Showing papers on "Minkowski addition published in 2010"
TL;DR: This study presents a blind and geometric technique which pursues the linear decomposition of the observations in bounded component signals with good performance in high SNR scenarios, even for a small number of samples.
Abstract: This study presents a blind and geometric technique which pursues the linear decomposition of the observations in bounded component signals. The bounded component analysis of the observations relies on the hypotheses of compactness and Cartesian decomposition of the convex support of the vector of component signals, and in the invertibility of the mixture. Assumptions, which in absence of noise, are able to guarantee the identifiability of the mixture and separability of the components, up to permutation, scaling, and phase ambiguities. Under these conditions, the convex perimeter of the normalized linear combination of the observations is shown to be a global contrast function whose minima correspond with the extraction of bounded components of the observations. Practical extraction and separation algorithms based on the minimization of this criterion are given. The experimental results with communications signals serve to illustrate the good performance of the proposed method in high SNR scenarios, even for a small number of samples.
70 citations
TL;DR: The Minkowski sum of edges corresponding to the column vectors of a matrix A with real entries is the same as the image of a unit cube under the linear transformation defined by A with respect to the standard bases as discussed by the authors.
69 citations
TL;DR: A novel framework for studying partially observable Markov decision processes (POMDPs) with finite state, action, observation sets, and discounted rewards based on future-reward vectors associated with future policies is presented, which is more parsimonious than the traditional framework based on belief vectors.
Abstract: This paper presents a novel framework for studying partially observable Markov decision processes (POMDPs) with finite state, action, observation sets, and discounted rewards. The new framework is solely based on future-reward vectors associated with future policies, which is more parsimonious than the traditional framework based on belief vectors. It reveals the connection between the POMDP problem and two computational geometry problems, i.e., finding the vertices of a convex hull and finding the Minkowski sum of convex polytopes, which can help solve the POMDP problem more efficiently. The new framework can clarify some existing algorithms over both finite and infinite horizons and shed new light on them. It also facilitates the comparison of POMDPs with respect to their degree of observability, as a useful structural result.
60 citations
TL;DR: The novelty of this work mainly lies in the introduction of the Transformed Minkowski Sum, which can be used to determine whether a moving bounding rectangle intersects a moving circular query region and enables us to use the traditional tree traversal algorithms to perform range and kNN searches.
56 citations
TL;DR: A novel technique for the efficient boundary evaluation of sweep operations applied to objects in polygonal boundary representation, which is exact for Minkowski sums and approximates swept volumes polygonally and significantly enhances the performance of the algorithm.
Abstract: We present a novel technique for the efficient boundary evaluation of sweep operations applied to objects in polygonal boundary representation. These sweep operations include Minkowski addition, offsetting, and sweeping along a discrete rigid motion trajectory. Many previous methods focus on the construction of a polygonal superset (containing self-intersections and spurious internal geometry) of the boundary of the volumes which are swept. Only few are able to determine a clean representation of the actual boundary, most of them in a discrete volumetric setting. We unify such superset constructions into a succinct common formulation and present a technique for the robust extraction of a polygonal mesh representing the outer boundary, i.e. it makes no general position assumptions and always yields a manifold, watertight mesh. It is exact for Minkowski sums and approximates swept volumes polygonally. By using plane-based geometry in conjunction with hierarchical arrangement computations we avoid the necessity of arbitrary precision arithmetics and extensive special case handling. By restricting operations to regions containing pieces of the boundary, we significantly enhance the performance of the algorithm.
55 citations
01 Sep 2010
TL;DR: It is shown that all Minkowski product and quotient operations may be represented implicitly as sublevel sets of the same real-valued convolution function.
Abstract: Group morphology is an extension of mathematical morphology with classical Minkowski sum and difference operations generalized respectively to Minkowski product and quotient operations over arbitrary groups. We show that group morphology is a proper setting for unifying, formulating and solving a number of important problems, including translational and rotational configuration space problems, mechanism workspace computation, and symmetry detection. The proposed computational approach is based on group convolution algebras, which extend classical convolutions and the Fourier transform to non-commutative groups. In particular, we show that all Minkowski product and quotient operations may be represented implicitly as sublevel sets of the same real-valued convolution function.
