Global Geometric Structure Research Articles

Globally and Locally Consistent Unsupervised Projection

In this paper, we propose an unsupervised projection method for feature extraction to preserve both global and local consistencies of the input data in the projected space. Traditional unsupervised feature extraction methods, such as principal component analysis (PCA) and locality preserving projections (LPP), can only explore either the global or local geometric structures of the input data, but not the both at the same time. In our new method, we introduce a new measurement using the neighborhood data variances to assess the data locality, by which we propose to learn an optimal projection by rewarding both the global and local structures of the input data. The formulated optimization problem is challenging to solve, because it ends up a trace ratio minimization problem. In this paper, as an important theoretical contribution, we propose a simple yet efficient optimization algorithm to solve the trace ratio problem with theoretically proved convergence. Extensive experiments have been performed on six benchmark data sets, where the promising results validate the proposed method.

Pairwise Three-Dimensional Shape Context for Partial Object Matching and Retrieval on Mobile Laser Scanning Data

A novel pairwise 3-D shape context for partial object matching and retrieval is developed for extracting 3-D light poles and trees from mobile laser scanning (MLS) point clouds in a typical urban street scene. Unlike the single-point shape context describing only the local topology of a shape, the pairwise 3-D shape context can simultaneously model the local and global geometric structures of a shape in manifold space. By using histogram descriptors, the pairwise 3-D shape context has such characteristics as invariance to scale, invariance to orientation, and partial insensitivity to topological changes. Our results show that 3-D light poles and individual trees can be extracted from the RIEGL VMX-450 MLS point clouds and the performance achieved using our algorithm is much more accurate and effective than those of the other two existing algorithms.

Perceptual relativity-based semi-supervised dimensionality reduction algorithm

As we all know, a well-designed graph tends to result in good performance for graph-based semi-supervised learning. Although most graph-based semi-supervised dimensionality reduction approaches perform very well on clean data sets, they usually cannot construct a faithful graph which plays an important role in getting a good performance, when performing on the high dimensional, sparse or noisy data. So this will generally lead to a dramatic performance degradation. To deal with these issues, this paper proposes a feasible strategy called relative semi-supervised dimensionality reduction (RSSDR) by utilizing the perceptual relativity to semi-supervised dimensionality reduction. In RSSDR, firstly, relative transformation will be performed over the training samples to build the relative space. It should be indicated that relative transformation improves the distinguishing ability among data points and diminishes the impact of noise on semi-supervised dimensionality reduction. Secondly, the edge weights of neighborhood graph will be determined through minimizing the local reconstruction error in the relative space such that it can preserve the global geometric structure as well as the local one of the data. Extensive experiments on face, UCI, gene expression, artificial and noisy data sets have been provided to validate the feasibility and effectiveness of the proposed algorithm with the promising results both in classification accuracy and robustness.

Global plus local: A complete framework for feature extraction and recognition

Linear discriminant analysis (LDA) is one of the most popular supervised feature extraction techniques used in machine learning and pattern classification. However, LDA only captures global geometrical structure information of the data and ignores the geometrical structure information of local data points. Though many articles have been published to address this issue, most of them are incomplete in the sense that only part of the local information is used. We show here that there are total three kinds of local information, namely, local similarity information, local intra-class pattern variation, and local inter-class pattern variation. We first propose a new method called enhanced within-class LDA (EWLDA) algorithm to incorporate the local similarity information, and then propose a complete framework called complete global–local LDA (CGLDA) algorithm to incorporate all these three kinds of local information. Experimental results on two image databases demonstrate the effectiveness of our algorithms.

Detecting, Grouping, and Structure Inference for Invariant Repetitive Patterns in Images

The efficient and robust extraction of invariant patterns from an image is a long-standing problem in computer vision. Invariant structures are often related to repetitive or near-repetitive patterns. The perception of repetitive patterns in an image is strongly linked to the visual interpretation and composition of textures. Repetitive patterns are products of both repetitive structures as well as repetitive reflections or color patterns. In other words, patterns that exhibit near-stationary behavior provide rich information about objects, their shapes, and their texture in an image. In this paper, we propose a new algorithm for repetitive pattern detection and grouping. The algorithm follows the classical region growing image segmentation scheme. It utilizes a mean-shift-like dynamic to group local image patches into clusters. It exploits a continuous joint alignment to: 1) match similar patches, and 2) refine the subspace grouping. We also propose an algorithm for inferring the composition structure of the repetitive patterns. The inference algorithm constructs a data-driven structural completion field, which merges the detected repetitive patterns into specific global geometric structures. The result of higher level grouping for image patterns can be used to infer the geometry of objects and estimate the general layout of a crowded scene.

