Similarity and symmetry measures for convex sets based on Minkowski addition

This paper discusses similarity and symmetry measures for convex shapes. Their definition is based on Minkowski addition and the Brunn-Minkowski inequality. All measures considered are invariant under translations; furthermore, they may also be invariant under rotations, multiplications, reflections, or the class of all affine transformations. The examples discussed in this paper allow efficient algorithms if one restricts oneselves to convex polygons. Although it deals exclusively with the 2-dimensional case, many of the theoretical results carry over almost directly to higher-dimensional spaces. Some results obtained in this paper are illustrated by experimental data

Cell shape analysis of random tessellations based on Minkowski tensors

To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic models. In the context of image analysis of synthetic and biological materials, this question is central to the problem of inferring information about formation processes from spatial measurements of resulting random structures. We address this question by a theory-based simulation study of shape indices derived from Minkowski tensors for a variety of tessellation models. We focus on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, and Gibbs hard-core and random sequential absorption processes as well as for Laguerre tessellations of polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data are complemented by mechanically stable crystalline sphere and disordered ellipsoid packings and area-minimising foam models. We find that shape indices of individual cells are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of the shape indices between many of these tessellation models. Given a realization of a tessellation, these shape indices can narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by density-resolved volume-shape correlations.Comment: Chapter of the forthcoming book "Tensor Valuations and their Applications in Stochastic Geometry and Imaging" in Lecture Notes in Mathematics edited by Markus Kiderlen and Eva B. Vedel Jense

Data depth and floating body

Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth

Segmentation And Spatial Depth Ridge Detection Of Unorganized Point Cloud Data

Visual 3D data are of interest to a number of fields: medical professionals, game designers, graphic designers, and (in the interest of this paper) ichthyologists interested in the taxonomy of fish. Since the release of the Kinect for the Microsoft XBox, game designers have been interested in using the 3D data returned by the device to understand human movement and translate that movement into an interface with which to interact with game systems. In the medical field, researchers must use computer vision tools to navigate through the data found in CT scans and MRI scans. These tools must segment images into the parts that are relevant to researchers and account for noise related to the scanning process all while ignoring other types of noise such as foreign elements in the body that might indicate signs of illness. 3D point cloud data represents some unique challenges. Consider an object scanned with a laser scanner. The scanner returns the surface points of the object, but nothing more. Using the tool Qhull, a researcher can quickly compute the convex hull of an object (which is an interesting challenge in itself), but the convex hull (obviously) leaves out any description of an object\u27s concave features. Several algorithms have been proposed to illustrate an object\u27s complete features based on unorganized 3D point cloud data as accurately as possible, most notably Boissonnat\u27s tetrahedral culling algorithm and The Power Crust algorithm. We introduce a new approach to the area partitioning problem that takes into consideration these algorithms\u27 strengths and weaknesses. In this paper we propose a methodology for approximating a shape\u27s solid geometry using the unorganized 3D point cloud data of that shape primarily by utilizing localized principal component analysis information. Our model accounts for three comissues that arise in the scanning of 3D objects: noise in surface points, poorly sampled surface area, and narrow corners. We explore each of these areas of concern and outline our approach to each. Our technique uses a growing algorithm that labels points as it progresses and uses those labels with a simple priority queue. We found that our approach works especially well for approximating surfaces under the condition where a local surface is poorly sampled (i.e a significant hole is present in the point cloud). We then turn to study the medial axis of a shape for the purposes of `unfolding\u27 that structure. Our approach uses a ridge formulation based on the spatial depth statistic to create the medial axis. We conclude the paper with visual results of our technique

Convex Geometry and its Applications

The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of other algorithms in computer science. High-dimensional geometry is an extremely active area of research: the participation of a considerable number of talented young mathematicians at this meeting is testament to the fact that the field is flourishing

The structure of random ellipsoid packings

Disordered packings of ellipsoidal particles are an important model for disordered granular matter and can shed light on geometric features and structural transitions in granular matter. In this thesis, the structure of experimental ellipsoid packings is analyzed in terms of contact numbers and measures from mathematical morphometry to characterize of Voronoi cell shapes. Jammed ellipsoid packings are prepared by vertical shaking of loose configurations in a cylindrical container. For approximately 50 realizations with packing fractions between 0.54 and 0.70 and aspect ratios from 0.40 to 0.97, tomographic images are recorded, from which positions and orientations of the ellipsoids are reconstructed. Contact numbers as well as discrete approximations of generalized Voronoi diagrams are extracted. The shape of the Voronoi cells is quantified by isotropy indexes b,r,s,n based on Minkowski tensors. In terms of the Voronoi cells, the behavior for jammed ellipsoids differs from that of spheres; the Voronoi Cells of spheres become isotropic with increasing packing fraction, whereas the shape of the Voronoi Cells of ellipsoids with high aspect ratio remains approximately constant. Contact numbers are discussed in the context of the jamming paradigm and it is found that the frictional ellipsoid packings are hyperstatic, i.e. have more contacts than are required for mechanical stability. It is observed, that the contact numbers of jammed ellipsoid packings predominantly depend on the packing fraction, but also a weaker dependence on the aspect ratio and the friction coefficient is found. The achieved packing fractions in the experiments lie within upper and lower limits expected from DEM simulations of jammed ellipsoid packings. Finally, the results are compared to Monte Carlo and Molecular Dynamics data of unjammed equilibrium ellipsoid ensembles. The Voronoi cell shapes of equilibrium ensembles of ellipsoidal particles with a low aspect ratio become more anisotropic by increasing the packing fraction, while the cell shape of particles with large aspect ratios does the opposite. The experimental jammed packings are always more anisotropic than the corresponding densest equilibrium configuration