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## Spectral grouping using the Nyström method (2004)

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Venue: | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 316 - 1 self |

### Citations

6608 |
Neural Networks for Pattern Recognition
- Bishop
- 1996
(Show Context)
Citation Context ...stering [15]. As discussed in [9], pairwise grouping methods present an appealing alternative to central grouping. Central grouping techniques such as k-means or Gaussian Mixture Model fitting via EM =-=[6]-=- tend to be computationally efficient since they only require one to compare the image pixels to a small set of cluster prototypes. However, they have the significant drawback of implicitly assuming t... |

3788 | Normalized cuts and image segmentation
- Shi, Malik
(Show Context)
Citation Context ...wo pixels captures their degree of similarity as measured by one or more cues. The pixels can then be grouped based on the set of pairwise affinities using methods such as spectral graph partitioning =-=[30]-=-, [32], [22], [26], [28], [20], deterministic annealing [25], or stochastic clustering [15]. As discussed in [9], pairwise grouping methods present an appealing alternative to central grouping. Centra... |

1709 | On Spectral clustering: analysis and an algorithm
- Ng, Jordan, et al.
- 2001
(Show Context)
Citation Context ...e of similarity as measured by one or more cues. The pixels can then be grouped based on the set of pairwise affinities using methods such as spectral graph partitioning [30], [32], [22], [26], [28], =-=[20]-=-, deterministic annealing [25], or stochastic clustering [15]. As discussed in [9], pairwise grouping methods present an appealing alternative to central grouping. Central grouping techniques such as ... |

1573 | Nonlinear component analysis as a kernel eigenvalue problem.
- Scholkopf, Smola, et al.
- 1998
(Show Context)
Citation Context ... by [34] for fast approximate Gaussian process classification and regression. As noted in [34], this approximation method directly corresponds to the kernel PCA features space projection technique of =-=[27]-=-. A generalization of these ideas on low-rank approximation to the SVD is studied in [14], [13]. One remaining detail is that the columns of U are not orthogonal. The process of orthogonalizing the so... |

1538 |
Spectral Graph Theory,
- Chung
- 1997
(Show Context)
Citation Context ...nd B is then given as follows: NCutðA; BÞ 2 cutðA; BÞ volðAÞkvolðBÞ ; wherekdenotes the harmonic mean. 2 We wish to find A and B such that NCutðA; BÞ is minimized. Appealing to spectral grap=-=h theory [11]-=-, Shi and Malik [30] showed that an approximate solution may be obtained by thresholding the eigenvector corresponding to the second smallest eigenvalue 2 of the normalized LaplacianL, which is define... |

904 | Spatiotemporal energy models for the perception of motion.
- Adelson, Bergen
- 1985
(Show Context)
Citation Context ...olume into regions that are coherent with respect to the various grouping cues. The insight of considering a video signal as three dimensional for purposes of analysis goes back to Adelson and Bergen =-=[2]-=- and Baker et al. [7] and is supported by evidence from psychophysics [16]. Unified treatment of the spatial and temporal domains is also appealing as it could solve some of the well known problems in... |

404 | Contour and texture analysis for image segmentation.
- Malik, Belongie, et al.
- 2001
(Show Context)
Citation Context ...ing 100 samples. In this case, each pixel in the image is associated with the nearest element in a small alphabet of prototypical linear filter responses using vector-quantitization (see Malik et al. =-=[18]). H-=-istograms of these “texton labels” are computed over an 9 9 pixel window and again compared with the 2 -distance. 5.2 Spatio-Temporal Segmentation One method for combining both static image cues a... |

380 | Segmentation using eigenvectors: A unifying view.
- Weiss
- 1999
(Show Context)
Citation Context ...els captures their degree of similarity as measured by one or more cues. The pixels can then be grouped based on the set of pairwise affinities using methods such as spectral graph partitioning [30], =-=[32]-=-, [22], [26], [28], [20], deterministic annealing [25], or stochastic clustering [15]. As discussed in [9], pairwise grouping methods present an appealing alternative to central grouping. Central grou... |