Joint Global and Local Structure Discriminant Analysis

Linear discriminant analysis (LDA) only considers the global Euclidean geometrical structure of data for dimensionality reduction. However, previous works have demonstrated that the local geometrical structure is effective for dimensionality reduction. In this paper, a novel approach is proposed, namely Joint Global and Local-structure Discriminant Analysis (JGLDA), for linear dimensionality reduction. To be specific, we construct two adjacency graphs to represent the local intrinsic structure, which characterizes both the similarity and diversity of data, and integrate the local intrinsic structure into Fisher linear discriminant analysis to build a stable discriminant objective function for dimensionality reduction. Experiments on several standard image databases demonstrate the effectiveness of our algorithm.

Kernel Fisher Discriminant Analysis with Locality Preserving for Feature Extraction and Recognition

Abstract Many previous studies have shown that class classification can be greatly improved by kernel Fisher discriminant analysis (KDA) technique. However, KDA only captures global geometrical structure and disregards local geometrical structure of the data. In this paper, we propose a new feature extraction algorithm, called locality preserving KDA (LPKDA) algorithm. LPKDA first casts KDA as a least squares problem in the kernel space and then explicitly incorporates the local geometrical structure information into the least squares problem via regularization technique. The fact that LPKDA can make full use of two kinds of discriminant information, global and local, makes it a more powerful discriminator. Experimental results on four image databases show that LPKDA outperforms other kernel-based algorithms.

Learning orthogonal projections for Isomap

We propose a dimensionality reduction technique in this paper, named Orthogonal Isometric Projection (OIP). In contrast with Isomap, which learns the low-dimension embedding, and solves problem under the classic Multidimensional Scaling (MDS) framework, we consider an explicit linear projection by capturing the geodesic distance, which is able to handle new data straightforward, and leads to a standard eigenvalue problem. We consider the orthogonal projection, and analyze the properties of orthogonal projection, and demonstrate the benefits, in which Euclidean distance, and angle at each pair in high-dimensional space are equivalent to ones in low-dimension, such that both global and local geometric structure are preserved. Numerical experiments are reported to demonstrate the performance of OIP by comparing with a few competing methods over different datasets.

Dynamical profile of a class of rank-one attractors

AbstractThis paper contains results on the geometric and ergodic properties of a class of strange attractors introduced by Wang and Young [Towards a theory of rank one attractors. Ann. of Math. (2) 167 (2008), 349–480]. These attractors can live in phase spaces of any dimension, and have been shown to arise naturally in differential equations that model several commonly occurring phenomena. Dynamically, such systems are chaotic; they have controlled non-uniform hyperbolicity with exactly one unstable direction, hence the name rank-one. In this paper we prove theorems on their Lyapunov exponents, Sinai–Ruelle–Bowen (SRB) measures, basins of attraction, and statistics of time series, including central limit theorems, exponential correlation decay and large deviations. We also present results on their global geometric and combinatorial structures, symbolic coding and periodic points. In short, we build a dynamical profile for this class of dynamical systems, proving that these systems exhibit many of the characteristics normally associated with ‘strange attractors’.

An Automatic and Adaptive Multi-manifolds Learning Algorithm

Isomap is a classic and representative manifold learning algorithm for nonlinear dimensionality reduction, which aims to circumvent the problem of “the curse of dimensionality” and attempts to recover the intrinsic structure hidden in high-dimensional data based on the assumption that data lie in or near a single manifold. However, Isomap fails to work when data set consists of multi-clusters or multiple sub-manifolds due to the requirement of preserving global geometric structure within data points. Several extended Isomap algorithms have been proposed to handle this issue, such as D-C Isoamp, M-Isomap, etc.. Here, we describe a new multi-manifolds learning algorithm by using a universal concept of neighbor: natural nearest neighbor (3N neighbor), which can be utilized automatically to construct a single or multiple adaptive neighborhood graphs that roughly coincide with the corresponding real multi-manifolds. The new algorithm preserves local adaptive neighborhood graph structure in terms of multi-manifolds as good as possible, in contrast to the previous algorithms, no category information about multi-manifolds are required. So it is a natural extension to the original Isomap, and also an adaptive multi-manifolds learning algorithm without the use of free parameter.

