Region In Euclidean Space Research Articles

CSWS—Suppress confusion subregion in indoor image fingerprint localization

AbstractImage data can provide rich content information, which has attracted a lot of attention in the field of indoor fingerprint positioning. However, indoor image information from different locations is characterized by high content repetition, and these repetitive regions can cause the problem of insufficient differentiation of adjacent image fingerprints or even fingerprint misclassification. To solve this problem, this article proposes a confusion subregion weighted suppression strategy in the image fingerprint database. First, duplicate regions (confusion subregions) are extracted from the fingerprint database using an appropriate salient region detection method. Then the similarity of these duplicate regions in Euclidean space is defined, and the degree of influence of these regions on the distinguishability of the fingerprint database is measured. Finally, the suppression of these duplicate regions is achieved in the original image fingerprint database by introducing weighted suppression coefficients. Experiments show that the algorithm proposed in this article can achieve significant results in the fingerprint localization task of real indoor scenes and effectively improve the localization accuracy.

Thermal corrections to Rényi entropies for free fermions

We calculate thermal corrections to R\'{e}nyi entropies for free massless fermions on a sphere. More specifically, we take a free fermion on $\mathbb{R}\times\mathbb{S}^{d-1}$ and calculate the leading thermal correction to the R\'{e}nyi entropies for a cap like region with opening angle $2\theta$. By expanding the density matrix in a Boltzmann sum, the problem of finding the R\'{e}nyi entropies can be mapped to the problem of calculating a two point function on an $n$ sheeted cover of the sphere. We follow previous work for conformal field theories to map the problem on the sphere to a conical region in Euclidean space. By using the method of images, we calculate the two point function and recover the R\'{e}nyi entropies.

A copy-move detection algorithm based on binary gradient contours

Copy-move is one of the most obvious ways of deliberate distortion of digital images in order to conceal the information contained in them. The process of duplicate embedding consists in copying an image fragment and pasting it within the same image. Prior to pasting, the fragment can be distorted using transformations such as contrast enhancement, noise adding, scaling, rotation, and combinations thereof. Existing approaches to copy-move forgery detection are based on calculating feature vectors for overlapping blocks of an image and then using these vectors to find the closest regions in Euclidean space. In this paper, we propose features based on binary gradient contours, which are resistant to contrast enhancement, additive noise and JPEG compression. We also present results of conducted experiments for demonstrating the proposed algorithm effectiveness for a range of distortion parameters. The research also involves comparing features based on binary gradient contours with features based on various forms of local binary patterns.

Spectral clustering for link prediction in social networks with positive and negative links

Online social networks (OSNs) recommend new friends to registered users based on local features of the graph (i.e. based on the number of common friends that two users share). Real OSNs (e.g. Facebook) do not exploit all network structure. Instead, they consider only pathways of maximum length 2 between a user and his candidate friends. This can limit the accuracy of prediction. On the other hand, there are global approaches, which detect the overall path structure in a network, being computationally prohibitive for huge-size social networks. In this paper, we provide friend recommendations, by performing multi-way spectral clustering, which uses information obtained from the top few eigenvectors and eigenvalues of the normalized Laplacian matrix and computes a multi-way partition of the data. As a result, it produces a less noisy matrix, which is smaller and more compact than the original one, focusing on main linking trends of the social network. Thus, we are able to provide fast and more accurate friend recommendations. Moreover, spectral clustering compared to traditional clustering algorithms, such as k-means and DBSCAN, which assume globular (convex) regions in Euclidean space, is more flexible, in capturing the non-connected components of a social graph and a wider range of cluster geometries. We perform an extensive experimental comparison of the proposed method against existing link prediction algorithms, the k-means and two-way nCut clustering algorithms, using synthetic and three real data sets (Hi5, Facebook, and Epinions). Our experimental results show that our SpectralLink algorithm outperforms the local approaches, the k-means and two-way nCut clustering algorithms in terms of effectiveness, whereas it is more efficient than the global approaches. We show that a significant accuracy improvement can be gained by using information about both positive and negative edges.