34 citations
01 Sep 2010
TL;DR: This work presents a new approach for computing the voxelized Minkowski sum of two polyhedral objects using programmable Graphics Processing Units (GPUs), which is at least one order of magnitude faster than existing boundary representation (B-rep) based algorithms for computing Minkingowski sums of objects with curved surfaces at similar accuracy.
Abstract: We present a new approach for computing the voxelized Minkowski sum of two polyhedral objects using programmable Graphics Processing Units (GPUs). We first cull out surface primitives that will not contribute to the final boundary of the Minkowski sum. The remaining surface primitives are then rendered to depth textures along six orthogonal directions to generate an initial solid voxelization of the Minkowski sum. Finally we employ fast flood fill to find all the outside voxels. We generate both solid and surface voxelizations of Minkowski sums without holes and support high volumetric resolution of 10243 with low video memory cost. The whole algorithm runs on the GPU and is at least one order of magnitude faster than existing boundary representation (B-rep) based algorithms for computing Minkowski sums of objects with curved surfaces at similar accuracy. It avoids complex 3D Boolean operations and is easy to implement. The voxelized Minkowski sums can be used in a variety of applications including motion planning and penetration depth computation.
29 citations
TL;DR: In this article, the authors give a characterization of the support map from classical convexity, and show that it is the unique additive transformation from the class of closed convex sets in R n which include 0 to a class of positive 1-homogeneous functions on R n.
27 citations
TL;DR: In this paper, the authors investigated the real geometric effects in real Minkowski space that are induced by and associated with complex world-lines in complex Minkowska space.
Abstract: In connection with the study of shear-free null geodesics in Minkowski space, we investigate the real geometric effects in real Minkowski space that are induced by and associated with complex world-lines in complex Minkowski space. It was already known, in a formal manner, that complex analytic curves in complex Minkowski space induce shear-free null geodesic congruences. Here we look at the direct geometric connections of the complex line and the real structures. Among other items, we show, in particular, how a complex world-line projects into the real Minkowski space in the form of a real shear-free null geodesic congruence.
16 citations
TL;DR: The main result of as discussed by the authors is that a closed convex set is Motzkin decomposable if and only if the set of extreme points of its intersection with the linear subspace orthogonal to its lineality is bounded.
15 citations
01 Sep 2010
TL;DR: In this article, it was shown that the spectrum of the complex Laplacian on a product of Hermitian manifolds is the Minkowski sum of the spectra of the L 1 on the factors.
Abstract: We show that the spectrum of the complex Laplacian □ on a product of Hermitian manifolds is the Minkowski sum of the spectra of the complex Laplacians on the factors. We use this to show that the range of the Cauchy-Riemann operator ∂ is closed on a product manifold, provided it is closed on each factor manifold.
01 Sep 2010
TL;DR: A novel method for fast retrieval of exact Minkowski sums of pairs of convex poly topes in R3, where one of the polytopes frequently rotates, based on pre-computing a so-called criticality map, which records the changes in the underlying graph-structure of the Minkowski sum, while one ofThe polytope rotates.
Abstract: We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in R3, where one of the polytopes frequently rotates. The algorithm is based on pre-computing a so-called criticality map, which records the changes in the underlying graph-structure of the Minkowski sum, while one of the polytopes rotates. We give tight combinatorial bounds on the complexity of the criticality map when the rotating polytope rotates about one, two, or three axes. The criticality map can be rather large already for rotations about one axis, even for summand polytopes with a moderate number of vertices each. We therefore focus on the restricted case of rotations about a single, though arbitrary, axis.Our work targets applications that require exact collision-detection such as motion planning with narrow corridors and assembly maintenance where high accuracy is required. Our implementation handles all degeneracies and produces exact results. It efficiently handles the algebra of exact rotations about an arbitrary axis in R3, and it well balances between preprocessing time and space on the one hand, and query time on the other.We use Cgal arrangements and in particular the support for spherical Gaussian-maps to efficiently compute the exact Minkowski sum of two polytopes. We conducted several experiments to verify the correctness of the algorithm and its implementation, and to compare its efficiency with an alternative (static) exact method. The results are reported.
