cvpr cvpr2013 cvpr2013-237 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Raviteja Vemulapalli, Jaishanker K. Pillai, Rama Chellappa
Abstract: In computer vision applications, features often lie on Riemannian manifolds with known geometry. Popular learning algorithms such as discriminant analysis, partial least squares, support vector machines, etc., are not directly applicable to such features due to the non-Euclidean nature of the underlying spaces. Hence, classification is often performed in an extrinsic manner by mapping the manifolds to Euclidean spaces using kernels. However, for kernel based approaches, poor choice of kernel often results in reduced performance. In this paper, we address the issue of kernelselection for the classification of features that lie on Riemannian manifolds using the kernel learning approach. We propose two criteria for jointly learning the kernel and the classifier using a single optimization problem. Specifically, for the SVM classifier, we formulate the problem of learning a good kernel-classifier combination as a convex optimization problem and solve it efficiently following the multiple kernel learning approach. Experimental results on image set-based classification and activity recognition clearly demonstrate the superiority of the proposed approach over existing methods for classification of manifold features.
Reference: text
sentIndex sentText sentNum sentScore
1 Pillai and Rama Chellappa Department of Electrical and Computer Engineering Center for Automation Research, UMIACS, University of Maryland, College Park, MD 20742 Abstract In computer vision applications, features often lie on Riemannian manifolds with known geometry. [sent-2, score-0.326]
2 Hence, classification is often performed in an extrinsic manner by mapping the manifolds to Euclidean spaces using kernels. [sent-5, score-0.509]
3 However, for kernel based approaches, poor choice of kernel often results in reduced performance. [sent-6, score-0.394]
4 In this paper, we address the issue of kernelselection for the classification of features that lie on Riemannian manifolds using the kernel learning approach. [sent-7, score-0.597]
5 We propose two criteria for jointly learning the kernel and the classifier using a single optimization problem. [sent-8, score-0.356]
6 Specifically, for the SVM classifier, we formulate the problem of learning a good kernel-classifier combination as a convex optimization problem and solve it efficiently following the multiple kernel learning approach. [sent-9, score-0.354]
7 Experimental results on image set-based classification and activity recognition clearly demonstrate the superiority of the proposed approach over existing methods for classification of manifold features. [sent-10, score-0.582]
8 For instance, popular features and models in computer vision like shapes [10], histograms, covariance features [22] , linear dynamical systems (LDS) [6], etc. [sent-14, score-0.269]
9 In such cases, one needs good classification techniques that make use of the underlying manifold structure. [sent-16, score-0.427]
10 Hence, classification is often performed in an extrinsic manner by first mapping the manifold to an Euclidean space, and then learning classifiers in the new space. [sent-19, score-0.587]
11 However, tangent spaces preserves only the local structure of the manifold and can often lead to sub-optimal performance. [sent-21, score-0.445]
12 An alternative approach is to map the manifold to a reproducing kernel Hilbert space (RKHS) [8, 9, 5] by using kernels. [sent-22, score-0.514]
13 Though kernel-based methods have been successfully used in many computer vi- sion applications, poor choice of kernel can often result in reduced classification performance. [sent-23, score-0.27]
14 This gives rise to an important question: How to find good kernels for Riemannian manifolds ? [sent-25, score-0.456]
15 In this paper, we answer this question using the kernel learning approach [14, 17], in which appropriate kernels are learned directly from the data. [sent-27, score-0.429]
16 Since we are interested in learning good kernels for the purpose of classification, we learn the kernel and the classifier jointly by solving a single optimization problem. [sent-28, score-0.526]
17 In order to preserve the manifold structure, we constrain the distances in the mapped space to be close to the manifold distances. [sent-35, score-0.77]
18 standard solvers such as SeDuMi [19], it is transductive in nature: both training and test data need to be present while learning the kernel matrix. [sent-38, score-0.26]
19 To solve both these issues, we follow the multiple kernel learning (MKL) approach [14, 17] and parametrize the kernel as a linear combination of known base kernels. [sent-40, score-0.549]
