局部保持的流形学习算法对比研究

局部保持的流形学习通过从局部到整体的思想保持观测空间和内在嵌入空间的局部几何共性,发现嵌入在高维欧氏空间中的内在低维流形。分析了局部保持的流形学习算法的基本实现框架,详细比较了一些局部保持的流形学习算法的特点,提出了几个有益的研究主题。     

A unified framework for semi-supervised dimensionality reduction

In practice, many applications require a dimensionality reduction method to deal with the partially labeled problem. In this paper, we propose a semi-supervised dimensionality reduction framework, which can efficiently handle the unlabeled data. Under the framework, several classical methods, such as principal component analysis (PCA), linear discriminant analysis (LDA), maximum margin criterion (MMC), locality preserving projections (LPP) and their corresponding kernel versions can be seen as special cases. For high-dimensional data, we can give a low-dimensional embedding result for both discriminating multi-class sub-manifolds and preserving local manifold structure. Experiments show that our algorithms can significantly improve the accuracy rates of the corresponding supervised and unsupervised approaches.     

黎曼流形上的保局投影在图像集匹配中的应用

目的提出了黎曼流形上局部结构特征保持的图像集匹配方法。方法该方法使用协方差矩阵建模图像集合,利用对称正定的非奇异协方差矩阵构成黎曼流形上的子空间,将图像集的匹配转化为流形上的点的匹配问题。通过基于协方差矩阵度量学习的核函数将黎曼流形上的协方差矩阵映射到欧几里德空间。不同于其他方法黎曼流形上的鉴别分析方法,考虑到样本分布的局部几何结构,引入了黎曼流形上局部保持的图像集鉴别分析方法,保持样本分布的局部邻域结构的同时提升样本的可分性。结果在基于图像集合的对象识别任务上测试了本文算法,在ETH80和YouTube Celebrities数据库分别进行了对象识别和人脸识别实验,分别达到91.5%和65.31%的识别率。结论实验结果表明,该方法取得了优于其他图像集匹配算法的效果。     

3D Motion recovery via affine Epipolar geometry

Algorithms to perform point-based motion estimation under orthographic and scaled orthographic projection abound in the literature. A key limitation of many existing algorithms is that they operate on the minimum amount of data required, often requiring the selection of a suitable minimal set from the available data to serve as a local coordinate frame. Such approaches are extremely sensitive to errors and noise in the minimal set, and forfeit the advantages of using the full data set. Furthermore, attention is seldom paid to the statistical performance of the algorithms.We present a new framework that allowsall available features to be used in the motion computations, without the need to select a frame explicitly. This theory is derived in the context of theaffine camera, which preserves parallelism and generalises the orthographic, scaled orthographic and para-perspective models. We define the affine epipolar geometry for two such cameras, giving the fundamental matrix in this case. The noise resistant computation of the epipolar geometry is discussed, and a statistical noise model constructed so that confidence in the results can be assessed.The rigid motion parameters are then determineddirectly from the epipolar geometry, using the novel rotation representation of Koenderink and van Doorn (1991). The two-view partial motion solution comprises the scale factor between views, the projection of the 3D axis of rotation and the cyclotorsion angle, while the addition of a third view allows the true 3D rotation axis to be computed (up to a Necker reversal). The computed uncertainties in these parameters permit optimal estimates to be obtained over time by means of a linear Kalman filter. Our theory extends work by Huang and Lee (1989), Harris (1990), and Koenderink and van Doorn (1991), and results are given on both simulated and real data.     

Manifold elastic net: a unified framework for sparse dimension reduction

It is difficult to find the optimal sparse solution of a manifold learning based dimensionality reduction algorithm. The lasso or the elastic net penalized manifold learning based dimensionality reduction is not directly a lasso penalized least square problem and thus the least angle regression (LARS) (Efron et al., Ann Stat 32(2):407–499, 2004), one of the most popular algorithms in sparse learning, cannot be applied. Therefore, most current approaches take indirect ways or have strict settings, which can be inconvenient for applications. In this paper, we proposed the manifold elastic net or MEN for short. MEN incorporates the merits of both the manifold learning based dimensionality reduction and the sparse learning based dimensionality reduction. By using a series of equivalent transformations, we show MEN is equivalent to the lasso penalized least square problem and thus LARS is adopted to obtain the optimal sparse solution of MEN. In particular, MEN has the following advantages for subsequent classification: (1) the local geometry of samples is well preserved for low dimensional data representation, (2) both the margin maximization and the classification error minimization are considered for sparse projection calculation, (3) the projection matrix of MEN improves the parsimony in computation, (4) the elastic net penalty reduces the over-fitting problem, and (5) the projection matrix of MEN can be interpreted psychologically and physiologically. Experimental evidence on face recognition over various popular datasets suggests that MEN is superior to top level dimensionality reduction algorithms.     