Efficient linear discriminant analysis with locality preserving for face recognition

Linear discriminant analysis (LDA) is one of the most popular techniques for extracting features in face recognition. LDA captures the global geometric structure. However, local geometric structure has recently been shown to be effective for face recognition. In this paper, we propose a novel feature extraction algorithm which integrates both global and local geometric structures. We first cast LDA as a least square problem based on the spectral regression, then regularization technique is used to model the global and local geometric structures. Furthermore, we impose penalty on parameters to tackle the singularity problem and design an efficient model selection algorithm to choose the optimal tuning parameter which balances the tradeoff between the global and local structures. Experimental results on four well-known face data sets show that the proposed integration framework is competitive with traditional face recognition algorithms, which use either global or local structure only.

Image Retargeting Quality Assessment

AbstractContent‐aware image retargeting is a technique that can flexibly display images with different aspect ratios and simultaneously preserve salient regions in images. Recently many image retargeting techniques have been proposed. To compare image quality by different retargeting methods fast and reliably, an objective metric simulating the human vision system (HVS) is presented in this paper. Different from traditional objective assessment methods that work in bottom‐up manner (i.e., assembling pixel‐level features in a local‐to‐global way), in this paper we propose to use a reverse order (top‐down manner) that organizes image features from global to local viewpoints, leading to a new objective assessment metric for retargeted images. A scale‐space matching method is designed to facilitate extraction of global geometric structures from retargeted images. By traversing the scale space from coarse to fine levels, local pixel correspondence is also established. The objective assessment metric is then based on both global geometric structures and local pixel correspondence. To evaluate color images, CIE L*a*b* color space is utilized. Experimental results are obtained to measure the performance of objective assessments with the proposed metric. The results show good consistency between the proposed objective metric and subjective assessment by human observers.

Global geometry of [formula omitted]-symmetric spacetimes with weak regularity

We define the class of weakly regular spacetimes with T 2 -symmetry, and investigate their global geometrical structure. We formulate the initial value problem for the Einstein vacuum equations with weak regularity, and establish the existence of a global foliation by the level sets of the area R of the orbits of symmetry, so that each leaf can be regarded as an initial hypersurface. Except for the flat Kasner spacetimes which are known explicitly, R takes all positive values. Our weak regularity assumptions only require that the gradient of R is continuous while the metric coefficients belong to the Sobolev space H 1 (or have even less regularity).

Feature extraction by learning Lorentzian metric tensor and its extensions

We develop a supervised dimensionality reduction method, called Lorentzian discriminant projection (LDP), for feature extraction and classification. Our method represents the structures of sample data by a manifold, which is furnished with a Lorentzian metric tensor. Different from classic discriminant analysis techniques, LDP uses distances from points to their within-class neighbors and global geometric centroid to model a new manifold to detect the intrinsic local and global geometric structures of data set. In this way, both the geometry of a group of classes and global data structures can be learnt from the Lorentzian metric tensor. Thus discriminant analysis in the original sample space reduces to metric learning on a Lorentzian manifold. We also establish the kernel, tensor and regularization extensions of LDP in this paper. The experimental results on benchmark databases demonstrate the effectiveness of our proposed method and the corresponding extensions.

Rapid Image Completion System Using Multiresolution Patch-Based Directional and Nondirectional Approaches

This study presents a rapid image completion system comprising a training (or analysis) process and an image completion (or synthesis) process. The proposed system adopts a multiresolution approach, which not only improves the convergence rate of the synthesis process, but also provides the ability to deal with large replaced regions. In the training process, a down-sampling approach is applied to create a patch-based texture eigenspace based on multiresolution background region information. In the image completion process, an up-sampling approach is applied to synthesize the replaced foreground regions. To ensure the continuity of the geometric texture structure between the original background scene regions and the replaced foreground regions, directional and nondirectional image completion approaches are developed to reconstruct the global geometric structure and to enhance the local detailed features of the replaced foreground regions in the lower and higher resolution level images, respectively. Moreover, the synthesis priority order of the individual patches and the appropriate choice of completion scheme (i.e., directional or nondirectional) are both determined in accordance with a Hessian matrix decision value (HMDV) parameter. Finally, a texture refinement process is performed to optimize the resolution of the synthesized result.

