & N. Petkov (Eds.), Computer analysis of images and patterns : 16th International Conference, CAIP 2015, Valletta, Malta, September 2-4, 2015. Proceedings. Part I (606-617). Springer Verlag. https://doi.org/10.1007/978-3-319-23192-1_51

2D images often contain irregular salient features and interest points with non-integer coordinates. Our skeletonization problem for such a noisy sparse cloud is to summarize the topology of a given 2D cloud across all scales in the form of a graph,... Read More about A Homologically Persistent Skeleton is a fast and robust descriptor of interest points in 2D images.