From biological to social networks: Link prediction based on multi-way spectral clustering

Link prediction in protein–protein interaction networks (PPINs) is an important task in biology, since the vast majority of biological functions involve such protein interactions. Link prediction is also important for online social networks (OSNs), which provide predictions about who is a friend of whom. Many link prediction methods for PPINs/OSNs are local-based and do not exploit all network structure, which limits prediction accuracy. On the other hand, there are global approaches to detect the overall path structure in a network, being computationally prohibitive for huge-size PPINs/OSNs. In this paper, we enhance a previously proposed multi-way spectral clustering method by introducing new ways to capture node proximity in both PPINs/OSNs. Our new enhanced method uses information obtained from the top few eigenvectors of the normalized Laplacian matrix. As a result, it produces a less noisy matrix, which is smaller and more compact than the original one. In this way, we are able to provide faster and more accurate link predictions. Moreover, our new spectral clustering model is based on the well-known Bray–Curtis coefficient to measure proximity between two nodes. Compared to traditional clustering algorithms, such as k-means and DBSCAN, which assume globular (convex) regions in Euclidean space, our approach is more flexible in capturing the non-connected components of a social graph and a wider range of cluster geometries. We perform an extensive experimental comparison of the proposed method against existing link prediction algorithms and k-means algorithm, using two synthetic data sets, three real social networks and three real human protein data sets. Our experimental results show that our SpectralLink algorithm outperforms the local approaches, the k-means algorithm and another spectral clustering method in terms of effectiveness, whereas it is more efficient than the global approaches.

The expressive power of first-order topological languages

We consider the expressive power of the first-order structure 〈Ω,C 〉 where Ω is either of two of different domains of extended regions in Euclidean space, and C(x,y) is the topological relation ‘Region x is in contact with region y.’ We prove two main theorems: Let P[ℚ] be the domain of bounded, non-empty, rational polyhedra in two- or three-dimensional Euclidean space. A relation Γ over P[ℚ] is definable in the structure 〈P[ℚ], C〉 if and only if Γ is arithmetic and invariant under rational PL-homeomorphisms of the space to itself. We also extend this result to a number of other domains, including the domain of all polyhedra and the domain of semi-algebraic regions. Let R be the space of bounded, non-empty, closed regular regions in n-dimensional Euclidean space. Any analytical relation over lower dimensional (i.e. empty interior) compact point sets that is invariant under homeomorphism is implicitly definable in the structure 〈R⁠,C〉.

Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds

We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty. We prove that a sequence of these manifolds whose ADM masses converge to zero must converge to Euclidean space in the pointed Intrinsic Flat sense. In fact we provide explicit bounds on the Intrinsic Flat Distance between annular regions in the manifold and annular regions in Euclidean space by constructing an explicit filling manifold and estimating its volume. In addition, we include a variety of propositions that can be used to estimate the Intrinsic Flat distance between Riemannian manifolds without rotationally symmetry. Conjectures regarding the Intrinsic Flat stability of the Positive Mass Theorem in the general case are proposed in the final section. Lee is partially supported by a PSC CUNY Research Grant and NSF DMS #0903467. Sormani is partially supported by a PSC CUNY Research Grant and NSF DMS #1006059. 1 ar X iv :1 10 4. 26 57 v1 [ m at h. D G ] 1 4 A pr 2 01 1 2 DAN A. LEE AND CHRISTINA SORMANI

Volume estimates and spectral asymptotics for a class of pseudo-differential operators

Two asymptotic expansions, known as the heat and Szego expansions, are studied for a class of operators defined on $$L^2(\Omega )$$ where $$\Omega $$ is a compact region in Euclidean space with smooth boundary. Pseudo-differential operator methods are combined with a version of a theorem originally due to H. Weyl on the volume of certain tubular neighborhoods to obtain significant new information about the higher order terms in the expansions.