Multiobjective scheduling algorithm for flexible manufacturing systems with Petri nets

In this work, we focus on general multi-objective scheduling problems that can be modeled using a Petri net framework. Due to their generality, Petri nets are a useful abstraction that captures multiple characteristics of real-life processes.To provide a general solution procedure for the abstraction, we propose three alternative approaches using an indirect scheme to represent the solution: (1) a genetic algorithm that combines two objectives through a weighted fitness function, (2) a non dominated sorting genetic algorithm (NSGA-II) that explicitly addresses the multi-objective nature of the problem and (3) a multi-objective local search approach that simultaneously explores multiple candidate solutions. These algorithms are tested in an extensive computational experiment showing the applicability of this general framework to obtain quality solutions.     

Nonlocal estimation of manifold structure

We claim and present arguments to the effect that a large class of manifold learning algorithms that are essentially local and can be framed as kernel learning algorithms will suffer from the curse of dimensionality, at the dimension of the true underlying manifold. This observation invites an exploration of nonlocal manifold learning algorithms that attempt to discover shared structure in the tangent planes at different positions. A training criterion for such an algorithm is proposed, and experiments estimating a tangent plane prediction function are presented, showing its advantages with respect to local manifold learning algorithms: it is able to generalize very far from training data (on learning handwritten character image rotations), where local nonparametric methods fail.     

新的流形学习方法统一框架及改进的拉普拉斯特征映射方法

流形学习是多个领域的重要研究课题.通过考察各种流形学习方法,提出了一种新的流形学习方法的统一框架,并在此框架下对拉普拉斯特征映射方法(Laplacian eigenmap,LE)进行了分析.进一步,基于此框架,提出了一种改进拉普拉斯特征映射方法(improved Laplacian eigenmap,ILE).它建立在LE方法和最大差异延展算法(maximum variance unfolding,MVU)的基础上,在保持流形谱图拉普拉斯特征的同时,以最大化任意两点之间的差异为目标.ILE有效地解决了拉普拉斯特征映射方法对邻域选择敏感以及MVU方法大计算量、局部限制过强等问题.且能够保持数据聚类性质,挖掘数据内蕴特征.通过实验说明了ILE的有效性.     

Stochastic Heat Kernel Estimation on Sampled Manifolds

The heat kernel is a fundamental geometric object associated to every Riemannian manifold, used across applications in computer vision, graphics, and machine learning. In this article, we propose a novel computational approach to estimating the heat kernel of a statistically sampled manifold (e.g. meshes or point clouds), using its representation as the transition density function of Brownian motion on the manifold. Our approach first constructs a set of local approximations to the manifold via moving least squares. We then simulate Brownian motion on the manifold by stochastic numerical integration of the associated Ito diffusion system. By accumulating a number of these trajectories, a kernel density estimation method can then be used to approximate the transition density function of the diffusion process, which is equivalent to the heat kernel. We analyse our algorithm on the 2‐sphere, as well as on shapes in 3D. Our approach is readily parallelizable and can handle manifold samples of large size as well as surfaces of high co‐dimension, since all the computations are local. We relate our method to the standard approaches in diffusion geometry and discuss directions for future work.     

Non‐iterative Second‐order Approximation of Signed Distance Functions for Any Isosurface Representation

Signed distance functions (SDF) to explicit or implicit surface representations are intensively used in various computer graphics and visualization algorithms. Among others, they are applied to optimize collision detection, are used to reconstruct data fields or surfaces, and, in particular, are an obligatory ingredient for most level set methods. Level set methods are common in scientific visualization to extract surfaces from scalar or vector fields. Usual approaches for the construction of an SDF to a surface are either based on iterative solutions of a special partial differential equation or on marching algorithms involving a polygonization of the surface. We propose a novel method for a non‐iterative approximation of an SDF and its derivatives in a vicinity of a manifold. We use a second‐order algebraic fitting scheme to ensure high accuracy of the approximation. The manifold is defined (explicitly or implicitly) as an isosurface of a given volumetric scalar field. The field may be given at a set of irregular and unstructured samples. Stability and reliability of the SDF generation is achieved by a proper scaling of weights for the Moving Least Squares approximation, accurate choice of neighbors, and appropriate handling of degenerate cases. We obtain the solution in an explicit form, such that no iterative solving is necessary, which makes our approach fast.     

