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37
Fast approximate nearest neighbor methods for nonEuclidean manifolds with applications to human activity analysis
 in videos,” in European Conference on Computer Vision, 2010
"... Approximate Nearest Neighbor (ANN) methods such as Locality Sensitive Hashing, Semantic Hashing, and Spectral Hashing, provide computationally efficient procedures for finding objects similar to a query object in large datasets. These methods have been successfully applied to search webscale datase ..."
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Approximate Nearest Neighbor (ANN) methods such as Locality Sensitive Hashing, Semantic Hashing, and Spectral Hashing, provide computationally efficient procedures for finding objects similar to a query object in large datasets. These methods have been successfully applied to search webscale datasets that can contain millions of images. Unfortunately, the key assumption in these procedures is that objects in the dataset lie in a Euclidean space. This assumption is not always valid and poses a challenge for several vision applications where data commonly lies in complex nonEuclidean manifolds. In particular, dynamic data such as human activities are commonly represented as distributions over bags of video words as a dynamical systems. In this paper, we propose two new algorithms that extend Spectral Hashing to nonEuclidean spaces. The first method considers the Riemannian geometry of the manifold and performs Spectral Hashing in the Tangent space of the manifold at several points. The second method divides the data into subsets and takes advantage of the kernel trick to perform nonEuclidean Spectral Hashing. For a data set of N samples the proposed methods are able to retrieve similar objects in as low as O (K) time complexity, where K is the number of clusters in the data. Since K muchlessthan N, our methods are extremely efficient. We test and evaluate our methods on synthetic data generated from the Unit Hypersphere and the Grassmann Manifold. Finally, we show promising results on a human action database.
Multiple Rotation Averaging
"... • Single Rotation Averaging: Several estimates are obtained of a single rotation, which are then averaged to give the best estimate. • Multiple Rotation Averaging: Relative rotations Rij are given, and absolute ..."
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Cited by 17 (4 self)
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• Single Rotation Averaging: Several estimates are obtained of a single rotation, which are then averaged to give the best estimate. • Multiple Rotation Averaging: Relative rotations Rij are given, and absolute
Projection Based MEstimators
, 2009
"... Random Sample Consensus (RANSAC) is the most widely used robust regression algorithm in computer vision. However, RANSAC has a few drawbacks which make it difficult to use for practical applications. Some of these problems have been addressed through improved sampling algorithms or better cost funct ..."
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Cited by 11 (3 self)
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Random Sample Consensus (RANSAC) is the most widely used robust regression algorithm in computer vision. However, RANSAC has a few drawbacks which make it difficult to use for practical applications. Some of these problems have been addressed through improved sampling algorithms or better cost functions, but an important difficulty still remains. The algorithm is not user independent, and requires knowledge of the scale of the inlier noise. We propose a new robust regression algorithm, the projection based Mestimator (pbM). The pbM algorithm is derived by building a connection to the theory of kernel density estimation and this leads to an improved cost function, which gives better performance. Furthermore, pbM is user independent and does not require any knowledge of the scale of noise corrupting the inliers. We propose a general framework for the pbM algorithm which can handle heteroscedastic data and multiple linear constraints on each data point through the use of Grassmann manifold theory. The performance of pbM is compared with RANSAC and MEstimator Sample Consensus (MSAC) on various real problems. It is shown that pbM gives better results than RANSAC and MSAC in spite of being user independent.
A new distance for scaleinvariant 3D shape recognition and registration
 In Proceedings of ICCV
, 2011
"... This paper presents a method for votebased 3D shape recognition and registration, in particular using mean shift on 3D pose votes in the space of direct similarity transforms for the first time. We introduce a new distance between poses in this space—the SRT distance. It is leftinvariant, unlike E ..."
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Cited by 11 (7 self)
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This paper presents a method for votebased 3D shape recognition and registration, in particular using mean shift on 3D pose votes in the space of direct similarity transforms for the first time. We introduce a new distance between poses in this space—the SRT distance. It is leftinvariant, unlike Euclidean distance, and has a unique, closedform mean, in contrast to Riemannian distance, so is fast to compute. We demonstrate improved performance over the state of the art in both recognition and registration on a real and challenging dataset, by comparing our distance with others in a mean shift framework, as well as with the commonly used Hough voting approach. 1.
Modedetection via medianshift
 ICCV
"... Medianshift is a mode seeking algorithm that relies on computing the median of local neighborhoods, instead of the mean. We further combine medianshift with Locality Sensitive Hashing (LSH) and show that the combined algorithm is suitable for clustering large scale, high dimensional data sets. In ..."
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Cited by 9 (1 self)
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Medianshift is a mode seeking algorithm that relies on computing the median of local neighborhoods, instead of the mean. We further combine medianshift with Locality Sensitive Hashing (LSH) and show that the combined algorithm is suitable for clustering large scale, high dimensional data sets. In particular, we propose a new mode detection step that greatly accelerates performance. In the past, LSH was used in conjunction with mean shift only to accelerate nearest neighbor queries. Here we show that we can analyze the density of the LSH bins to quickly detect potential mode candidates and use only them to initialize the medianshift procedure. We use the median, instead of the mean (or its discrete counterpart the medoid) because the median is more robust and because the median of a set is a point in the set. A median is well defined for scalars but there is no single agreed upon extension of the median to high dimensional data. We adopt a particular extension, known as the Tukey median, and show that it can be computed efficiently using random projections of the high dimensional data onto 1D lines, just like LSH, leading to a tightly integrated and efficient algorithm. 1.