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01 Dec 2010
TL;DR: An exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.
TL;DR: Two approximate Minkowski sum algorithms for planar regions bounded by line and circle segments that form a convolution curve, construct its arrangement, and use winding numbers to identify sum cells are presented.
Abstract: We present two approximate Minkowski sum algorithms for planar regions bounded by line and circle segments. Both algorithms form a convolution curve, construct its arrangement, and use winding numbers to identify sum cells. The first uses the kinetic convolution and the second uses our monotonic convolution. The asymptotic running times of the exact algorithms are increased by km log m with m the number of segments in the convolution and k the number of segment triples that are in cyclic vertical order due to approximate segment intersection. The approximate Minkowski sum is close to the exact sum of perturbation regions that are close to the input regions. We validate both algorithms on part packing tasks with industrial part shapes. The accuracy is near the floating point accuracy even after multiple iterated sums. The programs are 2% slower than direct floating point implementations of the exact algorithms. The monotonic algorithm is 42% faster than the kinetic algorithm.
01 Jan 2010
TL;DR: In this paper, the authors describe properties of the Resultant polytope of a given set of polynomial equations towards an outputsensitive algorithm for enumerating its vertices.
Abstract: We describe properties of the Resultant polytope of a given set of polynomial equations towards an outputsensitive algorithm for enumerating its vertices. In principle, one has to consider all regular ne mixed subdivisions of the Minkowski sum of the Newton polytopes of the given equations. By the Cayley trick, this is equivalent to computing all regular triangulations of another point set in higher dimension. However, the number of all regular triangulations is generally much larger than that of the vertices of the Resultant polytope, as illustrated by our experiments [3]. Thus, we study output-sensitive methods by dening classes of subdivisions, called congurations, which yield the same resultant vertex. Moreover, we oer algorithmic versions of certain results by Sturmfels [11], regarding the edges of the Resultant polytope. Lastly, we settle some easy cases, and discuss harder examples.
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TL;DR: In this article, the authors define the flag polynomial in commuting variables related to the well-known flag vector and describe how to express the flag polynomial of the Minkowski sum of standard simplices in a direct and canonical way in terms of the {\em $k$-th master polytope} $P(k)$ where $k\in
ats$ facilitates many direct computations.
Abstract: For a polytope we define the {\em flag polynomial}, a polynomial in commuting variables related to the well-known flag vector and describe how to express the the flag polynomial of the Minkowski sum of $k$ standard simplices in a direct and canonical way in terms of the {\em $k$-th master polytope} $P(k)$ where $k\in
ats$. The flag polynomial facilitates many direct computations. To demonstrate this we provide two examples; we first derive a formula for the $f$-polynomial and the maximum number of $d$-dimensional faces of the Minkowski sum of two simplices. We then compute the maximum discrepancy between the number of $(0,d)$-chains of faces of a Minkowski sum of two simplices and the number of such chains of faces of a simple polytope of the same dimension and on the same number of vertices.
01 Dec 2010
TL;DR: A complete buffer generation method based on the Minkowski sum of point, line and area is presented and the Jiangsu province 1∶1,000,000 basic geographic data is employed to verify the effectivity and practicability of the aforementioned method.
Abstract: From current study on buffer generation method, we find the research on buffer generation method in special shapes is insufficient This paper is a research on the general buffer generation and the specific buffer generation method of point, line and area, and finally presents a complete buffer generation method based on the Minkowski sum Generally, computing the Minkowski sum of a geometry with a disc of radius equal to the buffer distance And some special buffer regions can be created by calculating the Minkowski sum of the geometry and the other shapes In the end, we employ the Jiangsu province 1∶1,000,000 basic geographic data to verify the effectivity and practicability of the aforementioned method
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TL;DR: The linear relation implies that a sum of r polytopes in dimension d, where summands have n vertices in total, has less than n-1 vertices, even when r≥d, and a bound on the maximum possible number of vertices of the Minkowski sum is deduced.
