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Normalized Cuts and Image Segmentation
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2000
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Authoritative Sources in a Hyperlinked Environment
 JOURNAL OF THE ACM
, 1999
"... The network structure of a hyperlinked environment can be a rich source of information about the content of the environment, provided we have effective means for understanding it. We develop a set of algorithmic tools for extracting information from the link structures of such environments, and repo ..."
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Cited by 3615 (12 self)
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The network structure of a hyperlinked environment can be a rich source of information about the content of the environment, provided we have effective means for understanding it. We develop a set of algorithmic tools for extracting information from the link structures of such environments, and report on experiments that demonstrate their effectiveness in a variety of contexts on the World Wide Web. The central issue we address within our framework is the distillation of broad search topics, through the discovery of “authoritative ” information sources on such topics. We propose and test an algorithmic formulation of the notion of authority, based on the relationship between a set of relevant authoritative pages and the set of “hub pages ” that join them together in the link structure. Our formulation has connections to the eigenvectors of certain matrices associated with the link graph; these connections in turn motivate additional heuristics for linkbased analysis.
Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
 Neural Computation
, 2003
"... Abstract One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space. Drawing on the corr ..."
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Cited by 1213 (16 self)
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Abstract One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed. 1 Introduction In many areas of artificial intelligence, information retrieval and data mining, one is often confronted with intrinsically low dimensional data lying in a very high dimensional space. Consider, for example, gray scale images of an object taken under fixed lighting conditions with a moving camera. Each such image would typically be represented by a brightness value at each pixel. If there were n 2
Community detection in graphs
, 2009
"... The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of th ..."
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Cited by 805 (1 self)
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such
Contour and Texture Analysis for Image Segmentation
, 2001
"... This paper provides an algorithm for partitioning grayscale images into disjoint regions of coherent brightness and texture. Natural images contain both textured and untextured regions, so the cues of contour and texture differences are exploited simultaneously. Contours are treated in the interveni ..."
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Cited by 406 (29 self)
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This paper provides an algorithm for partitioning grayscale images into disjoint regions of coherent brightness and texture. Natural images contain both textured and untextured regions, so the cues of contour and texture differences are exploited simultaneously. Contours are treated in the intervening contour framework, while texture is analyzed using textons. Each of these cues has a domain of applicability, so to facilitate cue combination we introduce a gating operator based on the texturedness of the neighborhood at a pixel. Having obtained a local measure of how likely two nearby pixels are to belong to the same region, we use the spectral graph theoretic framework of normalized cuts to find partitions of the image into regions of coherent texture and brightness. Experimental results on a wide range of images are shown.
Face recognition using laplacianfaces
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2005
"... Abstract—We propose an appearancebased face recognition method called the Laplacianface approach. By using Locality Preserving Projections (LPP), the face images are mapped into a face subspace for analysis. Different from Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA) wh ..."
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Cited by 385 (40 self)
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Abstract—We propose an appearancebased face recognition method called the Laplacianface approach. By using Locality Preserving Projections (LPP), the face images are mapped into a face subspace for analysis. Different from Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA) which effectively see only the Euclidean structure of face space, LPP finds an embedding that preserves local information, and obtains a face subspace that best detects the essential face manifold structure. The Laplacianfaces are the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the face manifold. In this way, the unwanted variations resulting from changes in lighting, facial expression, and pose may be eliminated or reduced. Theoretical analysis shows that PCA, LDA, and LPP can be obtained from different graph models. We compare the proposed Laplacianface approach with Eigenface and Fisherface methods on three different face data sets. Experimental results suggest that the proposed Laplacianface approach provides a better representation and achieves lower error rates in face recognition. Index Terms—Face recognition, principal component analysis, linear discriminant analysis, locality preserving projections, face manifold, subspace learning. 1
Contour Detection and Hierarchical Image Segmentation
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2010
"... This paper investigates two fundamental problems in computer vision: contour detection and image segmentation. We present stateoftheart algorithms for both of these tasks. Our contour detector combines multiple local cues into a globalization framework based on spectral clustering. Our segmentati ..."
