## Statistical analysis on stiefel and grassmann manifolds with applications in computer vision (2008)

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Venue: | CVPR |

Citations: | 20 - 4 self |

### BibTeX

@TECHREPORT{Turaga08statisticalanalysis,

author = {Pavan Turaga and Ashok Veeraraghavan and Rama Chellappa},

title = {Statistical analysis on stiefel and grassmann manifolds with applications in computer vision},

institution = {CVPR },

year = {2008}

}

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### Abstract

Many applications in computer vision and pattern recognition involve drawing inferences on certain manifold-valued parameters. In order to develop accurate inference algorithms on these manifolds we need to a) understand the geometric structure of these manifolds b) derive appropriate distance measures and c) develop probability distribution functions (pdf) and estimation techniques that are consistent with the geometric structure of these manifolds. In this paper, we consider two related manifolds- the Stiefel manifold and the Grassmann manifold, which arise naturally in several vision applications such as spatio-temporal modeling, affine invariant shape analysis, image matching and learning theory. We show how accurate statistical characterization that reflects the geometry of these manifolds allows us to design efficient algorithms that compare favorably to the state of the art in these very different applications. In particular, we describe appropriate distance measures and parametric and non-parametric density estimators on these manifolds. These methods are then used to learn class conditional densities for applications such as activity recognition, video based face recognition and shape classification.

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Citation Context ...mation can then be performed in order to solve problems such as video classification, clustering and retrieval. 2. Shape Analysis: Representations and recognition of shapes is a well understood field =-=[15, 12]-=-. The shape observed in an image is a perspective projection of the original shape. In order to account for this, shape theory studies the equivalent class of all configurations that can be obtained b... |

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Citation Context ...0] where data naturally lies either on the Stiefel or the Grassmann manifold. The geometric properties of the Stiefel and Grassmann manifolds are well understood and we refer the interested reader to =-=[13, 1]-=- for specific details. The scope of this paper is not restricted to the differential geometry of these manifolds. Instead, we are interested in statistical modeling and inference tools on these manifo... |

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Citation Context ...nditional densities for applications such as activity recognition, video based face recognition and shape classification. 1. Introduction Many applications in computer vision such as dynamic textures =-=[24, 9]-=-, human activity modeling and recognition [7, 30], video based face recognition [2], shape analysis [14, 20] involve learning and recognition of patterns from exemplars which lie on certain manifolds.... |

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Citation Context ...mation can then be performed in order to solve problems such as video classification, clustering and retrieval. 2. Shape Analysis: Representations and recognition of shapes is a well understood field =-=[15, 12]-=-. The shape observed in an image is a perspective projection of the original shape. In order to account for this, shape theory studies the equivalent class of all configurations that can be obtained b... |

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Citation Context ...ar shape manifolds was presented in [27] where statistics are learnt on the manifold’s tangent space. Classification on Riemannian manifolds have also been explored in the vision community such as in =-=[28, 29]-=-. Organization of the paper: In section 2, we present motivating examples where Stiefel and Grassmann manifolds arise naturally in vision applications. In section 3, a brief review of statistical mode... |

117 | Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements
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Citation Context ...hese manifolds and describe learning algorithms for estimating the distribution from data. Prior Work: Statistical methods on manifolds have been studied for several years in the statistics community =-=[5, 21, 22]-=-. A compilation of research results on statistical analysis on the Stiefel and Grassmann manifolds can be found in [10]. Their utility in practical applications has not yet been fully explored. Theore... |

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Citation Context ...crete time Lyapunov equation. 4.1.3 Video-Based Face Recognition Video-based face recognition (FR) by modeling the ‘cropped video’ either as dynamical models ([2]) or as a collection of PCA subspaces =-=[16]-=- have recently gained popularity because of their ability to recognize faces from lowTest condition System Distance Procrustes Kernel density 1 Gallery1,Probe2 81.25 93.75 93.75 2 Gallery2,Probe1 68.... |

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Citation Context ...xm) be a configuration of m points where each xi ∈ R 2 .Letγ be a transformation on R 2 . For example, γ could belong to the affine group, linear group, projecAlgorithm Rank 1 Rank 2 Rank 3 Rank 4 SC =-=[17]-=- 20/40 10/40 11/40 5/40 IDSC [17] 40/40 34/40 35/40 27/40 Hashing [8] 40/40 38/40 33/40 20/40 Grassmann Pro- 38/40 30/40 23/40 17/40 crustes Table 3. Retrieval experiment on articulation dataset. Last... |