Maximum neighborhood margin criterion in face recognition

Feature extraction is a data analysis technique devoted to removing redundancy and extracting the most discriminative information. In face recognition, feature extractors are normally plagued with small sample size problems, in which the total number of training images is much smaller than the image dimensionality. Recently, an optimized facial feature extractor, maximum marginal criterion (MMC), was proposed. MMC computes an optimized projection by solving the generalized eigenvalue problem in a standard form that is free from inverse matrix operation, and thus it does not suffer from the small sample size problem. However, MMC is essentially a linear projection technique that relies on facial image pixel intensity to compute within- and between-class scatters. The nonlinear nature of faces restricts the discrimination of MMC. Hence, we propose an improved MMC, namely maximum neighborhood margin criterion (MNMC). Unlike MMC, which preserves global geometric structures that do not perfectly describe the underlying face manifold, MNMC seeks a projection that preserves local geometric structures via neighborhood preservation. This objective function leads to the enhancement of classification capability, and this is testified by experimental results. MNMC shows its performance superiority compared to MMC, especially in pose, illumination, and expression (PIE) and face recognition grand challenge (FRGC) databases.

One-sided invariant manifolds, recursive folding, and curvature singularity in area-preserving nonlinear maps with nonuniform hyperbolic behavior

Two-dimensional nonlinear models of conservative dynamics are typically nonuniformly hyperbolic in that there are nonhyperbolic trajectories that coexist with a “massive” hyperbolic region. We investigate the influence of nonhyperbolic points on the global geometric structure of a invariant manifolds associated with points of the hyperbolic region. As a case study, we consider a transformation of the Standard Map family and analyze the structure of invariant manifolds in the neighborhood of an isolated parabolic (fixed) point xp. This analysis shows the existence of lobes enclosing the parabolic point, that is, of simply connected regions containing xp whose boundary is formed by two continuous arcs of stable and unstable manifolds that intersect only at two points. From the existence of such regions, we derive that (i) there are points of the hyperbolic region where the local curvature of invariant manifolds is arbitrarily large and (ii) manifolds possess the recursively folding property. Property (ii) means that given an invariant manifold W and established an orientation on it, in the neighborhood of any point of the chaotic region there are nearby arcs of W that are traveled in opposite directions. We propose an archetypal model for which the existence of lobes and the recursive folding property can be derived analytically. The impact of nonuniform hyperbolicity on the evolution of physical processes that occur along with phase space mixing is also addressed.

On the global geometry of parametric models and information recovery

We examine the question of which statistic or statistics should be used in order to recover information important for inference. We take a global geometric viewpoint, developing the local geometry of Amari. By examining the behaviour of simple geometric models, we show how not only the local curvature properties of parametric families but also the global geometric structure can be of crucial importance in finite-sample analysis. The tool we use to explore this global geometry is the Karhunen-Loeve decomposition. Using global geometry, we show that the maximum likelihood estimate is the most important one-dimensional summary of information, but that traditional methods of information recovery beyond the maximum likelihood estimate can perform poorly. We also use the global geometry to construct better information summaries to be used with the maximum likelihood estimate.

BASIN CONFIGURATION OF A SIX-DIMENSIONAL MODEL OF AN ELECTRIC POWER SYSTEM

As part of an ongoing project on the stability of massively complex electrical power systems, we discuss the global geometric structure of contacts among the basins of attraction of a six-dimensional dynamical system. This system represents a simple model of an electrical power system involving three machines and an infinite bus. Apart from the possible occurrence of attractors representing pathological states, the contacts between the basins have a practical importance, from the point of view of the operation of a real electrical power system. With the aid of a global map of basins, one could hope to design an intervention strategy to boot the power system back into its normal state. Our method involves taking two-dimensional sections of the six-dimensional state space, and then determining the basins directly by numerical simulation from a dense grid of initial conditions. The relations among all the basins are given for a specific numerical example, that is, choosing particular values for the parameters in our model.

Global geometric structures associated with dynamical systems admitting normal shift of hypersurfaces in Riemannian manifolds

One of the ways of transforming hypersurfaces in Riemannian manifold is to move their points along some lines. In Bonnet construction of geodesic normal shift, these points move along geodesic lines. Normality of shift means that moving hypersurface keeps orthogonality to the trajectories of all its points. Geodesic lines correspond to the motion of free particles if the points of hypersurface are treated as physical entities obeying Newton′s second law. An attempt to introduce some external force F acting on the points of moving hypersurface in Bonnet construction leads to the theory of dynamical systems admitting a normal shift. As appears in this theory, the force field F of dynamical system should satisfy some system of partial differential equations. Recently, this system of equations was integrated, and explicit formula for F was obtained. But this formula is local. The main goal of this paper is to reveal global geometric structures associated with local expressions for F given by explicit formula.