111 research outputs found
Efficient view point selection for silhouettes of convex polyhedra
AbstractSilhouettes of polyhedra are an important primitive in application areas such as machine vision and computer graphics. In this paper, we study how to select view points of convex polyhedra such that the silhouette satisfies certain properties. Specifically, we give algorithms to find all projections of a convex polyhedron such that a given set of edges, faces and/or vertices appear on the silhouette.We present an algorithm to solve this problem in O(k2) time for k edges. For orthogonal projections, we give an improved algorithm that is fully adaptive in the number l of connected components formed by the edges, and has a time complexity of O(klogk+kl). We then generalize this algorithm to edges and/or faces appearing on the silhouette
Reconstruction of Orthogonal Polyhedra
In this thesis I study reconstruction of orthogonal polyhedral surfaces
and orthogonal polyhedra from partial information about their
boundaries. There are three main questions for which I provide novel
results. The first question is "Given the dual graph, facial angles and
edge lengths of an orthogonal polyhedral surface or polyhedron, is it
possible to reconstruct the dihedral angles?" The second question is
"Given the dual graph, dihedral angles and edge lengths of an
orthogonal polyhedral surface or polyhedron, is it possible to
reconstruct the facial angles?" The third question is "Given the
vertex coordinates of an orthogonal polyhedral surface or polyhedron, is
it possible to reconstruct the edges and faces, possibly after
rotating?"
For the first two questions, I show that the answer is "yes" for
genus-0 orthogonal polyhedra and polyhedral surfaces under some
restrictions, and provide linear time algorithms. For the third
question, I provide results and algorithms for orthogonally convex
polyhedra. Many related problems are studied as well
Statistical methods for polyhedral shape classification with incomplete data - application to cryo-electron tomographic images
A certain type of bacterial inclusion, known as a bacterial microcompartment, was recently identified and imaged through cryo-electron tomography. A reconstructed 3D object from single-axis limited angle tilt-series cryo-electron tomography contains missing regions and this problem is known as the missing wedge problem. Due to missing regions on the reconstructed images, analyzing their 3D structures is a challenging problem. The existing methods overcome this problem by aligning and averaging several similar shaped objects. These schemes work well if the objects are symmetric and several objects with almost similar shapes and sizes are available. Since the bacterial inclusions studied here are not symmetric, are deformed, and show a wide range of shapes and sizes, the existing approaches are not appropriate. This research develops new statistical methods for analyzing geometric properties, such as volume, symmetry, aspect ratio, polyhedral structures etc., of these bacterial inclusions in presence of missing data. These methods work with deformed and non-symmetric varied shaped objects and do not necessitate multiple objects for handling the missing wedge problem. The developed methods and contributions include: (a) an improved method for manual image segmentation, (b) a new approach to 'complete' the segmented and reconstructed incomplete 3D images, (c) a polyhedral structural distance model to predict the polyhedral shapes of these microstructures, (d) a new shape descriptor for polyhedral shapes, named as polyhedron profile statistic, and (e) the Bayes classifier, linear discriminant analysis and support vector machine based classifiers for supervised incomplete polyhedral shape classification. Finally, the predicted 3D shapes for these bacterial microstructures belong to the Johnson solids family, and these shapes along with their other geometric properties are important for better understanding of their chemical and biological characteristics
Cellular resolutions of noncommutative toric algebras from superpotentials
This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer and Sturmfels (1998)in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three-dimensional toric algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension. For abelian skew group algebras and algebraically consistent dimer model algebras, we introduce a cell complex Δ in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution ofA. We illustrate the general construction of Δ for an example in dimension four arising from a tilting bundle on a smooth toric Fano threefold to highlight the importance of the incidence function on Δ
A friendly introduction to Fourier analysis on polytopes
This book is an introduction to the nascent field of Fourier analysis on
polytopes, and cones. There is a rapidly growing number of applications of
these methods, so it is appropriate to invite students, as well as
professionals, to the field. We assume a familiarity with Linear Algebra, and
some Calculus. Of the many applications, we have chosen to focus on: (a)
formulations for the Fourier transform of a polytope, (b) Minkowski and
Siegel's theorems in the geometry of numbers, (c) tilings and multi-tilings of
Euclidean space by translations of a polytope, (d) Computing discrete volumes
of polytopes, which are combinatorial approximations to the continuous volume,
(e) Optimizing sphere packings and their densities, and (f) use iterations of
the divergence theorem to give new formulations for the Fourier transform of a
polytope, with an application. Throughout, we give many examples and exercises,
so that this book is also appropriate for a course, or for self-study.Comment: 204 pages, 46 figure
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by ErdËťos
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
Collection of abstracts of the 24th European Workshop on Computational Geometry
International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop
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