20 We performed experiments using two different manifold features: linear subspaces and covariance features, and three different applications: face recognition using image sets, object recognition using image sets and human activity recognition. [sent-42, score-0.903]
21 The superior performance of the proposed approach clearly shows that it can be successfully used in classification applications that use manifold features. [sent-43, score-0.39]
22 Contributions: 1) We introduce a general framework for developing extrinsic classifiers for features that lie on Riemannian manifolds using the kernel learning approach. [sent-44, score-0.643]
23 To the best of our knowledge, the proposed approach is the first one to use kernel learning techniques for classification of features that lie on Riemannian manifolds with known geometry. [sent-45, score-0.597]
24 2) We propose to use a geodesic distance-based regularizer for learning appropriate kernels directly from the data. [sent-46, score-0.358]
25 Section 4 briefly discusses the Riemannian geometry of two popularly used features, namely linear subspaces and covariance features. [sent-49, score-0.438]
26 Previous Work Existing classification methods for Riemannian manifolds (with known geometry) can be broadly grouped into three main categories: nearest-neighbor methods, Bayesian methods, and Euclidean-mapping methods. [sent-52, score-0.298]
27 In [27, 13, 12], image sets were modeled using linear subspaces and then compared using the largest canonical correlation in [27], the direct sum of canonical correlations in [13] and a weighted sum canonical correlations in [12]. [sent-56, score-0.264]
28 In [21] parametric pdfs like Gaussian were defined on the tangent space and then wrapped back on to the manifold to define intrinsic pdfs for the Grassmann manifold. [sent-58, score-0.542]
29 Both these approaches along with Bayes classifier were used for human activity recognition and video-based face recognition. [sent-60, score-0.3]
30 , can be extended to manifolds by mapping the manifolds to Euclidean spaces. [sent-63, score-0.546]
31 In [22], a LogitBoost classifier was developed using weak classifiers learned on tangent spaces, and then used for pedestrian detection with covariance features. [sent-65, score-0.324]
32 Tangent spaces only preserves the local structure of the manifold and can often lead to sub-optimal performance. [sent-66, score-0.389]
33 Alternatively, one can map manifolds to Euclidean spaces by defining Mercer kernels on them. [sent-67, score-0.509]
34 In [24], a kernel defined for the manifold of symmetric positive definite matrices was used with PLS for image setbased recognition tasks. [sent-69, score-0.69]
35 In [5], the Binet-Cauchy kernels defined on non-linear dynamical systems were used for human activity recognition. [sent-70, score-0.402]
36 Hence, in this paper we address the issue of kernel-selection for the classification of manifold features. [sent-72, score-0.39]
37 The idea of using manifold structure as a regularizer was previously explored in the context of data manifolds [4, 18], where the given high dimensional data samples were simply assumed to lie on a lower dimensional manifold. [sent-73, score-0.719]
38 Since the structure of the underlying manifold was unknown, a graph Laplacian-based empirical estimate of the data distribution was used in [4, 18]. [sent-74, score-0.373]
39 Contrary to this, in this paper, we are interested in analytical manifolds, like Grassmann manifold and manifold of symmetric positive definite matrices, whose underlying geometry is known. [sent-75, score-0.79]
40 Let M denote the Riemannian manifold on which the featLuertes M Mlie d. [sent-94, score-0.336]
41 (ii) Structure preservation: Since the features lie on a Riemannian manifold with a well defined structure, the mapping should be structure-preserving. [sent-117, score-0.476]
42 SVM classifier in the mapped space: The SVM classifier in the mapped space is given by f(x) = w? [sent-125, score-0.39]
43 Preserving the manifold structure: To preserve the manifold structure, we constrain the distances in the mapped space to be close to the manifold distances. [sent-167, score-1.106]
44 The squared Euclidean distance between two points xi and xj in the mapped space can be expressed in terms of kernel values as ? [sent-168, score-0.363]
45 at in (5) we are learning the entire kernel matrix K directly in a non-parametric finasghi tohen, e anntidr eth kee rcnlealss mifaietrri txer Km dhiraesc only Ktr,tr. [sent-217, score-0.217]
