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60
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 316 (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.
Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 148 (16 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusion-like operators, in any dimension, on manifolds, graphs, and in non-homogeneous media. In this case our construction can be viewed as a far-reaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the non-standard wavelet representation of Calderón-Zygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical Littlewood-Paley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
Learning spectral clustering
, 2003
"... Spectral clustering refers to a class of techniques which rely on the eigenstructure of a similarity matrix to partition points into disjoint clusters with points in the same cluster having high similarity and points in different clusters having low similarity. In this paper, we derive a new cost fu ..."
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Cited by 118 (4 self)
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Spectral clustering refers to a class of techniques which rely on the eigenstructure of a similarity matrix to partition points into disjoint clusters with points in the same cluster having high similarity and points in different clusters having low similarity. In this paper, we derive a new cost function for spectral clustering based on a measure of error between a given partition and a solution of the spectral relaxation of a minimum normalized cut problem. Minimizing this cost function with respect to the partition leads to a new spectral clustering algorithm. Minimizing with respect to the similarity matrix leads to an algorithm for learning the similarity matrix. We develop a tractable approximation of our cost function that is based on the power method of computing eigenvectors. 1
Spatio-Temporal Segmentation of Video by Hierarchical Mean Shift Analysis
- Center for Automat. Res., U. of Md, College Park
, 2002
"... We describe a simple new technique for spatio-temporal segmentation of video sequences. Each pixel of a 3D space-time video stack is mapped to a 7D feature point whose coordinates include three color components, two motion angle components and two motion position components. The clustering of these ..."
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Cited by 82 (4 self)
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We describe a simple new technique for spatio-temporal segmentation of video sequences. Each pixel of a 3D space-time video stack is mapped to a 7D feature point whose coordinates include three color components, two motion angle components and two motion position components. The clustering of these feature points provides color segmentation and motion segmentation, as well as a consistent labeling of regions over time which amounts to region tracking. For this task we have adopted a hierarchical clustering method which operates by repeatedly applying mean shift analysis over increasing large ranges, using at each pass the cluster centers of the previous pass, with weights equal to the counts of the points that contributed to the clusters. This technique has lower complexity for large mean shift radii than regular mean shift analysis because it can use binary tree structures more efficiently during range search. In addition, it provides a hierarchical segmentation of the data. Applications include video compression and compact descriptions of video sequences for video indexing and retrieval applications.
Learning spectral clustering, with application to speech separation
- JOURNAL OF MACHINE LEARNING RESEARCH
, 2006
"... Spectral clustering refers to a class of techniques which rely on the eigenstructure of a similarity matrix to partition points into disjoint clusters, with points in the same cluster having high similarity and points in different clusters having low similarity. In this paper, we derive new cost fun ..."
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Cited by 70 (6 self)
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Spectral clustering refers to a class of techniques which rely on the eigenstructure of a similarity matrix to partition points into disjoint clusters, with points in the same cluster having high similarity and points in different clusters having low similarity. In this paper, we derive new cost functions for spectral clustering based on measures of error between a given partition and a solution of the spectral relaxation of a minimum normalized cut problem. Minimizing these cost functions with respect to the partition leads to new spectral clustering algorithms. Minimizing with respect to the similarity matrix leads to algorithms for learning the similarity matrix from fully labelled datasets. We apply our learning algorithm to the blind one-microphone speech separation problem, casting the problem as one of segmentation of the spectrogram.
Spectral Partitioning with Indefinite Kernels Using the Nyström Extension
- European Conference on Computer Vision 2002
, 2002
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Geometric harmonics: a novel tool for multiscale out-of-sample extension of empirical functions
- Appl. Comp. Harm. Anal
"... Abstract We describe a simple scheme, based on the Nyström method, for extending empirical functions f defined on a set X to a larger setX. The extension process that we describe involves the construction of a specific family of functions that we term geometric harmonics. These functions constitute ..."