Kernel-view based discriminant approach for embedded feature extraction in high-dimensional space

Derived from the traditional manifold learning algorithms, local discriminant analysis methods identify the underlying submanifold structures while employing discriminative information for dimensionality reduction. Mathematically, they can all be unified into a graph embedding framework with different construction criteria. However, such learning algorithms are limited by the curse-of-dimensionality if the original data lie on the high-dimensional manifold. Different from the existing algorithms, we consider the discriminant embedding as a kernel analysis approach in the sample space, and a kernel-view based discriminant method is proposed for the embedded feature extraction, where both PCA pre-processing and the pruning of data can be avoided. Extensive experiments on the high-dimensional data sets show the robustness and outstanding performance of our proposed method.     

基于近邻元分析的半监督流形学习算法*

现有的大多数流形学习算法偏重保持流形的几何结构,并未考虑到样本点的标签信息,这在一定程度上限制了流形学习算法在数据分类中的应用.因此文中提出一种基于近邻元分析的半监督流形学习算法,采用近邻元分析学习距离度量矩阵,在距离度量方式下选择样本点的局部邻域点.基于距离度量方式构造样本点和邻域点的局部几何结构,并在样本点的低维嵌入坐标中保持这种局部几何结构不变.3个不同数据集上的分类实验验证了文中算法的有效性.     

Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition

In this paper, we examine image and video-based recognition applications where the underlying models have a special structure—the linear subspace structure. We discuss how commonly used parametric models for videos and image sets can be described using the unified framework of Grassmann and Stiefel manifolds. We first show that the parameters of linear dynamic models are finite-dimensional linear subspaces of appropriate dimensions. Unordered image sets as samples from a finite-dimensional linear subspace naturally fall under this framework. We show that an inference over subspaces can be naturally cast as an inference problem on the Grassmann manifold. To perform recognition using subspace-based models, we need tools from the Riemannian geometry of the Grassmann manifold. This involves a study of the geometric properties of the space, appropriate definitions of Riemannian metrics, and definition of geodesics. Further, we derive statistical modeling of inter and intraclass variations that respect the geometry of the space. We apply techniques such as intrinsic and extrinsic statistics to enable maximum-likelihood classification. We also provide algorithms for unsupervised clustering derived from the geometry of the manifold. Finally, we demonstrate the improved performance of these methods in a wide variety of vision applications such as activity recognition, video-based face recognition, object recognition from image sets, and activity-based video clustering.     

Feature extraction using constrained maximum variance mapping

In this paper, an efficient feature extraction method named as constrained maximum variance mapping (CMVM) is developed. The proposed algorithm can be viewed as a linear approximation of multi-manifolds learning based approach, which takes the local geometry and manifold labels into account. The CMVM and the original manifold learning based approaches have a point in common that the locality is preserved. Moreover, the CMVM is globally maximizing the distances between different manifolds. After the local scatters have been characterized, the proposed method focuses on developing a linear transformation that can maximize the dissimilarities between all the manifolds under the constraint of locality preserving. Compared to most of the up-to-date manifold learning based methods, this trick makes contribution to pattern classification from two aspects. On the one hand, the local structure in each manifold is still kept; on the other hand, the discriminant information between manifolds can be explored. Finally, FERET face database, CMU PIE face database and USPS handwriting data are all taken to examine the effectiveness and efficiency of the proposed method. Experimental results validate that the proposed approach is superior to other feature extraction methods, such as linear discriminant analysis (LDA), locality preserving projection (LPP), unsupervised discriminant projection (UDP) and maximum variance projection (MVP).     

基于相对流形的局部线性嵌入

局部线性嵌入算法极大地依赖于邻域是否真实地反映了流形的内在结构,现有方法构造的邻域结构是拓扑不稳定的,对噪音和稀疏数据敏感.根据认知的相对性规律提出了相对变换,并用其构造了相对空间和相对流形.相对变换可以提高数据之间的可区分性,并能抑制噪音和数据稀疏的影响.在构造的相对空间和相对流形上确定数据点的邻域能够更真实地反映流形的内在结构,由此提出了增强的局部线性嵌入算法,明显地提高了性能,特别是基于流形的方法还同时提高了速度.标准数据集上的实验结果验证了该方法的有效性.     

