In this work, we present a novel and practical approach to address one of the longstanding problems in computer vision: 2D and 3D affine invariant feature matching. Our Grassmannian Graph (GrassGraph) framework employs a two stage procedure that is capable of robustly recovering correspondences between two unorganized, affinely related feature (point) sets. The first stage maps the feature sets to an affine invariant Grassmannian representation, where the features are mapped into the same subspace. It turns out that coordinate representations extracted from the Grassmannian differ by an arbitrary orthonormal matrix. In the second stage, by approximating the Laplace-Beltrami operator (LBO) on these coordinates, this extra orthonormal factor is nullified, providing true affine-invariant coordinates which we then utilize to recover correspondences via simple nearest neighbor relations. The resulting GrassGraph algorithm is empirically shown to work well in non-ideal scenarios with noise, outliers, and occlusions. Our validation benchmarks use an unprecedented 440,000+ experimental trials performed on 2D and 3D datasets, with a variety of parameter settings and competing methods. State-of-the-art performance in the majority of these extensive evaluations confirm the utility of our method.
Begelfor and Werman @cite_12 popularized the use of Grassmannian subspaces as an affine invariant. Their work focused on developing clustering algorithms on the Grassmann manifold, but they did not utilize this invariance to solve the correspondence problem. A method that did try to address this correspondence problem through subspace invariance was @cite_34 . The robustness of their method was never evaluated, and, moreover, their approach of using QR factorizations of rank-deficient orthogonal projection matrices is quite different from our proposed two-stage approach. Finally, have shown the effectiveness of Grassmannian representations for object recognition @cite_22 and their use has expanded to other vision domains @cite_2 .
@cite_34 @cite_22 @cite_12 @cite_2
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We study the problem of online subspace learning in the context of sequential observations involving structured perturbations. In online subspace learning, the observations are an unknown mixture of two components presented to the model sequentially - the main effect which pertains to the subspace and a residual error term. If no additional requirement is imposed on the residual, it often corresponds to noise terms in the signal which were unaccounted for by the main effect. To remedy this, one may impose "structural" contiguity, which has the intended effect of leveraging the secondary terms as a covariate that helps the estimation of the subspace itself, instead of merely serving as a noise residual. We show that the corresponding online estimation procedure can be written as an approximate optimization process on a Grassmannian. We propose an efficient numerical solution, GOSUS, Grassmannian Online Subspace Updates with Structured-sparsity, for this problem. GOSUS is expressive enough in modeling both homogeneous perturbations of the subspace and structural contiguities of outliers, and after certain manipulations, solvable via an alternating direction method of multipliers (ADMM). We evaluate the empirical performance of this algorithm on two problems of interest: online background subtraction and online multiple face tracking, and demonstrate that it achieves competitive performance with the state-of-the-art in near real time. Recent advances in computer vision and machine learning suggest that a wide range of problems can be addressed more appropriately by considering non-Euclidean geometry. In this paper we explore sparse dictionary learning over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping, which enables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. Furthermore, to handle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. Experiments on several classification tasks (face recognition, action recognition, dynamic texture classification) show that the proposed approach achieves considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelised Affine Hull Method and graph-embedding Grassmann discriminant analysis. Recent advances in computer vision and machine learning suggest that a wide range of problems can be addressed more appropriately by considering non-Euclidean geometry. In this paper we explore sparse dictionary learning over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping, which enables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. Furthermore, to handle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. Experiments on several classification tasks (face recognition, action recognition, dynamic texture classification) show that the proposed approach achieves considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelised Affine Hull Method and graph-embedding Grassmann discriminant analysis. A convenient way of dealing with image sets is to represent them as points on Grassmannian manifolds. While several recent studies explored the applicability of discriminant analysis on such manifolds, the conventional formalism of discriminant analysis suffers from not considering the local structure of the data. We propose a discriminant analysis approach on Grassmannian manifolds, based on a graph-embedding framework. We show that by introducing within-class and between-class similarity graphs to characterise intra-class compactness and inter-class separability, the geometrical structure of data can be exploited. Experiments on several image datasets (PIE, BANCA, MoBo, ETH-80) show that the proposed algorithm obtains considerable improvements in discrimination accuracy, in comparison to three recent methods: Grassmann Discriminant Analysis (GDA), Kernel GDA, and the kernel version of Affine Hull Image Set Distance. We further propose a Grassmannian kernel, based on canonical correlation between subspaces, which can increase discrimination accuracy when used in combination with previous Grassmannian kernels. Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special type of Riemannian manifolds known as Grassmannian. To leverage the techniques developed for Euclidean spaces (e.g., support vector machines) with subspaces, several recent studies have proposed to embed the Grassmannian into a Hilbert space by making use of a positive definite kernel. Unfortunately, only two Grassmannian kernels are known, none of which -as we will show- is universal, which limits their ability to approximate a target function arbitrarily well. Here, we introduce several positive definite Grassmannian kernels, including universal ones, and demonstrate their superiority over previously-known kernels in various tasks, such as classification, clustering, sparse coding and hashing. 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. In this paper we propose a discriminant learning framework for problems in which data consist of linear subspaces instead of vectors. By treating subspaces as basic elements, we can make learning algorithms adapt naturally to the problems with linear invariant structures. We propose a unifying view on the subspace-based learning method by formulating the problems on the Grassmann manifold, which is the set of fixed-dimensional linear subspaces of a Euclidean space. Previous methods on the problem typically adopt an inconsistent strategy: feature extraction is performed in the Euclidean space while non-Euclidean distances are used. In our approach, we treat each sub-space as a point in the Grassmann space, and perform feature extraction and classification in the same space. We show feasibility of the approach by using the Grassmann kernel functions such as the Projection kernel and the Binet-Cauchy kernel. Experiments with real image databases show that the proposed method performs well compared with state-of-the-art algorithms. In video based face recognition, great success has been made by representing videos as linear subspaces, which typically lie in a special type of non-Euclidean space known as Grassmann manifold. To leverage the kernel-based methods developed for Euclidean space, several recent methods have been proposed to embed the Grassmann manifold into a high dimensional Hilbert space by exploiting the well established Project Metric, which can approximate the Riemannian geometry of Grassmann manifold. Nevertheless, they inevitably introduce the drawbacks from traditional kernel-based methods such as implicit map and high computational cost to the Grassmann manifold. To overcome such limitations, we propose a novel method to learn the Projection Metric directly on Grassmann manifold rather than in Hilbert space. From the perspective of manifold learning, our method can be regarded as performing a geometry-aware dimensionality reduction from the original Grassmann manifold to a lower-dimensional, more discriminative Grassmann manifold where more favorable classification can be achieved. Experiments on several real-world video face datasets demonstrate that the proposed method yields competitive performance compared with the state-of-the-art algorithms. In this paper, we propose a novel space-time geometric representation of human landmark configurations and derive tools for comparison and classification. We model the temporal evolution of landmarks as parametrized trajectories on the Riemannian manifold of positive semidefinite matrices of fixed-rank. Our representation has the benefit to bring naturally a second desirable quantity when comparing shapes — the spatial covariance — in addition to the conventional affine-shape representation. We derived then geometric and computational tools for rate-invariant analysis and adaptive re-sampling of trajectories, grounding on the Riemannian geometry of the underlying manifold. Specifically, our approach involves three steps: (1) landmarks are first mapped into the Riemannian manifold of positive semidefinite matrices of fixed-rank to build time-parameterized trajectories; (2) a temporal warping is performed on the trajectories, providing a geometry-aware (dis-)similarity measure between them; (3) finally, a pairwise proximity function SVM is used to classify them, incorporating the (dis-)similarity measure into the kernel function. We show that such representation and metric achieve competitive results in applications as action recognition and emotion recognition from 3D skelet al data, and facial expression recognition from videos. Experiments have been conducted on several publicly available up-to-date benchmarks. We introduce the notion of subspace learning from image gradient orientations for appearance-based object recognition. As image data are typically noisy and noise is substantially different from Gaussian, traditional subspace learning from pixel intensities very often fails to estimate reliably the low-dimensional subspace of a given data population. We show that replacing pixel intensities with gradient orientations and the l2 norm with a cosine-based distance measure offers, to some extend, a remedy to this problem. Within this framework, which we coin Image Gradient Orientations (IGO) subspace learning, we first formulate and study the properties of Principal Component Analysis of image gradient orientations (IGO-PCA). We then show its connection to previously proposed robust PCA techniques both theoretically and experimentally. Finally, we derive a number of other popular subspace learning techniques, namely, Linear Discriminant