Abstract: It is known that in the Minkowski sum of $r$ polytopes in dimension $d$, with $r
TL;DR: In this article, the authors studied a special family of Minkowski sums, which is of polytopes relatively in general position, and showed that the maximum number of faces in the sum can be attained by this family.
Abstract: The objective of this paper is to study a special family of Minkowski sums, that is of polytopes relatively in general position. We show that the maximum number of faces in the sum can be attained by this family. We present a new linear equation that is satisfied by f-vectors of the sum and the summands. We study some of the implications of this equation.
21 Jun 2010
TL;DR: This work presents a new algorithm, based on the concept of contributing vertices, for the exact and efficient computation of the Minkowski difference of convex polyhedra, and compared it to a Nef polyhedron-based approach using Minkowsky addition, complement, transposition, and union operations.
Abstract: We present a new algorithm, based on the concept of contributing vertices, for the exact and efficient computation of the Minkowski difference of convex polyhedra. First, we extend the concept of contributing vertices for the Minkowski difference case. Then, we generate a Minkowski difference facets superset by exploiting the information provided by the computed contributing vertices. Finally, we compute the Minkowski difference polyhedron through the trimming of the generated superset. We compared our Contributing Vertices-based Minkowski Difference (CVMD) algorithm to a Nef polyhedra-based approach using Minkowski addition, complement, transposition, and union operations. The performance benchmark shows that our CVMD algorithm outperforms the indirect Nef polyhedra-based approach. All our implementations use exact number types, produce exact results, and are based on CGAL, the Computational Geometry Algorithms Library.
01 Jan 2010
TL;DR: In this paper, the Nakanishi integral representation of the Bethe-Salpeter amplitude and subsequent projection of the equation on the light-front plane were used to solve the two-body Bethe Salpeter equation in Minkowski space.
Abstract: We present a new method for solving the two-body Bethe-Salpeter equation in Minkowski space. It is based on the Nakanishi integral representation of the Bethe-Salpeter amplitude and on subsequent projection of the equation on the light-front plane. The method is valid for any kernel given by the irreducible Feynman graphs and for systems of spinless particles or fermions. The Bethe-Salpeter amplitudes in Minkowski space are obtained. The electromagnetic form factors are computed and compared to the Euclidean results.
DOI•
01 Jan 2010
TL;DR: This work proposes a method based on an iterative model that generates directly planarquadrilateral meshes, and expands the possibilities for creating forms, working in a 4D homogeneous coordinate system, and projecting these forms in the 3D modelling space.
Abstract: Iterative models are widely used today in CAD. They allow, with a limited number of parameters, to represent relatively complex forms through a subdivision algorithm. There is a wide variety of such models (Catmull-Clark, Doo-Sabin, L-Systems...). Most iterative models used in CAD can represent smooth shapes, such as polynomial or rational. The IFS model (Iterated Function System) is a mathematical model allowing to represent objects that can be smooth, in particular cases, or fractal, in more general cases. An IFS is defined by a set of geometric operators called "subdivision operators". These operators define an object iteratively, by successively applying this set of subdivision operators on a geometric base object. Classical subdivision schemes take as parameters a set of control points, that can be moved anywhere in space. These control points are the entry parameters of the subdivision algorithm, which uses predefined subdivision matrices to calculate the new points. In the IFS model, subdivision operators are not predefined, but customizable. These new parameters are graphically represented as movable points in space, like the control points. Each of these points, referred to as "subdivision point" is the image of a control point through a subdivision operator. The position of the control points allow to control the global aspect of the modelled object. Moving subdivision points affects the object at each level of subdivision, and therefore at smaller and smaller scales. The generated objects are not necessarily smooth, but are generally fractal. The constraints due to construction require some precise geometric properties of the modelled objects. As part of the wooden building, we want to achieve particular surface structures by assembly of wood panels. This requires modelling meshes composed of planar faces. We are particularly interested in modelling quadrangular mesh. We discard the solution of triangular meshes. This comes from constraints related to construction, and is more particularly due to the complexity of realizing assemblies around high valence vertices. The vertices in triangulated meshes have a valence of six, while in quadrangular meshes they have a valence of four. The development and implementation of solutions are relatively expensive in terms of the valence of the node. The valence of the nodes of a mesh has a direct influence on the geometry of faces ; the higher the valence of a vertex, the higher angles of faces around this vertex will be acute. Faces with acute angles are not desirable for a constructive application, because constructive elements have fragile parts and handling them during the implementation process is a delicate operation. We propose a method based on an iterative model that generates directly planarquadrilateral meshes. We start from a Minkowski sum of two curves. This process is rather limited, because it generates meshes only composed by parallelograms. We expand the possibilities for creating forms, working in a 4D homogeneous coordinate system, and projecting these forms in the 3D modelling space. Using projective geometry allows to extend the method by additional parameters such as the weight of points. This allows to reach a relatively general range of surface meshes.