3-D Reconstruction of Shaded Objects from Multiple Images Under Unknown Illumination

We propose a variational algorithm to jointly estimate the shape, albedo, and light configuration of a Lambertian scene from a collection of images taken from different vantage points. Our work can be thought of as extending classical multi-view stereo to cases where point correspondence cannot be established, or extending classical shape from shading to the case of multiple views with unknown light sources. We show that a first naive formalization of this problem yields algorithms that are numerically unstable, no matter how close the initialization is to the true geometry. We then propose a computational scheme to overcome this problem, resulting in provably stable algorithms that converge to (local) minima of the cost functional. We develop a new model that explicitly enforces positivity in the light sources with the assumption that the object is Lambertian and its albedo is piecewise constant and show that the new model significantly improves the accuracy and robustness relative to existing approaches.     

Real-time manifold regularized context-aware correlation tracking

Despite the demonstrated success of numerous correlation filter(CF)based tracking approaches,their assumption of circulant structure of samples introduces significant redundancy to learn an effective classifier.In this paper,we develop a fast manifold regularized context-aware correlation tracking algorithm that mines the local manifold structure information of different types of samples.First,different from the traditional CF based tracking that only uses one base sample,we employ a set of contextual samples near to the base sample,and impose a manifold structure assumption on them.Afterwards,to take into account the manifold structure among these samples,we introduce a linear graph Laplacian regularized term into the objective of CF learning.Fortunately,the optimization can be efficiently solved in a closed form with fast Fourier transforms(FFTs),which contributes to a highly efficient implementation.Extensive evaluations on the OTB100 and VOT2016 datasets demonstrate that the proposed tracker performs favorably against several state-of-the-art algorithms in terms of accuracy and robustness.Especially,our tracker is able to run in real-time with 28 fps on a single CPU.     

Displacement-Correlated XFEM for Simulating Brittle Fracture

We present a remeshing-free brittle fracture simulation method under the assumption of quasi-static linear elastic fracture mechanics (LEFM). To achieve this, we devise two algorithms. First, we develop an approximate volumetric simulation, based on the extended Finite Element Method (XFEM), to initialize and propagate Lagrangian crack-fronts. We model the geometry of fracture explicitly as a surface mesh, which allows us to generate high-resolution crack surfaces that are decoupled from the resolution of the deformation mesh. Our second contribution is a mesh cutting algorithm, which produces fragments of the input mesh using the fracture surface. We do this by directly operating on the half-edge data structures of two surface meshes, which enables us to cut general surface meshes including those of concave polyhedra and meshes with abutting concave polygons. Since we avoid triangulation for cutting, the connectivity of the resulting fragments is identical to the (uncut) input mesh except at edges introduced by the cut. We evaluate our simulation and cutting algorithms and show that they outperform state-of-the-art approaches both qualitatively and quantitatively.     

Fully automatic face recognition framework based on local and global features

Face recognition algorithms can be divided into two categories: holistic and local feature-based approaches. Holistic methods are very popular in recent years due to their good performance and high efficiency. However, they depend on careful positioning of the face images into the same canonical pose, which is not an easy task. On the contrary, some local feature-based approaches can achieve good recognition performances without additional alignment. But their computational burden is much heavier than holistic approaches. To solve these problems in holistic and local feature-based approaches, we propose a fully automatic face recognition framework based on both the local and global features. In this work, we propose to align the input face images using multi-scale local features for the holistic approach, which serves as a filter to narrow down the database for further fine matching. The computationally heavy local feature-based approach is then applied on the narrowed database. This fully automatic framework not only speeds up the local feature-based approach, but also improves the recognition accuracy comparing with the holistic and local approaches as shown in the experiments.     

TRASMIL: A local anomaly detection framework based on trajectory segmentation and multi-instance learning

Local anomaly detection refers to detecting small anomalies or outliers that exist in some subsegments of events or behaviors. Such local anomalies are easily overlooked by most of the existing approaches since they are designed for detecting global or large anomalies. In this paper, an accurate and flexible three-phase framework TRASMIL is proposed for local anomaly detection based on TRAjectory Segmentation and Multi-Instance Learning. Firstly, every motion trajectory is segmented into independent sub-trajectories, and a metric with Diversity and Granularity is proposed to measure the quality of segmentation. Secondly, the segmented sub-trajectories are modeled by a sequence learning model. Finally, multi-instance learning is applied to detect abnormal trajectories and sub-trajectories which are viewed as bags and instances, respectively. We validate the TRASMIL framework in terms of 16 different algorithms built on the three-phase framework. Substantial experiments show that algorithms based on the TRASMIL framework outperform existing methods in effectively detecting the trajectories with local anomalies in terms of the whole trajectory. In particular, the MDL-C algorithm (the combination of HDP-HMM with MDL segmentation and Citation kNN) achieves the highest accuracy and recall rates. We further show that TRASMIL is generic enough to adopt other algorithms for identifying local anomalies.