Three-dimensional alpha shapes

Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the ``shape'' of the set. For that purpose, this paper introduces the formal notion of the family of $\alpha$-shapes of a finite point set in \Real^3. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter \alpha \in \Real controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size $n$ in time $O(n^2)$, worst case. A robust implementation of the algorithm is discussed and several applications in the area of scientific computing are mentioned.Comment: 32 page

Slime mould computes planar shapes

Computing a polygon defining a set of planar points is a classical problem of modern computational geometry. In laboratory experiments we demonstrate that a concave hull, a connected alpha-shape without holes, of a finite planar set is approximated by slime mould Physarum polycephalum. We represent planar points with sources of long-distance attractants and short-distance repellents and inoculate a piece of plasmodium outside the data set. The plasmodium moves towards the data and envelops it by pronounced protoplasmic tubes

Application of the morphological alpha shape method to the extraction of topographical features from engineering surfaces

In contrast to the mean-line based filters, morphological filters are function oriented and more suitable for the functional prediction of component performance. This paper presents a novel morphological method based on the alpha shape for the extraction of topographical features from engineering surfaces. Compared to the traditional implementation of morphological filters, the alpha shape method is more efficient in performance for large structuring element. The resulting envelope follows the form of the surface all over such that the distortions caused the end effects are avoided. A series of measured surfaces from the automotive cylinder liner and the bioengineering femoral heads are analyzed using the morphological alpha shape method. The topographical features are successfully extracted, enabling further analysis to the components

Distributed boundary tracking using alpha and Delaunay-Cech shapes

For a given point set $S$ in a plane, we develop a distributed algorithm to compute the $\alpha-$shape of $S$. $\alpha-$shapes are well known geometric objects which generalize the idea of a convex hull, and provide a good definition for the shape of $S$. We assume that the distances between pairs of points which are closer than a certain distance $r>0$ are provided, and we show constructively that this information is sufficient to compute the alpha shapes for a range of parameters, where the range depends on $r$. Such distributed algorithms are very useful in domains such as sensor networks, where each point represents a sensing node, the location of which is not necessarily known. We also introduce a new geometric object called the Delaunay-\v{C}ech shape, which is geometrically more appropriate than an $\alpha-$shape for some cases, and show that it is topologically equivalent to $\alpha-$shapes

A Fully Coupled Formulation For Incompressible Fluid-Elastic Structure-Interactions

We present a general formulation for analysis of fluid flows with structural interactions using the particle finite element method (PFEM). The fluids are fully coupled to the structures that can undergo highly non-linear response due to large deformations. The key feature of the PFEM is the use of an updated Lagrangian description to model the motion of nodes(particles) in both the fluid and the structure domains. A mesh connects the nodes defining the discretized domains where the governing equations, expressed in an integral form are solved as in the standard FEM. A fractional step scheme for the transient coupled fluid-structure solution is described. Examples of application of the PFEM method to solve a number of fluidstructure interaction problems including free-fluid-surfaces, breaking waves and fluid particle separation may be easily solved with this formulation are presented.Fil: Marti, Julio Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Idelsohn, Sergio Rodolfo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Limache, Alejandro Cesar. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; Argentin

Computational Topology Techniques for Characterizing Time-Series Data

Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and holes - could be useful for real-world data, TDA lets us compare different systems, and even do membership testing or change-point detection. However, TDA is computationally expensive and involves a number of free parameters. This complexity can be obviated by coarse-graining, using a construct called the witness complex. The parametric dependence gives rise to the concept of persistent homology: how shape changes with scale. Its results allow us to distinguish time-series data from different systems - e.g., the same note played on different musical instruments.Comment: 12 pages, 6 Figures, 1 Table, The Sixteenth International Symposium on Intelligent Data Analysis (IDA 2017

Reconstruction of freeform surfaces for metrology

The application of freeform surfaces has increased since their complex shapes closely express a product's functional specifications and their machining is obtained with higher accuracy. In particular, optical surfaces exhibit enhanced performance especially when they take aspheric forms or more complex forms with multi-undulations. This study is mainly focused on the reconstruction of complex shapes such as freeform optical surfaces, and on the characterization of their form. The computer graphics community has proposed various algorithms for constructing a mesh based on the cloud of sample points. The mesh is a piecewise linear approximation of the surface and an interpolation of the point set. The mesh can further be processed for fitting parametric surfaces (Polyworks® or Geomagic®). The metrology community investigates direct fitting approaches. If the surface mathematical model is given, fitting is a straight forward task. Nonetheless, if the surface model is unknown, fitting is only possible through the association of polynomial Spline parametric surfaces. In this paper, a comparative study carried out on methods proposed by the computer graphics community will be presented to elucidate the advantages of these approaches. We stress the importance of the pre-processing phase as well as the significance of initial conditions. We further emphasize the importance of the meshing phase by stating that a proper mesh has two major advantages. First, it organizes the initially unstructured point set and it provides an insight of orientation, neighbourhood and curvature, and infers information on both its geometry and topology. Second, it conveys a better segmentation of the space, leading to a correct patching and association of parametric surfaces.EMR

Repairing triangle meshes built from scanned point cloud

The Reverse Engineering process consists of a succession of operations that aim at creating a digital representation of a physical model. The reconstructed geometric model is often a triangle mesh built from a point cloud acquired with a scanner. Depending on both the object complexity and the scanning process, some areas of the object outer surface may never be accessible, thus inducing some deficiencies in the point cloud and, as a consequence, some holes in the resulting mesh. This is simply not acceptable in an integrated design process where the geometric models are often shared between the various applications (e.g. design, simulation, manufacturing). In this paper, we propose a complete toolbox to fill in these undesirable holes. The hole contour is first cleaned to remove badly-shaped triangles that are due to the scanner noise. A topological grid is then inserted and deformed to satisfy blending conditions with the surrounding mesh. In our approach, the shape of the inserted mesh results from the minimization of a quadratic function based on a linear mechanical model that is used to approximate the curvature variation between the inner and surrounding meshes. Additional geometric constraints can also be specified to further shape the inserted mesh. The proposed approach is illustrated with some examples coming from our prototype software