Theory and algorithms for hop-count-based localization with random geometric graph models of dense sensor networks

Wireless sensor networks can often be viewed in terms of a uniform deployment of a large number of nodes in a region of Euclidean space. Following deployment, the nodes self-organize into a mesh topology with a key aspect being self-localization . Having obtained a mesh topology in a dense, homogeneous deployment, a frequently used approximation is to take the hop distance between nodes to be proportional to the Euclidean distance between them. In this work, we analyze this approximation through two complementary analyses. We assume that the mesh topology is a random geometric graph on the nodes; and that some nodes are designated as anchors with known locations. First, we obtain high probability bounds on the Euclidean distances of all nodes that are h hops away from a fixed anchor node. In the second analysis, we provide a heuristic argument that leads to a direct approximation for the density function of the Euclidean distance between two nodes that are separated by a hop distance h . This approximation is shown, through simulation, to very closely match the true density function. Localization algorithms that draw upon the preceding analyses are then proposed and shown to perform better than some of the well-known algorithms present in the literature. Belief-propagation-based message-passing is then used to further enhance the performance of the proposed localization algorithms. To our knowledge, this is the first usage of message-passing for hop-count-based self-localization.

Elementarily equivalent structures for topological languages over regions in Euclidean space

We prove that the class of rational polyhedra and the class of topologically regular regions definable in an o-minimal structure over the reals are each elementarily equivalent to the class of polyhedra for topological languages.

Topological change in mean convex mean curvature flow

Consider the mean curvature flow of an (n+1)-dimensional compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking S k×R n−k singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R n+1 with smooth boundary ∂K for which the resulting mean curvature flow has a shrinking S m×R n−m singularity.

Multivariate Monge-Kantorovich Transportation Problem

Monge-Kantorovich transportation problem in a bounded region of Euclidean space is transferred into a partial differential equations group. And then the explicit formula of the optimal coupling is achieved. The proofs are based on variational method from probability point of view.

Computing Geodesic Distances in Tree Space

We present two algorithms for computing the geodesic distance between phylogenetic trees in tree space, as introduced by Billera, Holmes, and Vogtmann [Adv. Appl. Math., 27 (2001), pp. 733–767]. We show that the possible combinatorial types of shortest paths between two trees can be compactly represented by a partially ordered set. We calculate the shortest distance along each candidate path by converting the problem into one of finding the shortest path through a certain region of Euclidean space. In particular, we show there is a linear time algorithm for finding the shortest path between a point in the all-positive orthant and a point in the all-negative orthant of $\mathbb{R}^k$ contained in the subspace of $\mathbb{R}^k$ consisting of all orthants with the first i coordinates nonpositive and the remaining coordinates nonnegative for $0 \leq i \leq k$.

Constrained CVT meshes and a comparison of triangular mesh generators

Mesh generation in regions in Euclidean space is a central task in computational science, and especially for commonly used numerical methods for the solution of partial differential equations, e.g., finite element and finite volume methods. We focus on the uniform Delaunay triangulation of planar regions and, in particular, on how one selects the positions of the vertices of the triangulation. We discuss a recently developed method, based on the centroidal Voronoi tessellation (CVT) concept, for effecting such triangulations and present two algorithms, including one new one, for CVT-based grid generation. We also compare several methods, including CVT-based methods, for triangulating planar domains. To this end, we define several quantitative measures of the quality of uniform grids. We then generate triangulations of several planar regions, including some having complexities that are representative of what one may encounter in practice. We subject the resulting grids to visual and quantitative comparisons and conclude that all the methods considered produce high-quality uniform grids and that the CVT-based grids are at least as good as any of the others.