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Citation Context ...or each test instance using these class conditional distributions (denoted as Kernel-Stiefel). 4.1.2 Activity Recognition We performed a recognition experiment on the publicly available INRIA dataset =-=[31]-=-. The dataset consists of 10 actors performing 11 actions, each action executed 3 times at varying rates while freely changing orientation. We used the view-invariant representation and features as pr... |

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Citation Context ...ARMA model closed form solutions for learning the model parameters have been proposed in [19, 24] and are widely used. The parameters of the model are known to lie on the Stiefel manifold as noted in =-=[23]-=-. Given several instances, current approaches involve computing the distance between them using well-known distance measures [11] followed by nearest neighbor classification. Instead, given several in... |

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Citation Context ...vity recognition, video based face recognition and shape classification. 1. Introduction Many applications in computer vision such as dynamic textures [24, 9], human activity modeling and recognition =-=[7, 30]-=-, video based face recognition [2], shape analysis [14, 20] involve learning and recognition of patterns from exemplars which lie on certain manifolds. Given a database of examples and a query, the fo... |

64 | Matching shape sequences in video with an application to human movement analysis
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Citation Context ...vity recognition, video based face recognition and shape classification. 1. Introduction Many applications in computer vision such as dynamic textures [24, 9], human activity modeling and recognition =-=[7, 30]-=-, video based face recognition [2], shape analysis [14, 20] involve learning and recognition of patterns from exemplars which lie on certain manifolds. Given a database of examples and a query, the fo... |

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Citation Context ...0] where data naturally lies either on the Stiefel or the Grassmann manifold. The geometric properties of the Stiefel and Grassmann manifolds are well understood and we refer the interested reader to =-=[13, 1]-=- for specific details. The scope of this paper is not restricted to the differential geometry of these manifolds. Instead, we are interested in statistical modeling and inference tools on these manifo... |

49 | Statistical shape analysis: Clustering, learning, and testing
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Citation Context ... exploited the geometry of the Grassmann manifold for subspace tracking in array signal processing applications. Statistical learning of shape classes using nonlinear shape manifolds was presented in =-=[27]-=- where statistics are learnt on the manifold’s tangent space. Classification on Riemannian manifolds have also been explored in the vision community such as in [28, 29]. Organization of the paper: In ... |

43 |
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Citation Context ...hese manifolds and describe learning algorithms for estimating the distribution from data. Prior Work: Statistical methods on manifolds have been studied for several years in the statistics community =-=[5, 21, 22]-=-. A compilation of research results on statistical analysis on the Stiefel and Grassmann manifolds can be found in [10]. Their utility in practical applications has not yet been fully explored. Theore... |

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Citation Context ...nditional densities for applications such as activity recognition, video based face recognition and shape classification. 1. Introduction Many applications in computer vision such as dynamic textures =-=[24, 9]-=-, human activity modeling and recognition [7, 30], video based face recognition [2], shape analysis [14, 20] involve learning and recognition of patterns from exemplars which lie on certain manifolds.... |

39 | A system identification approach for video-based face recognition
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Citation Context ...ognition and shape classification. 1. Introduction Many applications in computer vision such as dynamic textures [24, 9], human activity modeling and recognition [7, 30], video based face recognition =-=[2]-=-, shape analysis [14, 20] involve learning and recognition of patterns from exemplars which lie on certain manifolds. Given a database of examples and a query, the following two questions are usually ... |

31 |
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Citation Context ...ar shape manifolds was presented in [27] where statistics are learnt on the manifold’s tangent space. Classification on Riemannian manifolds have also been explored in the vision community such as in =-=[28, 29]-=-. Organization of the paper: In section 2, we present motivating examples where Stiefel and Grassmann manifolds arise naturally in vision applications. In section 3, a brief review of statistical mode... |

30 |
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Citation Context ...rix. Thus, the C matrix lies on the Stiefel manifold. For comparison of models, the most commonly used distance metric is based on subspace angles between columns spaces of the observability matrices =-=[11]-=- (denoted as Subspace Angles). The extended observability matrix for a model (A, C) is given by O T ∞ = [ C T , (CA) T , (CA 2 ) T ,...(CA n ) T ... ] (10) Thus, a linear dynamical system can be alter... |

