Pointwise convergence of the Lloyd algorithm in higher dimension

We establish the pointwise convergence of the iterative Lloyd algorithm, also known as $k$-means algorithm, when the quadratic quantization error of the starting grid (with size $N\ge 2$) is lower than the minimal quantization error with respect to the input distribution is lower at level $N-1$. Such a protocol is known as the splitting method and allows for convergence even when the input distribution has an unbounded support. We also show under very light assumption that the resulting limiting grid still has full size $N$. These results are obtained without continuity assumption on the input distribution. A variant of the procedure taking advantage of the asymptotic of the optimal quantizer radius is proposed which always guarantees the boundedness of the iterated grids

Well-Centered Triangulation

Meshes composed of well-centered simplices have nice orthogonal dual meshes (the dual Voronoi diagram). This is useful for certain numerical algorithms that prefer such primal-dual mesh pairs. We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations. We present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. The cost function is a direct result of a new characterization of well-centeredness in arbitrary dimensions that we present. Ours is the first optimization-based heuristic for well-centeredness, and the first one that applies in both two and three dimensions. We show the results of applying our algorithm to small and large two-dimensional meshes, some with a complex boundary, and obtain a well-centered tetrahedralization of the cube. We also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method.Comment: Content has been added to experimental results section. Significant edits in introduction and in summary of current and previous results. Minor edits elsewher

Stable topological summaries for analyzing the organization of cells in a packed tissue

We use topological data analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify the lifetime of homology classes (persistent homology) along different filtrations (increasing nested sequences of simplicial complexes) that are built from the regions representing the cells in the tissue. We use a complete and well-grounded set of numerical variables over those persistence barcodes, also known as topological summaries. A novel combination of normalization methods for both the set of input segmented images and the produced barcodes allows for the proven stability results for those variables with respect to small changes in the input, as well as invariance to image scale. Our study provides new insights to this problem, such as a possible novel indicator for the development of the drosophila wing disc tissue or the importance of centroids’ distribution to differentiate some tissues from their CVT-path counterpart (a mathematical model of epithelia based on Voronoi diagrams). We also show how the use of topological summaries may improve the classification accuracy of epithelial images using a Random Forest algorithm.Ministerio de Ciencia e Innovación PID2019-107339GB-I0

Visualization of implicit geographic information through map-like graphics

Viele von Web 2.0-Benutzern gesammelte Daten sind ortsbezogen, wobei der Ort meist nur eine Information unter vielen ist und dem Ortsbezug häufig keine besondere Bedeutung zugesprochen wird. Jedoch werden auch mehr und mehr geographische Informationen von Laien gesammelt und im Netz veröffentlicht. Trotz der technischen Möglichkeiten, die im Web-2.0 geboten werden, ist es für Nutzer ohne entsprechendes Expertenwissen meist nicht möglich, gut lesbare und ansprechende Karten zu erzeugen. Dieses Problem besteht, da der Nutzer den darzustellenden Inhalt bestimmt, ohne dass überprüft wird, ob die daraus resultierende Karte den kartographischen Ansprüchen genügt. Des Weiteren hat der Nutzer keinerlei Möglichkeiten, die Standardkartenbilder zu bearbeiten, um beispielsweise für seine Thematik irrelevante Objekte auszublenden oder durch Verdrängung relevante Objekte freizustellen. Besonders bei der Darstellung von POIs treten Überlappungen zwischen Signaturen häufig auf. Trotz des Bedarfs existiert aus verschiedenen Gründen keine etablierte Methode zur Verdrängung von Punktdaten. Daher liegt der Schwerpunkt dieser Arbeit auf der Entwicklung von Verfahren zur Verdrängung von Punktsignaturen. Als Hilfsstrukturen werden dazu Voronoi-Diagramme benutzt und als nutzergenerierte Information werden Sentiments visualisiert. Für den Entwurf der Visualisierungen werden relevante kartographische Bedingungen berücksichtigt und durch zugehörige Qualitätsmaße bewertet. Für die Darstellung von Sentiments werden neben der Verwendung von Punktsignaturen zwei weitere Darstellungsarten erstellt: Anpassung gegebener Signaturen und die Darstellung von Sentiments als Kontinua. Es werden Verdrängungsverfahren für Punktsignaturen entworfen und implementiert. Zur Bestimmung der Verschiebungsrichtung werden zwei verschieden Heuristiken vorgeschlagen und untersucht. Des Weiteren wird eine Möglichkeit zur Steigerung der Effizienz durch Aufteilung der Punktmenge aufgezeigt. Die Bewertung der entworfenen Punktsignaturen erfolgt durch eine Umfrage. Anschließend wird das realisierte Verfahren für gleich große Kreissignaturen in drei Aspekten evaluiert: Grad der Reduzierung der Iterationsschritte durch Zerlegung der Punktmenge, erreichte Verminderung der Überlappungsfläche und Veränderung der relativen Lage der Punkte.A lot of user generated information accumulated in the web is related to a place, with the location usually being just one piece of information among many, which gets no special attention. However, more and more geographic information is collected by laymen and published on the web. Despite the technical possibilities that are offered in Web 2.0, it is usually not possible for users without expert knowledge to produce legible and appealing maps. This problem exists because the user determines the content to be displayed without checking whether the resulting map meets the cartographic requirements. Furthermore, the user has no possibilities to edit the standard map images, for example, hide for his subject irrelevant objects or reducing overlap of relevant objects by displacement. Especially when displaying POIs, overlaps between point symbols often occur. Despite the need, there is no established method for displacing point data for various reasons. Therefore, the focus of this work is the development of methods for the displacement of point signatures. Voronoi diagrams are used as auxiliary structures and sentiment is visualized as user-generated information. For the design of the visualizations relevant cartographic requirements are taken into account and evaluated by quality measures. For the depiction of sentiments, in addition to the use of point symbols, two further types of visualizations are created: adaptation of given map symbols and the representation of sentiments as continua. Displacement techniques for point symbols are designed and implemented. To determine the direction of displacement two different heuristics are proposed and examined. Furthermore, a way to increase the efficiency by dividing the point set is shown. The evaluation of the designed point symbols is done by a survey. Subsequently, the realized method for circular symbols with equal size is evaluated in three aspects: degree of reduction of the iteration steps by decomposition of the point set, achieved reduction of the overlap area and change of the relative position of the points