Coverage in sensor networks via persistent homology

We introduce a topological approach to a problem of covering a region in Euclidean space by balls of fixed radius at unknown locations (this problem being motivated by sensor networks with minimal sensing capabilities). In particular, we give a homological criterion to rigorously guarantee that a collection of balls covers a bounded domain based on the homology of a certain simplicial pair. This pair of (Vietoris‐Rips) complexes is derived from graphs representing a coarse form of distance estimation between nodes and a proximity sensor for the boundary of the domain. The methods we introduce come from persistent homology theory and are applicable to nonlocalized sensor networks with ad hoc wireless communications. 55M25, 93A15; 55N35

The Expressivity of Quantifying over Regions

We categorize in recursion-theoretic terms the expressivity of a number of first-order languages that allow quantification over regions in Euclidean space. Specifically we show the following: 1. Let U be any class of closed regions in Euclidean space that includes all simple polygons. Let C(x, y) be the relation, “Region x is connected to region y,” and let Convex(x) be the property, “Region x is convex.” Then any relation over U that is analytical and invariant under affine transformations is first-order definable in the structure 〈U , C, Convex〉. 2. Let U be as in (1), and let Closer(x, y, z) be the relation “Region x is closer to y than to z.” Then any relation over U that is analytical and invariant under orthogonal transformations is first-order definable in the structure 〈U , Closer〉. 3. Let U be the class of finite unions of intervals in the real line. Then any relation over U that is analytical and invariant under linear transformations is first-order definable in the structure 〈U , Closer〉. 4. If the class of regions is restricted to be polygons with rational vertices, then results analogous to (1-3) hold, substituting “arithmetical relation” for “analytical relation”.

Wireless ad hoc lattice computers (WAdL)

We propose an architecture to harness the comparatively low computational power of geographically concentrated mobile devices (such as in a wireless ad hoc network, especially a sensor network) to build a wireless ad hoc lattice computer (WAdL). The primary contribution of the WAdL design is the ability to maintain, despite the mobility of the participating devices, a virtual lattice where the devices represent lattice points. WAdL is a cellular automaton-like architecture designed to carry out simulations of the unfolding of physical phenomena (e.g., fluid flow, system of moving, interacting objects, etc.) in the bounded region of Euclidean space represented by the underlying virtual lattice of WAdL. We present the design and algorithms of the WAdL architecture, and demonstrate its use with an example application (lift and drag on an airplane wing in flight) implemented on a simulated WAdL environment. We also discuss current issues and future directions of work on the WAdL architecture.

Volumes of symmetric spaces via lattice points

We show how to use elementary methods to compute the volume of Slk R/ Slk Z. We compute the volumes of certain unbounded regions in Euclidean space by counting lattice points and then appeal to the machinery of Dirichlet series to get estimates of the growth rate of the number of lattice points appearing in the region as the lattice spacing decreases. We also present a proof of the closely related result that the Tamagawa number is 1.

Constrained Centroidal Voronoi Tessellations for Surfaces

Centroidal Voronoi tessellations are useful for subdividing a region in Euclidean space into Voronoi regions whose generators are also the centers of mass, with respect to a prescribed density function, of the regions. Their extensions to general spaces and sets are also available; for example, tessellations of surfaces in a Euclidean space may be considered. In this paper, a precise definition of such constrained centroidal Voronoi tessellations (CCVTs) is given and a number of their properties are derived, including their characterization as minimizers of an "energy." Deterministic and probabilistic algorithms for the construction of CCVTs are presented and some analytical results for one of the algorithms are given. Computational examples are provided which serve to illustrate the high quality of CCVT point sets. Finally, CCVT point sets are applied to polynomial interpolation and numerical integration on the sphere.

Reciprocity and Polynomial Properties for Even Flows and Potentials on Directed Graphs

We consider special types of mod-λ flows, called odd and even mod-λ flows, for directed graphs, and prove that the numbers of such flows can be interpolated by polynomials in λ with the degree given by the cycle rank of the graph. The proofs involve computation of the number of integer solutions in a polyhedral region of Euclidean space using theorems due to Ehrhart. The resulting reciprocity properties of the interpolating polynomials for even flows are considered. The analogous properties of odd and even mod-λ potential differences and their associated potentials are also developed.