1,783 research outputs found
Axiomatic Digital Topology
The paper presents a new set of axioms of digital topology, which are easily
understandable for application developers. They define a class of locally
finite (LF) topological spaces. An important property of LF spaces satisfying
the axioms is that the neighborhood relation is antisymmetric and transitive.
Therefore any connected and non-trivial LF space is isomorphic to an abstract
cell complex. The paper demonstrates that in an n-dimensional digital space
only those of the (a, b)-adjacencies commonly used in computer imagery have
analogs among the LF spaces, in which a and b are different and one of the
adjacencies is the "maximal" one, corresponding to 3n\"i1 neighbors. Even these
(a, b)-adjacencies have important limitations and drawbacks. The most important
one is that they are applicable only to binary images. The way of easily using
LF spaces in computer imagery on standard orthogonal grids containing only
pixels or voxels and no cells of lower dimensions is suggested
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
Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images
We present an efficient algorithm to compute Euler characteristic curves of
gray scale images of arbitrary dimension. In various applications the Euler
characteristic curve is used as a descriptor of an image.
Our algorithm is the first streaming algorithm for Euler characteristic
curves. The usage of streaming removes the necessity to store the entire image
in RAM. Experiments show that our implementation handles terabyte scale images
on commodity hardware. Due to lock-free parallelism, it scales well with the
number of processor cores. Our software---CHUNKYEuler---is available as open
source on Bitbucket.
Additionally, we put the concept of the Euler characteristic curve in the
wider context of computational topology. In particular, we explain the
connection with persistence diagrams
Polyhedral Surface Approximation of Non-Convex Voxel Sets and Improvements to the Convex Hull Computing Method
In this paper we introduce an algorithm for the creation of polyhedral approximations for objects represented as strongly connected sets of voxels in three-dimensional binary images. The algorithm generates the convex hull of a given object and modifies the hull afterwards by recursive repetitions of generating convex hulls of subsets of the given voxel set or subsets of the background voxels. The result of this method is a polyhedron which separates object voxels from background voxels. The objects processed by this algorithm and also the background voxel components inside the convex hull of the objects are restricted to have genus 0. The second aim of this paper is to present some improvements to our convex hull algorithm to reduce computation time
An exact general remeshing scheme applied to physically conservative voxelization
We present an exact general remeshing scheme to compute analytic integrals of
polynomial functions over the intersections between convex polyhedral cells of
old and new meshes. In physics applications this allows one to ensure global
mass, momentum, and energy conservation while applying higher-order polynomial
interpolation. We elaborate on applications of our algorithm arising in the
analysis of cosmological N-body data, computer graphics, and continuum
mechanics problems.
We focus on the particular case of remeshing tetrahedral cells onto a
Cartesian grid such that the volume integral of the polynomial density function
given on the input mesh is guaranteed to equal the corresponding integral over
the output mesh. We refer to this as "physically conservative voxelization".
At the core of our method is an algorithm for intersecting two convex
polyhedra by successively clipping one against the faces of the other. This
algorithm is an implementation of the ideas presented abstractly by Sugihara
(1994), who suggests using the planar graph representations of convex polyhedra
to ensure topological consistency of the output. This makes our implementation
robust to geometric degeneracy in the input. We employ a simplicial
decomposition to calculate moment integrals up to quadratic order over the
resulting intersection domain.
We also address practical issues arising in a software implementation,
including numerical stability in geometric calculations, management of
cancellation errors, and extension to two dimensions. In a comparison to recent
work, we show substantial performance gains. We provide a C implementation
intended to be a fast, accurate, and robust tool for geometric calculations on
polyhedral mesh elements.Comment: Code implementation available at https://github.com/devonmpowell/r3
Morphological Image Analysis of Quantum Motion in Billiards
Morphological image analysis is applied to the time evolution of the
probability distribution of a quantum particle moving in two and
three-dimensional billiards. It is shown that the time-averaged Euler
characteristic of the probability density provides a well defined quantity to
distinguish between classically integrable and non-integrable billiards. In
three dimensions the time-averaged mean breadth of the probability density may
also be used for this purpose.Comment: Major revision. Changes include a more detailed discussion of the
theory and results for 3 dimensions. Now: 10 pages, 9 figures (some are
colored), 3 table
Fully Deformable 3D Digital Partition Model with Topological Control
International audienceWe propose a purely discrete deformable partition model for segmenting 3D images. Its main ability is to maintain the topology of the partition during the minimization process. To do so, our main contribution is a new definition of multi-label simple points (ML simple point) that is easily computable. An ML simple point can be relabeled without modifying the overall topology of the partition. The definition is based on intervoxel properties, and uses the notion of collapse on cubical complexes. This work is an extension of a former restricted definition [DupasAl09] that prohibits the move of intersections of boundary surfaces. A deformation process is carried out with a greedy energy minimization algorithm. A discrete area estimator is used to approach at best standard regularizers classically used in continuous energy minimizing methods. We illustrate the potential of our approach with the segmentation of 3D medical images with known expected topology
Combinatorial Boundary Tracking of a 3D Lattice Point Set
Boundary tracking and surface generation are ones of main topological topics for three-dimensional digital image analysis. However, there is no adequate theory to make relations between these different topological properties in a completely discrete way. In this paper, we present a new boundary tracking algorithm which gives not only a set of border points but also the surface structures by using the concepts of combinatorial/algebraic topologies. We also show that our boundary becomes a triangulation of border points (in the sense of general topology), that is, we clarify relations between border points and their surface structures
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