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Cited by 384 (23 self)
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This paper investigates two fundamental problems in computer vision: contour detection and image segmentation. We present stateoftheart algorithms for both of these tasks. Our contour detector combines multiple local cues into a globalization framework based on spectral clustering. Our segmentation algorithm consists of generic machinery for transforming the output of any contour detector into a hierarchical region tree. In this manner, we reduce the problem of image segmentation to that of contour detection. Extensive experimental evaluation demonstrates that both our contour detection and segmentation methods significantly outperform competing algorithms. The automatically generated hierarchical segmentations can be interactively refined by userspecified annotations. Computation at multiple image resolutions provides a means of coupling our system to recognition applications.
Segmentation using eigenvectors: A unifying view
 In ICCV
, 1999
"... Automatic grouping and segmentation of images remains a challenging problem in computer vision. Recently, a number of authors have demonstrated good performance on this task using methods that are based on eigenvectors of the a nity matrix. These approaches are extremely attractive in that they are ..."
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Cited by 383 (1 self)
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Automatic grouping and segmentation of images remains a challenging problem in computer vision. Recently, a number of authors have demonstrated good performance on this task using methods that are based on eigenvectors of the a nity matrix. These approaches are extremely attractive in that they are based on simple eigendecomposition algorithms whose stability is well understood. Nevertheless, the use of eigendecompositions in the context of segmentation is far from well understood. In this paper we give a unied treatment of these algorithms, and show the close connections between them while highlighting their distinguishing features. We then prove results on eigenvectors of block matrices that allow us to analyze the performance of these algorithms in simple grouping settings. Finally, we use our analysis to motivate a variation on the existing methods that combines aspects from di erent eigenvector segmentation algorithms. We illustrate our analysis with results on real and synthetic images. Human perceiving a scene can often easily segment it into coherent segments or groups. There has been a tremendous amount of e ort devoted to achieving the same level of performance in computer vision. In many cases, this is done by associating with each pixel a feature vector (e.g. color, motion, texture, position) and using a clustering or grouping algorithm on these feature vectors. Perhaps the cleanest approach to segmenting points in feature space is based on mixture models in which one assumes the data were generated by multiple processes and estimates the parameters of the processes and the number of components in the mixture. The assignment of points to clusters can then be easily performed by calculating the posterior probability ofa point belonging to a cluster. Despite the elegance of this approach, the estimation process leads to a notoriously di cult optimization. The frequently used EM algorithm [3] often converges to a local maximum that depends on the initial conditions. Recently, anumber of authors [11, 10, 8, 9, 2] have suggested alternative segmentation methods that are based on eigenvectors of the (possibly normalized) \a nity matrix". Figure 1a shows two clusters of points and gure 1b shows the a nity matrix de ned by:
Expander Flows, Geometric Embeddings and Graph Partitioning
 IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING
, 2004
"... We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a ..."
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Cited by 325 (18 self)
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We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.
Spectral grouping using the Nyström method
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2004
"... Spectral graph theoretic methods have recently shown great promise for the problem of image segmentation. However, due to the computational demands of these approaches, applications to large problems such as spatiotemporal data and high resolution imagery have been slow to appear. The contribution ..."
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Cited by 314 (1 self)
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Spectral graph theoretic methods have recently shown great promise for the problem of image segmentation. However, due to the computational demands of these approaches, applications to large problems such as spatiotemporal data and high resolution imagery have been slow to appear. The contribution of this paper is a method that substantially reduces the computational requirements of grouping algorithms based on spectral partitioning making it feasible to apply them to very large grouping problems. Our approach is based on a technique for the numerical solution of eigenfunction problems knownas the Nyström method. This method allows one to extrapolate the complete grouping solution using only a small number of "typical" samples. In doing so, we leverage the fact that there are far fewer coherent groups in a scene than pixels.