Getting topological information for a 80-adjacency doxel-based 4D volume through a polytopal cell complex

Given an 80-adjacency doxel-based digital four-dimensional hypervolume V, we construct here an associated oriented 4–dimensional polytopal cell complex K(V), having the same integer homological information (that related to n-dimensional holes that object has) than V. This is the first step toward the construction of an algebraic-topological representation (AT-model) for V, which suitably codifies it mainly in terms of its homological information. This AT-model is especially suitable for global and local topological analysis of digital 4D images

On Volumetric Shape Reconstruction from Implicit Forms

International audienceIn this paper we report on the evaluation of volumetric shape reconstruction methods that consider as input implicit forms in 3D. Many visual applications build implicit representations of shapes that are converted into explicit shape representations using geometric tools such as the Marching Cubes algorithm. This is the case with image based reconstructions that produce point clouds from which implicit functions are computed, with for instance a Poisson reconstruction approach. While the Marching Cubes method is a versatile solution with proven efficiency, alternative solutions exist with different and complementary properties that are of interest for shape modeling. In this paper, we propose a novel strategy that builds on Centroidal Voronoi Tessellations (CVTs). These tessellations provide volumetric and surface representations with strong regularities in addition to provably more accurate approximations of the implicit forms considered. In order to compare the existing strategies, we present an extensive evaluation that analyzes various properties of the main strategies for implicit to explicit volumetric conversions: Marching cubes, Delaunay refinement and CVTs, including accuracy and shape quality of the resulting shape mesh

A continuous analog for 4-dimensional objects

In this paper, we follow up on the studies developed by Kovalevsky (Comput Vis Graph Image Process 46:141–161, 1989) and Kenmochi et al. (Comput Vis Image Underst 71:281–293, 1998), which defined a continuous analog for a 4-dimensional digital object. Here, we construct a cell complex that has the same topological information as the original 4-dimensional digital object

Advanced homology computation of digital volumes via cell complexes

Given a 3D binary voxel-based digital object V, an algorithm for computing homological information for V via a polyhedral cell complex is designed. By homological information we understand not only Betti numbers, representative cycles of homology classes and homological classification of cycles but also the computation of homology numbers related additional algebraic structures defined on homology (coproduct in homology, product in cohomology, (co)homology operations,...). The algorithm is mainly based on the following facts: a) a local 3D-polyhedrization of any 2×2×2 configuration of mutually 26-adjacent black voxels providing a coherent cell complex at global level; b) a description of the homology of a digital volume as an algebraic-gradient vector field on the cell complex (see Discrete Morse Theory [5], AT-model method [7,5]). Saving this vector field, we go further obtaining homological information at no extra time processing cost

A Lagrangian meshless finite element method applied to fluid–structure interaction problems

A method is presented for the solution of the incompressible fluid flow equations using a Lagrangian formulation. The interpolation functions are those used in the meshless finite element method and the time integration is introduced in a semi-implicit way by a fractional step method. Classical stabilization terms used in the momentum equations are unnecessary due to the lack of convective terms in the Lagrangian formulation. Furthermore, the Lagrangian formulation simplifies the connections with fixed or moving solid structures, thus providing a very easy way to solve fluid–structure interaction problems

Properties of Gauss digitized sets and digital surface integration

International audienceThis paper presents new topological and geometrical properties of Gauss digitizations of Euclidean shapes, most of them holding in arbitrary dimension $d$. We focus on $r$-regular shapes sampled by Gauss digitization at gridstep $h$. The digitized boundary is shown to be close to the Euclidean boundary in the Hausdorff sense, the minimum distance $\frac{\sqrt{d}}{2}h$ being achieved by the projection map $\xi$ induced by the Euclidean distance. Although it is known that Gauss digitized boundaries may not be manifold when $d \ge 3$, we show that non-manifoldness may only occur in places where the normal vector is almost aligned with some digitization axis, and the limit angle decreases with $h$. We then have a closer look at the projection of the digitized boundary onto the continuous boundary by $\xi$. We show that the size of its non-injective part tends to zero with $h$. This leads us to study the classical digital surface integration scheme, which allocates a measure to each surface element that is proportional to the cosine of the angle between an estimated normal vector and the trivial surface element normal vector. We show that digital integration is convergent whenever the normal estimator is multigrid convergent, and we explicit the convergence speed. Since convergent estimators are now available in the litterature, digital integration provides a convergent measure for digitized objects

Computer simulations of realistic three-dimensional microstructures

A novel and efficient methodology is developed for computer simulations of realistic two-dimensional (2D) and three-dimensional (3D) microstructures. The simulations incorporate realistic 2D and 3D complex morphologies/shapes, spatial patterns, anisotropy, volume fractions, and size distributions of the microstructural features statistically similar to those in the corresponding real microstructures. The methodology permits simulations of sufficiently large 2D as well as 3D microstructural windows that incorporate short-range (on the order of particle/feature size) as well as long-range (hundred times the particle/feature size) microstructural heterogeneities and spatial patterns at high resolution. The utility of the technique has been successfully demonstrated through its application to the 2D microstructures of the constituent particles in wrought Al-alloys, the 3D microstructure of discontinuously reinforced Al-alloy (DRA) composites containing SiC particles that have complex 3D shapes/morphologies and spatial clustering, and 3D microstructure of boron modified Ti-6Al-4V composites containing fine TiB whiskers and coarse primary TiB particles. The simulation parameters are correlated with the materials processing parameters (such as composition, particle size ratio, extrusion ratio, extrusion temperature, etc.), which enables the simulations of rational virtual 3D microstructures for the parametric studies on microstructure-properties relationships. The simulated microstructures have been implemented in the 3D finite-elements (FE)-based framework for simulations of micro-mechanical response and stress-strain curves. Finally, a new unbiased and assumption free dual-scale virtual cycloids probe for estimating surface area of 3D objects constructed by 2D serial section images is also presented.Ph.D.Committee Chair: Arun M. Gokhale; Committee Member: David Frost; Committee Member: Meilin Liu; Committee Member: Burton R Patterson; Committee Member: Min Zho