Analysis (LDA), Locally Linear Embedding (LLE), and Laplacian Eigenmaps (LE). Experimental results show that our algorithms significantly outperform popular methods such as Gabor features and Local Binary Patterns and achieve state-of-the-art performance for difficult problems such as illumination and occlusion-robust face recognition. In addition to this, the proposed IGO-methods require the eigendecomposition of simple covariance matrices and are as computationally efficient as their corresponding l2 norm intensity-based counterparts. Matlab code for the methods presented in this paper can be found at http: ibug.doc.ic.ac.uk resources. In this paper, we study the problem of facial expression recognition using a novel space-time geometric representation. We describe the temporal evolution of facial landmarks as parametrized trajectories on the Riemannian manifold of positive semidefinite matrices of fixed-rank. Our representation has the advantage to bring naturally a second desirable quantity when comparing shapes -- the spatial covariance -- in addition to the conventional affine-shape representation. We derive then geometric and computational tools for rate-invariant analysis and adaptive re-sampling of trajectories, grounding on the Riemannian geometry of the manifold. Specifically, our approach involves three steps: 1) facial landmarks are first mapped into the Riemannian manifold of positive semidefinite matrices of rank 2, to build time-parameterized trajectories; 2) a temporal alignment is performed on the trajectories, providing a geometry-aware (dis-)similarity measure between them; 3) finally, pairwise proximity function SVM (ppfSVM) is used to classify them, incorporating the latter (dis-)similarity measure into the kernel function. We show the effectiveness of the proposed approach on four publicly available benchmarks (CK+, MMI, Oulu-CASIA, and AFEW). The results of the proposed approach are comparable to or better than the state-of-the-art methods when involving only facial landmarks. We consider the problem of approximating a set P of η points in Rd by a j-dimensional subspace under the lp measure, in which we wish to minimize the sum of lp distances from each point of P to this subspace. More generally, the Fq (lp)-subspace approximation problem asks for a j-subspace that minimizes the sum of qth powers of lp-distances to this subspace, up to a multiplicative factor of (1 + e). We develop techniques for subspace approximation, regression, and matrix approximation that can be used to deal with massive data sets in high dimensional spaces. In particular, we develop coresets and sketches, i.e. small space representations that approximate the input point set P with respect to the subspace approximation problem. Our results are: • A dimensionality reduction method that can be applied to Fq (lp)-clustering and shape fitting problems, such as those in [8, 15]. • The first strong coreset for F1 (l2)-subspace approximation in high-dimensional spaces, i.e. of size polynomial in the dimension of the space. This coreset approximates the distances to any j-subspace (not just the optimal one). • A (1 + e)-approximation algorithm for the j-dimensional F1 (l2)-subspace approximation problem with running time ηd(j e)O(1) + (η + d)2poly(j e). • A streaming algorithm that maintains a coreset for the F1 (l2)-subspace approximation problem and uses a space of d [EQUATION] (weighted) points. • Streaming algorithms for the above problems with bounded precision in the turnstile model, i.e, when coordinates appear in an arbitrary order and undergo multiple updates. We show that bounded precision can lead to further improvements. We extend results of [7] for approximate linear regression, distances to subspace approximation, and optimal rank-j approximation, to error measures other than the Frobenius norm. We discuss a new robust convergence analysis of the well-known subspace iteration algorithm for computing the dominant singular vectors of a matrix, also known as simultaneous iteration or power method. The result characterizes the convergence behavior of the algorithm when a large amount noise is introduced after each matrix-vector multiplication. While interesting in its own right, the main motivation comes from the problem of privacy-preserving spectral analysis where noise is added in order to achieve the privacy guarantee known as differential privacy. This result leads to nearly tight worst-case bounds for the problem of computing a differentially private low-rank approximation in the spectral norm. Our results extend to privacy-preserving principal component analysis. We obtain improvements for several variants of differential privacy that have been considered. The running time of our algorithm is nearly linear in the input sparsity leading to strong improvements in running time over previous work. Complementing our worst-case bounds, we show that the error dependence of our algorithm on the matrix dimension can be replaced by a tight dependence on the coherence of the matrix. This parameter is always bounded by the matrix dimension but often much smaller. Indeed, the assumption of low coherence is essential in several machine learning and signal processing applications. Low-rank representation (LRR) has recently attracted great interest due to its pleasing efficacy in exploring low-dimensional subspace structures embedded in data. One of its successful applications is subspace clustering which means data are clustered according to the subspaces they belong to. In this paper, at a higher level, we intend to cluster subspaces into classes of subspaces. This is naturally described as a clustering problem on Grassmann manifold. The novelty of this paper is to generalize LRR on Euclidean space into the LRR model on Grassmann