253 | Epipolar-plane image analysis: an approach to determining structure from motion,
- Bolles, Baker, et al.
- 1987
(Show Context)
Citation Context ...at are coherent with respect to the various grouping cues. The insight of considering a video signal as three dimensional for purposes of analysis goes back to Adelson and Bergen [2] and Baker et al. =-=[7]-=- and is supported by evidence from psychophysics [16]. Unified treatment of the spatial and temporal domains is also appealing as it could solve some of the well known problems in grouping schemes bas... |

237 | Fast Monte-Carlo algorithms for finding low rank approximations,
- Frieze, Kannan, et al.
- 2004
(Show Context)
Citation Context ... [34], this approximation method directly corresponds to the kernel PCA features space projection technique of [27]. A generalization of these ideas on low-rank approximation to the SVD is studied in =-=[14]-=-, [13]. One remaining detail is that the columns of U are not orthogonal. The process of orthogonalizing the solution can proceed in two different ways depending on whether A is positive definite. 3.3... |

203 |
Harmonic Analysis on Semigroups,
- Berg, Christensen, et al.
- 1984
(Show Context)
Citation Context ...onstant times e Qij . Since the exponential of as224 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 2, FEBRUARY 2004 positive definite function is also positive definite =-=[5]-=-, we have established that e 2 ij is positive definite. tu ACKNOWLEDGMENTS The authors would like to thank Gianluca Donato, William Kahan, Andrew Ng, Jianbo Shi, Yair Weiss, Stella Yu, and Alice Zheng... |

179 | Motion segmentation and tracking using normalized cuts
- Shi, Malik
- 1998
(Show Context)
Citation Context ...problem has been to use a sparse, approximate version of W in which each element is connected only to a few of its nearby neighbors in the image plane and all other connections are assumed to be zero =-=[29]-=-. While this makes it possible to employ efficient, sparse eigensolvers (e.g., Lanczos), the effects of this process are not well understood. Our proposed alternative based on sampling allows all affi... |

175 | A unified mixture framework for motion segmentation: incorporating spatial coherence and estimating the number of models,”
- Weiss, Adelson
- 1996
(Show Context)
Citation Context ... [16]. Unified treatment of the spatial and temporal domains is also appealing as it could solve some of the well known problems in grouping schemes based on motion alone (e.g., layered motion models =-=[33]-=-, [31]). For example, color or brightness cues can help to segment untextured regions for which the motion cues are ambiguous and contour cues can impose sharp boundaries where optical flow algorithms... |

173 | A factorization approach to grouping.
- Perona, Freeman
- 1998
(Show Context)
Citation Context ...ptures their degree of similarity as measured by one or more cues. The pixels can then be grouped based on the set of pairwise affinities using methods such as spectral graph partitioning [30], [32], =-=[22]-=-, [26], [28], [20], deterministic annealing [25], or stochastic clustering [15]. As discussed in [9], pairwise grouping methods present an appealing alternative to central grouping. Central grouping t... |

172 | Smoothness in layers: Motion segmentation using nonparametric mixture estimation.
- Weiss
- 1997
(Show Context)
Citation Context ... Unified treatment of the spatial and temporal domains is also appealing as it could solve some of the well known problems in grouping schemes based on motion alone (e.g., layered motion models [33], =-=[31]-=-). For example, color or brightness cues can help to segment untextured regions for which the motion cues are ambiguous and contour cues can impose sharp boundaries where optical flow algorithms tend ... |

155 |
Numerical Treatment of Integral Equations
- Baker
- 1977
(Show Context)
Citation Context ... proposed alternative based on sampling allows all affinities to be retained at the expense of some numerical accuracy in their values. 3 THE NYSTRöM EXTENSION 3.1 Background The Nyström method [21]=-=, [3], [23-=-] is a technique for finding numerical approximations to eigenfunction problems of the form Z b a Wðx; yÞ ðyÞdy ðxÞ: We can approximate this integral equation by evaluating it at a set of evenly... |

133 |
An algorithm for associating the features of two images.
- Scott, Lonquiet-Higgins
- 1991
(Show Context)
Citation Context ... degree of similarity as measured by one or more cues. The pixels can then be grouped based on the set of pairwise affinities using methods such as spectral graph partitioning [30], [32], [22], [26], =-=[28]-=-, [20], deterministic annealing [25], or stochastic clustering [15]. As discussed in [9], pairwise grouping methods present an appealing alternative to central grouping. Central grouping techniques su... |