46 Once the kernel matrix K is obtained, the SVM classifier in the mapped space can bKe sob otabitnaiende by solving tchlea SsiVfieMr ndu tahle ( m4)a. [sent-224, score-0.373]
47 pNedote s tahcaet ctahen above formulation is transductive in nature: both training and test data need to be present while learning the kernel matrix. [sent-225, score-0.26]
48 Extrinsic SVM Using MKL Framework Instead of learning a non-parametric kernel matrix K, following [ o1f4, l 1a7rn],i we parametrize trihce keernrneell as a l Kin-, ear combination of fixed base kernels K1, K2 , . [sent-230, score-0.583]
49 for both training and test data, the weights μ can be learned using only the training data, and the kernel values for test data can be computed using the known base kernels and the learned weights. [sent-242, score-0.466]
50 Let pimj denote the squared distance between samples xi and xj induced by the base kernel Km, i. [sent-282, score-0.391]
51 2, (7) Let Φm be the mapping corresponding to the kernel Km and ? [sent-311, score-0.236]
52 ) is also convex and differentiable if all the base kernel matrices Km are strictly positive tdieafbilnei itfe. [sent-351, score-0.398]
53 Once the optimal μ∗ is computed, the classifier in the mapped spa? [sent-373, score-0.195]
54 Riemannian Manifolds in Computer Vision In this section we briefly discuss the Riemannian geometry of two popularly used features, namely linear subspaces and covariance features, and show how these features are used in various computer vision applications. [sent-381, score-0.438]
55 yT nhe × geodesic dnoisrtamnacle mbae-tween two subspaces S1 and S2 on the Grassmann manifold is given by where = [θ1, . [sent-386, score-0.533]
56 O ∈th e[0r, popularly used distances for the Grassmann manifold are the Procrustes metric given by (θi/2))1/2, and the Projection metric given by sin2θi) 1/2. [sent-392, score-0.456]
57 rassmann kernels: Grassmann manifold can be mapped to Euclidean spaces by using Mercer kernels [8]. [sent-396, score-0.699]
58 One popularly used kernel [8, 9, 24] is the Projection kernel given by ? [sent-397, score-0.476]
59 Various kernels can be generated from KP and ΦP using KKrPPpbofl(yY(Y1,1,YY2)2) = = e (xγpK? [sent-412, score-0.212]
60 , (Y) We refer to the family of kernels (12) as projection-RBF × kernels and the family of kernels polynomial kernels. [sent-417, score-0.76]
61 ,d non-singular ciocv aporisainticvee m deaftirnicitees, can ) be m aftorri-mulated as a Riemannian manifold [16], and the resulting affine-invariant geodesic distance (AID) is given by (C1, C2)) 1/2 , where λi (C1, C2) are the gener? [sent-422, score-0.386]
62 Kernels for SPD matrices: Similar to the Grassmann manifold, we can define kernels for the set of SPD matrices. [sent-435, score-0.212]
63 We refer to the family of kernels Klrobgf as LED-RBF , (13) ker- nels and the family of kernels kernels. [sent-446, score-0.535]
64 Since covariance matrices are positive semi-definite in general, a small ridge may be added to their diagonals to make them positive definite. [sent-461, score-0.262]
65 Activity recognition using dynamical models: The autoregressive and moving average (ARMA) model is a dynamical model widely used in computer vision for modeling various kinds of time-series data [6, 3] and has been successfully used for activity recognition [21, 23, 5]. [sent-462, score-0.38]
66 Experimental Evaluation In this section, we evaluate the proposed approach using three applications where manifold features are used: (i) × Face recognition using image sets, (ii) Object recognition using image sets and (iii) Human activity recognition from videos. [sent-484, score-0.566]
67 We use two different manifold features, namely linear subspaces and covariance features. [sent-485, score-0.654]
68 For both of these datasets, we performed experiments with two different manifold features: covariance matrices and linear subspaces. [sent-499, score-0.556]
69 3, to avoid matrix singularity, we added a small ridge δI to each covariance matrix C, where δ = 10−3 trace(C) and I the is identity matrix. [sent-501, score-0.213]
70 ed as a convex combination of fixed base kernels (K = ? [sent-515, score-0.386]
71 iii) Statistical modeling (SM) [21]: This approach uses parametric (SM-P) and non-parametric (SM-NP) probability density estimation on the manifold followed by Bayes b? [sent-518, score-0.409]
72 (iv) Grassmann discriminant analysis (GDA) [8]: Performs discriminant analysis followed by NN classification for the Grassmann manifold using the Projection kernel. [sent-523, score-0.554]
73 (v) PLS with the Projection kernel (Proj+PLS) [24]: Uses PLS combined with the Projection kernel for the Grassmann manifold. [sent-524, score-0.356]