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Cited by 50 (11 self)
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Abstract We describe a simple scheme, based on the Nyström method, for extending empirical functions f defined on a set X to a larger setX. The extension process that we describe involves the construction of a specific family of functions that we term geometric harmonics. These functions constitute a generalization of the prolate spheroidal wave functions of Slepian in the sense that they are optimally concentrated on X. We study the case when X is a submanifold of R n in greater detail. In this situation, any empirical function f on X can be characterized by its decomposition over the intrinsic Fourier modes, i.e., the eigenfunctions of the LaplaceBeltrami operator, and we show that this intrinsic frequency spectrum determines the largest domain of extension of f to the entire space R n . Our analysis relates the complexity of the function on the training set to the scale of extension off this set. This approach allows us to present a novel multiscale extension scheme for empirical functions.
Probabilistic Space-Time Video Modeling via Piecewise GMM
- IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2004
"... Abstract—In this paper, we describe a statistical video representation and modeling scheme. Video representation schemes are needed to segment a video stream into meaningful video-objects, useful for later indexing and retrieval applications. In the proposed methodology, unsupervised clustering via ..."
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Cited by 46 (0 self)
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Abstract—In this paper, we describe a statistical video representation and modeling scheme. Video representation schemes are needed to segment a video stream into meaningful video-objects, useful for later indexing and retrieval applications. In the proposed methodology, unsupervised clustering via Gaussian mixture modeling extracts coherent space-time regions in feature space, and corresponding coherent segments (video-regions) in the video content. A key feature of the system is the analysis of video input as a single entity as opposed to a sequence of separate frames. Space and time are treated uniformly. The probabilistic space-time video representation scheme is extended to a piecewise GMM framework in which a succession of GMMs are extracted for the video sequence, instead of a single global model for the entire sequence. The piecewise GMM framework allows for the analysis of extended video sequences and the description of nonlinear, nonconvex motion patterns. The extracted space-time regions allow for the detection and recognition of video events. Results of segmenting video content into static versus dynamic video regions and video content editing are presented. Index Terms—Video representation, video segmentation, detection of events in video, Gaussian mixture model. 1
Evaluation of Super-Voxel Methods for Early Video Processing
"... Supervoxel segmentation has strong potential to be incorporated into early video analysis as superpixel segmentation has in image analysis. However, there are many plausible supervoxel methods and little understanding as to when and where each is most appropriate. Indeed, we are not aware of a singl ..."
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Cited by 43 (7 self)
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Supervoxel segmentation has strong potential to be incorporated into early video analysis as superpixel segmentation has in image analysis. However, there are many plausible supervoxel methods and little understanding as to when and where each is most appropriate. Indeed, we are not aware of a single comparative study on supervoxel segmentation. To that end, we study five supervoxel algorithms in the context of what we consider to be a good supervoxel: namely, spatiotemporal uniformity, object/region boundary detection, region compression and parsimony. For the evaluation we propose a comprehensive suite of 3D volumetric quality metrics to measure these desirable supervoxel characteristics. We use three benchmark video data sets with a variety of content-types and varying amounts of human annotations. Our findings have led us to conclusive evidence that the hierarchical graph-based and segmentation by weighted aggregation methods perform best and almost equally-well on nearly all the metrics and are the methods of choice given our proposed assumptions. 1.
Normalized cuts in 3-d for spinal mri segmentation
- IEEE Trans Med Imaging
, 2004
"... Abstract—Segmentation of medical images has become an indispensable process to perform quantitative analysis of images of human organs and their functions. Normalized Cuts is a spectral graph theoretic method that readily admits combinations of different features for image segmentation. The computat ..."
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Cited by 28 (0 self)
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Abstract—Segmentation of medical images has become an indispensable process to perform quantitative analysis of images of human organs and their functions. Normalized Cuts is a spectral graph theoretic method that readily admits combinations of different features for image segmentation. The computational demand imposed by Normalized Cuts has been successfully alleviated with the Nyström approximation method for applications different than medical imaging. In this paper we discuss the application of Normalized Cuts with the Nyström approximation method to segment vertebral bodies from sagittal T1-weighted magnetic resonance images of the spine. The magnetic resonance images were pre-processed by the anisotropic diffusion algorithm, and 3D local histograms of brightness was chosen as the segmentation feature. Results of the segmentation as well as limitations and challenges in this area are presented. Index Terms — Magnetic resonance imaging (MRI), Normalized Cuts (NCut), Nyström approximation method, segmentation, spine.