105 |
Quantitative measures of change based on feature organization: Eigenvalues and eigenvectors.
- Sarkar, Boyer
- 1996
(Show Context)
Citation Context ... their degree of similarity as measured by one or more cues. The pixels can then be grouped based on the set of pairwise affinities using methods such as spectral graph partitioning [30], [32], [22], =-=[26]-=-, [28], [20], deterministic annealing [25], or stochastic clustering [15]. As discussed in [9], pairwise grouping methods present an appealing alternative to central grouping. Central grouping techniq... |

102 | Non-parametric similarity measures for unsupervised texture segmentation and image retrieval.
- Puzicha, Hofmann, et al.
- 1997
(Show Context)
Citation Context ... one or more cues. The pixels can then be grouped based on the set of pairwise affinities using methods such as spectral graph partitioning [30], [32], [22], [26], [28], [20], deterministic annealing =-=[25]-=-, or stochastic clustering [15]. As discussed in [9], pairwise grouping methods present an appealing alternative to central grouping. Central grouping techniques such as k-means or Gaussian Mixture Mo... |

64 | Sampling techniques for kernel methods.
- Scholkopf, Achlioptas, et al.
- 2002
(Show Context)
Citation Context ...re is an oracle that allows one to avoid computing small entries, this still requires OðN 2 Þ affinity calculations which can be quite expensive. A more likely alternative, analyzed in some detail b=-=y [1]-=- is to zero out random entries in the matrix. Both of these options allow one to employ a sparse matrix representation and corresponding sparse eigensolver (Lanczos/Arnoldi) which can improve signific... |

60 | Efficient spatiotemporal grouping using the Nyström method.
- Fowlkes, Belongie, et al.
- 2001
(Show Context)
Citation Context ...mage size is 128 192 and the histogram window size is 5 5. Color quantization was performed as in [24] with eight bins. Since the e 2 ij kernel is positive definite, we can use the one-shot method of =-=[12].-=- (a) Original image. (b) Nyström-NCut leading eigenvectors using 100 random samples. Eigenvector images are sorted by eigenvalue. (c) Segment-label image obtained via k-means clustering on the eigenv... |

56 | Spectral partitioning with indefinite kernels using the Nystrom extension, in:
- Belongie, Fowlkes, et al.
- 2002
(Show Context)
Citation Context ...nd S and then we orthogonalize US to produce V and . Although this approach is applicable in general, the additional Oðn3Þ step required leads to an increased loss of significant figures. As noted i=-=n [4],-=- it is therefore expedient to know when the one-shot method can be applied, i.e., when a given kernel is positive definite. 3.4 Application to Normalized Cut To apply the Nyström approximation to NCu... |

34 |
The emergence of visual objects in space–time.
- Gepshtein, Kubovy
- 2000
(Show Context)
Citation Context ... cues. The insight of considering a video signal as three dimensional for purposes of analysis goes back to Adelson and Bergen [2] and Baker et al. [7] and is supported by evidence from psychophysics =-=[16]-=-. Unified treatment of the spatial and temporal domains is also appealing as it could solve some of the well known problems in grouping schemes based on motion alone (e.g., layered motion models [33],... |

30 |
Stochastic image segmentation by typical cuts
- Gdalyahu, Weinshall, et al.
- 1999
(Show Context)
Citation Context ...an then be grouped based on the set of pairwise affinities using methods such as spectral graph partitioning [30], [32], [22], [26], [28], [20], deterministic annealing [25], or stochastic clustering =-=[15]-=-. As discussed in [9], pairwise grouping methods present an appealing alternative to central grouping. Central grouping techniques such as k-means or Gaussian Mixture Model fitting via EM [6] tend to ... |

25 |
Svms for histogram based image classification
- Chapelle, Haffner, et al.
- 1999
(Show Context)
Citation Context ...e shown in the left column. Each column shows slices through a space-time segment. APPENDIX B PROOF OF POSITIVE DEFINITENESS OF e We now prove that e 2 ij 2 ij is positive definite (as conjectured by =-=[10]). We assume throug-=-hout that hiðkÞ > 0 8i; k. We begin by considering the 2 ij term by itself. Noting that ðhiðkÞ hjðkÞÞ 2 ðhiðkÞþhjðkÞÞ 2 write 2 ij as 2 ij 1 2XK k1 4hiðkÞhiðkÞ, we can rehiðkÞhjð... |