manifold. The new method has many applications in computer vision tasks. The paper conducts the experiments over two real world examples, clustering handwritten digits and clustering dynamic textures. The experiments show the proposed method outperforms a number of existing methods. This paper proposes an affine invariant matching algorithm for shape correspondence problems in arbitrary dimensions. Formulating shapes by configuration matrices of landmarks, and using the fact that subspaces (e.g. range spaces) of these matrices are invariant to affine transformations, the shape correspondence is modelled as a permutation relation between orthogonal projection matrices of the subspaces. Then the matching result is solved by an efficient factorization procedure for rank-deficient matrices. The algorithm is compact, fast, and independent of dimensions. Experimental results for 1D, 2D and 3D matchings of synthetic and real data are provided, which demonstrate potential applications of the algorithm to shape analysis, and to other related problems like wide baseline stereo matching and range data registration. In this paper we address the problem of classifying vector sets. We motivate and introduce a novel method based on comparisons between corresponding vector subspaces. In particular, there are two main areas of novelty: (i) we extend the concept of principal angles between linear subspaces to manifolds with arbitrary nonlinearities; (ii) it is demonstrated how boosting can be used for application-optimal principal angle fusion. The strengths of the proposed method are empirically demonstrated on the task of automatic face recognition (AFR), in which it is shown to outperform state-of-the-art methods in the literature. We consider the problem of segmenting multiple rigid-body motions from point correspondences in multiple affine views. We cast this problem as a subspace clustering problem in which point trajectories associated with each motion live in a linear subspace of dimension two, three or four. Our algorithm involves projecting all point trajectories onto a 5-dimensional subspace using the SVD, the PowerFactorization method, or RANSAC, and fitting multiple linear subspaces representing different rigid-body motions to the points in ?5 using GPCA. Unlike previous work, our approach does not restrict the motion subspaces to be four-dimensional and independent. Instead, it deals gracefully with all the spectrum of possible affine motions: from two-dimensional and partially dependent to four-dimensional and fully independent. Our algorithm can handle the case of missing data, meaning that point tracks do not have to be visible in all images, by using the PowerFactorization method to project the data. In addition, our method can handle outlying trajectories by using RANSAC to perform the projection. We compare our approach to other methods on a database of 167 motion sequences with full motions, independent motions, degenerate motions, partially dependent motions, missing data, outliers, etc. On motion sequences with complete data our method achieves a misclassification error of less that 5 for two motions and 29 for three motions. We describe a new algorithm, termed subspace evolution and transfer (SET), for solving consistent low-rank matrix completion problems. The algorithm takes as its input a subset of entries of a low-rank matrix and outputs one low-rank matrix consistent with the given observations. The completion task is accomplished by searching for a column space in the Grassmann manifold that matches the incomplete observations. The SET algorithm consists of two parts-subspace evolution and subspace transfer. In the evolution part, we use a gradient descent method on the Grassmann manifold to refine our estimate of the column space. Since the gradient descent algorithm is not guaranteed to converge due to the existence of barriers along the search path, we design a new mechanism for detecting barriers and transferring the estimated column space across the barriers. This mechanism constitutes the core of the transfer step of the algorithm. The SET algorithm exhibits excellent empirical performance for a large range of sampling rates. We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R n . In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications - computing an invariant subspace of a matrix and the mean of subspaces - are worked out. Many real world datasets exhibit an embedding of low-dimensional structure in a high-dimensional manifold. Examples include images, videos and internet traffic data. It is of great significance to estimate and track the low-dimensional structure with small storage requirements and computational complexity when the data dimension is high. Therefore we consider the problem of reconstructing a data stream from a small subset of its entries, where the data is assumed to lie in a low-dimensional linear subspace, possibly corrupted by noise. We further consider tracking the change of the underlying subspace, which can be applied to applications such as video denoising, network monitoring and anomaly detection. Our setting can be viewed as a sequential low-rank matrix completion problem in which the subspace is learned in an online fashion. The proposed algorithm, dubbed Parallel Estimation and Tracking by REcursive Least Squares (PETRELS), first identifies the underlying low-dimensional subspace, and then reconstructs the missing entries via least-squares estimation if required. Subspace identification is performed via a recursive procedure for each row of the subspace matrix in parallel with discounting for previous observations. Numerical examples are provided for direction-of-arrival estimation and matrix completion, comparing PETRELS with state of the art batch algorithms.