A Krasnosel’skii-type theorem for certain orthogonal polytopes starshaped via k-staircase paths

Let 𝒞 be a finite family of distinct axis-parallel boxes in ℝd whose intersection graph is a tree, and let S = ⋃{C : C in 𝒞}. If every two points of S see a common point of S via k-staircase paths, then S is starshaped via k-staircase paths. Moreover, the k-staircase kernel of S will be convex via k-staircases.

A Krasnosel’skii-type theorem for an enlarged class of orthogonal polytopes

Let 𝓒 be a finite family of distinct boxes in ℝ

Concise and Accessible Representations for Multidimensional Datasets: Introducing a Framework Based on thenD-EVM and Kohonen Networks

A new framework intended for representing and segmenting multidimensional datasets resulting in low spatial complexity requirements and with appropriate access to their contained information is described. Two steps are going to be taken in account. The first step is to specify (n-1)D hypervoxelizations,n≥2, as Orthogonal Polytopes whosenth dimension corresponds to color intensity. Then, thenD representation is concisely expressed via the Extreme Vertices Model in then-Dimensional Space (nD-EVM). Some examples are presented, which, under our methodology, have storing requirements minor than those demanded by their original hypervoxelizations. In the second step, 1-Dimensional Kohonen Networks (1D-KNs) are applied in order to segment datasets taking in account their geometrical and topological properties providing a non-supervised way to compact even more the proposedn-Dimensional representations. The application of our framework shares compression ratios, for our set of study cases, in the range 5.6496 to 32.4311. Summarizing, the contribution combines the power of thenD-EVM and 1D-KNs by producing very concise datasets’ representations. We argue that the new representations also provide appropriate segmentations by introducing some error functions such that our 1D-KNs classifications are compared against classifications based only in color intensities. Along the work, main properties and algorithms behind thenD-EVM are introduced for the purpose of interrogating the final representations in such a way that it efficiently obtains useful geometrical and topological information.

Computing the Discrete Compactness of Orthogonal Pseudo-Polytopes via Their D-EVM Representation

This work is devoted to present a methodology for the computation of Discrete Compactness in -dimensional orthogonal pseudo-polytopes. The proposed procedures take in account compactness' definitions originally presented for the 2D and 3D cases and extend them directly for considering the D case. There are introduced efficient algorithms for computing discrete compactness which are based on an orthogonal polytopes representation scheme known as the Extreme Vertices Model in the -Dimensional Space (D-EVM). It will be shown the potential of the application of Discrete Compactness in higher-dimensional contexts by applying it, through EVM-based algorithms, in the classification of video sequences, associated to the monitoring of a volcano's activity, which are expressed as 4D orthogonal polytopes in the space-color-time geometry.