Statistical analysis on manifolds and its applications to video analysis
, 2010
"... The analysis and interpretation of video data is an important component of modern vision applications such as biometrics, surveillance, motionsynthesis and webbased user interfaces. A common requirement among these very different applications is the ability to learn statistical models of appearance ..."
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Cited by 7 (3 self)
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The analysis and interpretation of video data is an important component of modern vision applications such as biometrics, surveillance, motionsynthesis and webbased user interfaces. A common requirement among these very different applications is the ability to learn statistical models of appearance and motion from a collection of videos, and then use them for recognizing actions or persons in a new video. These applications in video analysis require statistical inference methods to be devised on nonEuclidean spaces or more formally on manifolds. This chapter outlines a broad survey of applications in video analysis that involve manifolds. We develop the required mathematical tools needed to perform statistical inference on manifolds and show their effectiveness in real videounderstanding applications.
Semiintrinsic mean shift on riemannian manifolds
 Proc. European Conference on Computer Vision (ECCV), Lecture Notes in Computer Science
, 2012
"... Abstract. The original mean shift algorithm [1] on Euclidean spaces (MS) was extended in [2] to operate on general Riemannian manifolds. This extension is extrinsic (ExtMS) since the mode seeking is performed on the tangent spaces [3], where the underlying curvature is not fully considered (tangen ..."
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Abstract. The original mean shift algorithm [1] on Euclidean spaces (MS) was extended in [2] to operate on general Riemannian manifolds. This extension is extrinsic (ExtMS) since the mode seeking is performed on the tangent spaces [3], where the underlying curvature is not fully considered (tangent spaces are only valid in a small neighborhood). In [3] was proposed an intrinsic mean shift designed to operate on two particular Riemannian manifolds (IntGSMS), i.e. Grassmann and Stiefel manifolds (using manifolddedicated density kernels). It is then natural to ask whether mean shift could be intrinsically extended to work on a large class of manifolds. We propose a novel paradigm to intrinsically reformulate the mean shift on general Riemannian manifolds. This is accomplished by embedding the Riemannian manifold into a Reproducing Kernel Hilbert Space (RKHS) by using a general and mathematically wellfounded Riemannian kernel function, i.e. heat kernel [4]. The key issue is that when the data is implicitly mapped to the Hilbert space, the curvature of the manifold is taken into account (i.e. exploits the underlying information of the data). The inherent optimization is then performed on the embedded space. Theoretic analysis and experimental results demonstrate the promise and effectiveness of this novel paradigm. 1
CLUSTERING ON GRASSMANN MANIFOLDS VIA KERNEL EMBEDDING WITH APPLICATION TO ACTION ANALYSIS
"... With the aim of improving the clustering of data (such as image sequences) lying on Grassmann manifolds, we propose to embed the manifolds into Reproducing Kernel Hilbert Spaces. To this end, we define a measure of cluster distortion and embed the manifolds such that the distortion is minimised. We ..."
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Cited by 5 (2 self)
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With the aim of improving the clustering of data (such as image sequences) lying on Grassmann manifolds, we propose to embed the manifolds into Reproducing Kernel Hilbert Spaces. To this end, we define a measure of cluster distortion and embed the manifolds such that the distortion is minimised. We show that the optimal solution is a generalised eigenvalue problem that can be solved very efficiently. Experiments on several clustering tasks (including human action clustering) show that in comparison to the recent intrinsic Grassmann kmeans algorithm, the proposed approach obtains notable improvements in clustering accuracy, while also being several orders of magnitude faster.
Clusters and water flows: a novel approach to modal clustering through Morse theory
, 2014
"... The problem of finding groups in data (cluster analysis) has been extensively studied by researchers from the fields of Statistics and Computer Science, among others. However, despite its popularity it is widely recognized that the investigation of some theoretical aspects of clustering has been re ..."
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Cited by 5 (2 self)
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The problem of finding groups in data (cluster analysis) has been extensively studied by researchers from the fields of Statistics and Computer Science, among others. However, despite its popularity it is widely recognized that the investigation of some theoretical aspects of clustering has been relatively sparse. One of the main reasons for this lack of theoretical results is surely the fact that, unlike the situation with other statistical problems as regression or classification, for some of the cluster methodologies it is quite difficult to specify a population goal to which the databased clustering algorithms should try to get close. This paper aims to provide some insight into the theoretical foundations of the usual nonparametric approach to clustering, which understands clusters as regions of high density, by presenting an explicit formulation for the ideal population clustering.
Rolling riemannian manifolds to solve the multiclass classification problem
 In CVPR
"... Abstract In the past few years there has been a growing interest on geometric frameworks to learn supervised classification models on Riemannian manifolds ..."
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Abstract In the past few years there has been a growing interest on geometric frameworks to learn supervised classification models on Riemannian manifolds