22 |
Using the Nyström Method to Speed Up Kernel
- Williams, Seeger
(Show Context)
Citation Context ...ified as the norm of the Schur complementkC B T A 1 Bk. The size of this norm is governed by the extent to which C is spanned by the rows of B. The Nyström approximation has been used in this form by=-= [34]-=- for fast approximate Gaussian process classification and regression. As noted in [34], this approximation method directly corresponds to the kernel PCA features space projection technique of [27]. A ... |

19 |
Über die praktische auflösung von linearen integralgleichungen mit anwendungen auf randwertaufgaben der potentialtheorie
- Nyström
- 1928
(Show Context)
Citation Context ...d. Our proposed alternative based on sampling allows all affinities to be retained at the expense of some numerical accuracy in their values. 3 THE NYSTRöM EXTENSION 3.1 Background The Nyström metho=-=d [21], [3]-=-, [23] is a technique for finding numerical approximations to eigenfunction problems of the form Z b a Wðx; yÞ ðyÞdy ðxÞ: We can approximate this integral equation by evaluating it at a set of e... |

18 |
Convolution kernels on discrete structure (Tech. Rep
- Haussler
- 1999
(Show Context)
Citation Context ... pairwise methods can readily employ a suitable affinity function such as the 2 -distance. Affinities can even be defined between features with no natural vector space structure (e.g., string kernels =-=[17]-=-). The drawback of pairwise methods is the requirement of comparing all possible pairs of pixels in an image. Processing short video sequences or the output of inexpensive multimegapixel digital camer... |

12 | Model-Based Halftoning for Color Image Segmentation
- Puzicha, Belongie
- 2000
(Show Context)
Citation Context ...egmentation of tiger image based on Gaussian weighted 2-distance between local color histograms. The image size is 128 192 and the histogram window size is 5 5. Color quantization was performed as in =-=[24] -=-with eight bins. Since the e 2 ij kernel is positive definite, we can use the one-shot method of [12]. (a) Original image. (b) Nyström-NCut leading eigenvectors using 100 random samples. Eigenvector ... |

8 |
Data clustering and learning. In: The handbook of brain theory and neural networks (Arbib MA, ed), pp 278–282. Cambridege
- Buhmann
- 1998
(Show Context)
Citation Context ...ed on the set of pairwise affinities using methods such as spectral graph partitioning [30], [32], [22], [26], [28], [20], deterministic annealing [25], or stochastic clustering [15]. As discussed in =-=[9]-=-, pairwise grouping methods present an appealing alternative to central grouping. Central grouping techniques such as k-means or Gaussian Mixture Model fitting via EM [6] tend to be computationally ef... |

3 | POCV99 breakout session report: Spatiotemporal grouping - Boyer, Fagerstrom, et al. - 2000 |

2 | SVMs for histogram based image classication - Chapelle, ner, et al. - 1999 |

1 |
Quick Approximation to Matricies and
- Frieze, Kannan
- 1999
(Show Context)
Citation Context ... this approximation method directly corresponds to the kernel PCA features space projection technique of [27]. A generalization of these ideas on low-rank approximation to the SVD is studied in [14], =-=[13]-=-. One remaining detail is that the columns of U are not orthogonal. The process of orthogonalizing the solution can proceed in two different ways depending on whether A is positive definite. 3.3 Metho... |

1 | Quick approximation to matricies and applications. Combinatorica - Frieze, Kannan - 1999 |

1 |
The Numerical Treatment of Integral Equations.Oxford
- Baker
- 1977
(Show Context)
Citation Context ...roposed alternative based on sampling allows all affinities to be retained at the expense of some numerical accuracy in their values. 3 THE NYSTRöM EXTENSION 3.1 Background The Nyström method [21], =-=[3]-=-, [23] is a technique for finding numerical approximations to eigenfunction problems of the form Z b a Wðx; yÞðyÞdy ðxÞ: We can approximate this integral equation by evaluating it at a set of eve... |