29 | Incremental PCA for on-line visual learning and recognition - Artac, Jogan, et al. - 2002 |

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Citation Context ...o articulation invariant descriptors were used. tive group etc. Let A(γ,(x1,...,xm)) = (γ(x1),...,γ(xm)) (12) be the action of γ on the point configuration. In particular, the affine shape space [14] =-=[25]-=- is very important because the effect of the camera location and orientation can be approximated as affine transformations on the original base shape. The affine transforms of the shape can be derived... |

24 |
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Citation Context ...hese manifolds and describe learning algorithms for estimating the distribution from data. Prior Work: Statistical methods on manifolds have been studied for several years in the statistics community =-=[5, 21, 22]-=-. A compilation of research results on statistical analysis on the Stiefel and Grassmann manifolds can be found in [10]. Their utility in practical applications has not yet been fully explored. Theore... |

21 |
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Citation Context ...ions has not yet been fully explored. Theoretical foundations for manifolds based shape analysis were described in [14, 20]. The Grassmann manifold structure of the affine shape space is exploited in =-=[4]-=- to perform affine invariant clustering of shapes. [26] exploited the geometry of the Grassmann manifold for subspace tracking in array signal processing applications. Statistical learning of shape cl... |

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Citation Context ...pular dynamical model for such time-series data is the autoregressive and moving average (ARMA) model. For the ARMA model closed form solutions for learning the model parameters have been proposed in =-=[19, 24]-=- and are widely used. The parameters of the model are known to lie on the Stiefel manifold as noted in [23]. Given several instances, current approaches involve computing the distance between them usi... |

13 |
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Citation Context ...ng probability density functions) provide far richer tools for pattern classification problems. In this context, we describe the Procrustes distance measures on the Stiefel and the Grassmann manifold =-=[10]-=-. Further, we describe parametric and non-parametric kernel based density functions 1on these manifolds and describe learning algorithms for estimating the distribution from data. Prior Work: Statist... |

10 |
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Citation Context ...lassification. 1. Introduction Many applications in computer vision such as dynamic textures [24, 9], human activity modeling and recognition [7, 30], video based face recognition [2], shape analysis =-=[14, 20]-=- involve learning and recognition of patterns from exemplars which lie on certain manifolds. Given a database of examples and a query, the following two questions are usually addressed – a) what is th... |

9 |
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Citation Context ...dations for manifolds based shape analysis were described in [14, 20]. The Grassmann manifold structure of the affine shape space is exploited in [4] to perform affine invariant clustering of shapes. =-=[26]-=- exploited the geometry of the Grassmann manifold for subspace tracking in array signal processing applications. Statistical learning of shape classes using nonlinear shape manifolds was presented in ... |

8 | Efficient indexing for articulation invariant shape matching and retrieval
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Citation Context ...sformation on R 2 . For example, γ could belong to the affine group, linear group, projecAlgorithm Rank 1 Rank 2 Rank 3 Rank 4 SC [17] 20/40 10/40 11/40 5/40 IDSC [17] 40/40 34/40 35/40 27/40 Hashing =-=[8]-=- 40/40 38/40 33/40 20/40 Grassmann Pro- 38/40 30/40 23/40 17/40 crustes Table 3. Retrieval experiment on articulation dataset. Last row is the results obtained using Grassmann manifold Procrustes repr... |

6 |
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Citation Context ...lassification. 1. Introduction Many applications in computer vision such as dynamic textures [24, 9], human activity modeling and recognition [7, 30], video based face recognition [2], shape analysis =-=[14, 20]-=- involve learning and recognition of patterns from exemplars which lie on certain manifolds. Given a database of examples and a query, the following two questions are usually addressed – a) what is th... |

4 | Incremental PCA: An alternative approach for novelty detection,” Towards Autonomous Robotic Systems
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Citation Context ...learn a better representational model as the appearance of the target dynamically changes as in [3]. Incremental PCA has also been used to recognize abnormalities in the visual field of a robot as in =-=[18]-=-. In an unrelated domain, the theory of subspace tracking on the Grassmann manifold [26] has been developed for array signal processing applications. Since PCA basis vectors represent a subspace which... |