Hierarchical interpolative factorization for elliptic operators: integral equations

This paper introduces the hierarchical interpolative factorization for integral equations (HIF-IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretized operator and its inverse. HIF-IE is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features: (1) a matrix factorization framework for sparsifying structured dense matrices and (2) a recursive dimensional reduction strategy to decrease the cost. Thus, higher-dimensional problems are effectively mapped to one dimension, and we conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity. Numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method, direct solver, and preconditioner. HIF-IE is compatible with geometric adaptivity and can handle both boundary and volume problems. MATLAB codes are freely available.Comment: 39 pages, 14 figures, 13 tables; to appear, Comm. Pure Appl. Mat

3D Geometric Analysis of Tubular Objects based on Surface Normal Accumulation

This paper proposes a simple and efficient method for the reconstruction and extraction of geometric parameters from 3D tubular objects. Our method constructs an image that accumulates surface normal information, then peaks within this image are located by tracking. Finally, the positions of these are optimized to lie precisely on the tubular shape centerline. This method is very versatile, and is able to process various input data types like full or partial mesh acquired from 3D laser scans, 3D height map or discrete volumetric images. The proposed algorithm is simple to implement, contains few parameters and can be computed in linear time with respect to the number of surface faces. Since the extracted tube centerline is accurate, we are able to decompose the tube into rectilinear parts and torus-like parts. This is done with a new linear time 3D torus detection algorithm, which follows the same principle of a previous work on 2D arc circle recognition. Detailed experiments show the versatility, accuracy and robustness of our new method.Comment: in 18th International Conference on Image Analysis and Processing, Sep 2015, Genova, Italy. 201

Combined 3D thinning and greedy algorithm to approximate realistic particles with corrected mechanical properties

The shape of irregular particles has significant influence on micro- and macro-scopic behavior of granular systems. This paper presents a combined 3D thinning and greedy set-covering algorithm to approximate realistic particles with a clump of overlapping spheres for discrete element method (DEM) simulations. First, the particle medial surface (or surface skeleton), from which all candidate (maximal inscribed) spheres can be generated, is computed by the topological 3D thinning. Then, the clump generation procedure is converted into a greedy set-covering (SCP) problem. To correct the mass distribution due to highly overlapped spheres inside the clump, linear programming (LP) is used to adjust the density of each component sphere, such that the aggregate properties mass, center of mass and inertia tensor are identical or close enough to the prototypical particle. In order to find the optimal approximation accuracy (volume coverage: ratio of clump's volume to the original particle's volume), particle flow of 3 different shapes in a rotating drum are conducted. It was observed that the dynamic angle of repose starts to converge for all particle shapes at 85% volume coverage (spheres per clump < 30), which implies the possible optimal resolution to capture the mechanical behavior of the system.Comment: 34 pages, 13 figure

Gap Processing for Adaptive Maximal Poisson-Disk Sampling

In this paper, we study the generation of maximal Poisson-disk sets with varying radii. First, we present a geometric analysis of gaps in such disk sets. This analysis is the basis for maximal and adaptive sampling in Euclidean space and on manifolds. Second, we propose efficient algorithms and data structures to detect gaps and update gaps when disks are inserted, deleted, moved, or have their radius changed. We build on the concepts of the regular triangulation and the power diagram. Third, we will show how our analysis can make a contribution to the state-of-the-art in surface remeshing.Comment: 16 pages. ACM Transactions on Graphics, 201

ToScA North America (6 – 8 June 2017, The University of Texas, Austin, TX) Program

ToScA North America will address key areas of science, including Multi-modal Imaging, Geosciences, Forensics, Increasing Contrast, Educational Outreach, Data, Materials Science and Medical and Biological Science.University of Texas High-Resolution X-ray CT Facility (UTCT); Jackson School of Geosciences, The University of Texas at Austin; Natural History Museum (London); Royal Microscopical Society (Oxford, UK)Geological Science