TL;DR: It is proved that for “not far from convex” regions this measure of non-convexity for a simple polygonal region in the plane does not decrease under the Minkowski sum operation, and guarantees that the MINKowski sum has no “holes”.
Abstract: In this paper a measure of non-convexity for a simple polygonal region in the plane is introduced It is proved that for “not far from convex” regions this measure does not decrease under the Minkowski sum operation, and guarantees that the Minkowski sum has no “holes”
TL;DR: In this article, a set of periodic lattices in (1+1)-dimensional Minkowski space is described, where each lattice has an associated symmetry group consisting of inhomogeneous Lorentz transformations that map the lattice onto itself.
Abstract: We describe a set of periodic lattices in (1+1)-dimensional Minkowski space, where each lattice has an associated symmetry group consisting of inhomogeneous Lorentz transformations that map the lattice onto itself. Our results show how ideas of crystal structure in Euclidean space generalize to Minkowski space and provide an example that illustrates basic concepts of spacetime symmetry.
TL;DR: In this paper, ruled surfaces in a Minkowski 3-space satisfying some equation in terms of a position vector field and Laplacian operator with respect to non-degenerate third fundamental form are studied.
TL;DR: In this article, the structure of polytopes in which are Minkowski sums of de Gua simplex is discussed. And the shape of their faces is characterized by describing a notion of denegeneracy or general position.
Abstract: Within this article we discuss the structure of those polytopes in which are Minkowski sums of de Gua simplexes. A de Gua simplex is the convex hull of the origin and n positive multiples of the unit vectors (see [J.J. Gray, Algebra in geometry from Newton to Plucker (German), Math. Semesterberichte 36 (1989), pp. 175–204.]). We characterize these polytopes by describing the shape of their (outward) faces. Given some notion of ‘nondegeneracy’ or ‘general position’ for our polytopes, we present a recursive procedure that yields all maximal faces. Also, we derive a formula indicating the number of maximal faces, which depends on the dimension and the number of de Gua simplexes involved only.
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TL;DR: In this article, it was shown that any given spacelike polygonal curve in generic position in Minkowski 3-space bounds at least one maxface of disk-type.
Abstract: We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the existence of singularities, the appropriate framework is to consider maxfaces -- generalized maximal surfaces without branch points, introduced by M. Umehara and K. Yamada. We prove that any given spacelike polygonal curve in generic position in Minkowski 3-space bounds at least one maxface of disk-type. This is a new result, since the only known result for the Plateau problem in Minkowski space (due to N. Quien) deals with boundary curves of regularity $C^{3,\alpha}$.
TL;DR: Lower bounds for the Minkowski and Hausdorff dimensions of the algebraic sum E+K of two sets E,K⊂ℝ d were studied in this paper.
Abstract: We study lower bounds for the Minkowski and Hausdorff dimensions of the algebraic sum E+K of two sets E,K⊂ℝ d .
28 Jan 2010