Karol Mikula

Karol Mikula, Daniel Sevcovic, Martin Balazovjech

A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial differential equation for updating the position vector of evolving family of plane curves. A curve can be evolved in the normal direction by a combination of fourth order terms related to the intrinsic Laplacian of the curvature, second order terms related to the curvature, first order terms related to anisotropy and by a given external velocity field. The evolution is numerically stabilized by an asymptotically uniform tangential redistribution of grid points yielding the first order intrinsic advective terms in the governing system of equations. By using a semi-implicit in time discretization it can be numerically approximated by a solution to linear penta-diagonal systems of equations (in presence of the fourth order terms) or tri-diagonal systems (in the case of the second order terms). Various numerical experiments of plane curve evolutions, including, in particular, nonlinear, anisotropic and regularized backward curvature flows, surface diffusion and Willmore flows, are presented and discussed.

Michal Benes, Karol Mikula, Tomas Oberhuber et al.

The main goal of this paper is to present results of comparison study for the level set and direct Lagrangian methods for computing evolution of the Willmore flow of embedded planar curves. To perform such a study we construct new numerical approximation schemes for both Lagrangian as well as level set methods based on semi-implicit in time and finite/complementary volume in space discretizations. The Lagrangian scheme is stabilized in tangential direction by the asymptotically uniform grid point redistribution. Both methods are experimentally second order accurate. Moreover, we show precise coincidence of both approaches in case of various elastic curve evolutions provided that solving the linear systems in semi-implicit level set method is done in a precise way, redistancing is performed occasionally and the influence of boundary conditions on the level set function is eliminated.

Seol Ah Park, Tamara Sipka, Zuzana Kriva et al.

The automated segmentation and tracking of macrophages during their migration are challenging tasks due to their dynamically changing shapes and motions. This paper proposes a new algorithm to achieve automatic cell tracking in time-lapse microscopy macrophage data. First, we design a segmentation method employing space-time filtering, local Otsu's thresholding, and the SUBSURF (subjective surface segmentation) method. Next, the partial trajectories for cells overlapping in the temporal direction are extracted in the segmented images. Finally, the extracted trajectories are linked by considering their direction of movement. The segmented images and the obtained trajectories from the proposed method are compared with those of the semi-automatic segmentation and manual tracking. The proposed tracking achieved 97.4% of accuracy for macrophage data under challenging situations, feeble fluorescent intensity, irregular shapes, and motion of macrophages. We expect that the automatically extracted trajectories of macrophages can provide pieces of evidence of how macrophages migrate depending on their polarization modes in the situation, such as during wound healing.

Peter Frolkovič, Karol Mikula

A new class of semi-implicit numerical schemes for linear advection equation on Cartesian grids is derived that is inspired by so-called $κ$-schemes used with fully explicit discretizations for this type of problems. Opposite to fully explicit $κ$-scheme the semi-implicit variant is unconditionally stable in one-dimensional case and it preserves second order accuracy for dimension by dimension extension in higher dimensional cases. We discuss von Neumann stability conditions numerically for all numerical schemes. Using so-called Corner Transport Upwind extension of two-dimensional semi-implicit scheme with a special choice of $κ$ parameters, a second order accurate method is obtained for which numerical unconditional stability can be shown for variable velocity and the third order accuracy can be proved for constant velocity. Several numerical experiments illustrate the properties of semi-implicit schemes for chosen examples.

Karol Mikula, Daniel Sevcovic

We discuss the role of tangential stabilization in a curvature driven flow of planar curves. The governing system of nonlinear parabolic equations includes a nontrivial tangential velocity functional yielding a uniform redistribution of grid points along the evolving family of curves preventing numerically computed curves from forming various instabilities.

Karol Mikula, Mariana Sarkociová Remešíková

In this paper, we present an algorithm for evaluating lexical similarity between a given language and several reference language clusters. As an input, we have a list of concepts and the corresponding translations in all considered languages. Moreover, each reference language is assigned to one of $c$ language clusters. For each of the concepts, the algorithm computes the distance between each pair of translations. Based on these distances, it constructs a weighted directed graph, where every vertex represents a language. After, it solves a graph diffusion equation with a Dirichlet boundary condition, where the unknown is a map from the vertex set to $\mathbb{R}^c$. The resulting coordinates are values from the interval $[0,1]$ and they can be interpreted as probabilities of belonging to each of the clusters or as a lexical similarity distribution with respect to the reference clusters. The distances between translations are calculated using phonetic transcriptions and a modification of the Damerau-Levenshtein distance. The algorithm can be useful in analyzing relationships between languages spoken in multilingual territories with a lot of mutual influences. We demonstrate this by presenting a case study regarding various European languages.

Balazs Kosa, Karol Mikula

In this paper, we introduce novel methods for computing middle surfaces between various 3D data sets such as point clouds and/or discrete surfaces. Traditionally the middle surface is obtained by detecting singularities in computed distance function such as ridges, triple junctions, etc. It requires to compute second order differential characteristics and also some kinds of heuristics must be applied. Opposite to that, we determine the middle surface just from computing the distance function itself which is a fast and simple approach. We present and compare the results of the fast sweeping method, the vector distance transform algorithm, the fast marching method, and the Dijkstra-Pythagoras method in finding the middle surface between 3D data sets.

Karol Mikula, Michal Kollar, Aneta A. Ozvat et al.

Natural numerical networks are introduced as a new classification algorithm based on the numerical solution of nonlinear partial differential equations of forward-backward diffusion type on complete graphs. The proposed natural numerical network is applied to open important environmental and nature conservation task, the automated identification of protected habitats by using satellite images. In the natural numerical network, the forward diffusion causes the movement of points in a feature space toward each other. The opposite effect, keeping the points away from each other, is caused by backward diffusion. This yields the desired classification. The natural numerical network contains a few parameters that are optimized in the learning phase of the method. After learning parameters and optimizing the topology of the network graph, classification necessary for habitat identification is performed. A relevancy map for each habitat is introduced as a tool for validating the classification and finding new Natura 2000 habitat appearances.

Karol Mikula, Daniel Sevcovic

In this review paper we present a stable Lagrangian numerical method for computing plane curves evolution driven by the fourth order geometric equation. The numerical scheme and computational examples are presented.