74 (vi) Covariance discriminative learning (CDL) [24]: Uses discriminant analysis and PLS for covariance features using a kernel derived from the LED metric. [sent-525, score-0.47]
75 For the experiments with linear subspaces, we used multiple projection-RBF and projectionpolynomial kernels defined in (12). [sent-537, score-0.212]
76 Specifically, for the INRIA IXMAS dataset, we used 6 projection-polynomial kernels and 13 projection-RBF kernels. [sent-539, score-0.212]
77 For the YouTube dataset, we used 10 projection-polynomial kernels and 15 projection-RBF kernels. [sent-540, score-0.212]
78 For the ETH80 dataset, we used 10 projection-polynomial kernels and 13 projection-RBF kernels. [sent-541, score-0.212]
79 Polynomial kernels were generated by taking γ = n1 and varying the degree from 1to 6 for the INRIA IXMAS dataset, and from 1to 10 for the other two datasets. [sent-557, score-0.212]
80 For the experiments with covariance features, we used multiple LED-RBF and LED-polynomial kernels defined in (13), whose parameters were chosen based on their individual crossvalidation performance. [sent-558, score-0.383]
81 Specifically, for the YouTube dataset, we used 10 LED-polynomial kernels and 15 LED-RBF kernels. [sent-559, score-0.212]
82 For the ETH80 dataset, we used 10 LED-polynomial kernels and 20 LED-RBF kernels. [sent-560, score-0.212]
83 For both linear subspaces and covariance features, geodesic distances were used in the distance preserving constraints. [sent-573, score-0.368]
84 Results Table 1 shows the recognition rates for human activity recognition using dynamical models. [sent-577, score-0.338]
85 Tables 2 and 3 show the recognition rates for image set-based object and face recognition tasks using linear subspaces and covariance features respectively. [sent-578, score-0.531]
86 The horizontal axis corresponds to the kernel index Table 1: Recognition rates for human activity recognition on the INRIA IXMAS dataset using dynamical models IdNXatRMasIAe tS8N0 . [sent-588, score-0.47]
87 o08rpoascehd Table 2: Recognition rates for image set-based face and object recognition tasks using linear subspaces Table 3: Recognition rates for image set-based face and object recognition tasks using covariance features dEYaTotuaHsT8eu0tbe94N20. [sent-598, score-0.652]
88 In the case of other datasets, the S-MKL approach mostly picked few base kernels (usually RBF kernels with high γ value or polynomial kernels of high degree d), whereas the weights for the proposed approach were distributed over many kernels. [sent-606, score-0.766]
89 In the case of SM-NP, the poor performance could be due to the sub-optimal choice of the kernel width used in [21]. [sent-610, score-0.216]
90 In general, non-parametric density estimation methods are sensitive to the choice of kernel width and a sub-optimal choice often results in poor performance [21]. [sent-611, score-0.251]
91 The relatively lower performance of the other kernel-based methods suggests that, it is effective to jointly learn the kernel and the classifier directly from the data using the proposed framework. [sent-612, score-0.275]
92 Recently, covariance feature combined with PLS has been shown [24] to perform better than various other recent methods for image set-based recognition tasks. [sent-613, score-0.217]
93 Our 111777888866 results show that the classification performance can be further improved by combining the covariance feature with the proposed approach. [sent-614, score-0.225]
94 Conclusion and Future Work In this paper, we introduced a general framework for developing extrinsic classifiers for features that lie on Riemannian manifolds using the kernel learning approach. [sent-616, score-0.643]
95 We proposed two criteria for learning a good kernel-classifier combination for manifold features. [sent-617, score-0.456]
96 In the case of SVM classifier, based on the proposed criteria, we showed that the problem of learning a good kernel-classifier combination can be formulated as a convex optimization problem and efficiently solved following the multiple kernel learning approach. [sent-618, score-0.354]
97 We performed experiments using two popularly used manifold features and obtained superior performance compared to other relevant approaches. [sent-619, score-0.456]
98 In this paper, the manifold structure has been used as a regularizer using simple distance-preserving constraints. [sent-621, score-0.393]
99 Another possible direction of future work is to explore more sophisticated regularizers that can make use of the underlying manifold structure. [sent-622, score-0.373]
100 Riemannian geometry of Grassmann manifolds with a view on algorithmic computation. [sent-629, score-0.244]
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