jmlr jmlr2012 jmlr2012-50 knowledge-graph by maker-knowledge-mining

50 jmlr-2012-Human Gesture Recognition on Product Manifolds

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Author: Yui Man Lui

Abstract: Action videos are multidimensional data and can be naturally represented as data tensors. While tensor computing is widely used in computer vision, the geometry of tensor space is often ignored. The aim of this paper is to demonstrate the importance of the intrinsic geometry of tensor space which yields a very discriminating structure for action recognition. We characterize data tensors as points on a product manifold and model it statistically using least squares regression. To this aim, we factorize a data tensor relating to each order of the tensor using Higher Order Singular Value Decomposition (HOSVD) and then impose each factorized element on a Grassmann manifold. Furthermore, we account for underlying geometry on manifolds and formulate least squares regression as a composite function. This gives a natural extension from Euclidean space to manifolds. Consequently, classiﬁcation is performed using geodesic distance on a product manifold where each factor manifold is Grassmannian. Our method exploits appearance and motion without explicitly modeling the shapes and dynamics. We assess the proposed method using three gesture databases, namely the Cambridge hand-gesture, the UMD Keck body-gesture, and the CHALEARN gesture challenge data sets. Experimental results reveal that not only does the proposed method perform well on the standard benchmark data sets, but also it generalizes well on the one-shot-learning gesture challenge. Furthermore, it is based on a simple statistical model and the intrinsic geometry of tensor space. Keywords: gesture recognition, action recognition, Grassmann manifolds, product manifolds, one-shot-learning, kinect data

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 The aim of this paper is to demonstrate the importance of the intrinsic geometry of tensor space which yields a very discriminating structure for action recognition. [sent-5, score-0.586]

2 We characterize data tensors as points on a product manifold and model it statistically using least squares regression. [sent-6, score-0.576]

3 To this aim, we factorize a data tensor relating to each order of the tensor using Higher Order Singular Value Decomposition (HOSVD) and then impose each factorized element on a Grassmann manifold. [sent-7, score-0.465]

4 Consequently, classiﬁcation is performed using geodesic distance on a product manifold where each factor manifold is Grassmannian. [sent-10, score-0.935]

5 We assess the proposed method using three gesture databases, namely the Cambridge hand-gesture, the UMD Keck body-gesture, and the CHALEARN gesture challenge data sets. [sent-12, score-0.784]

6 Keywords: gesture recognition, action recognition, Grassmann manifolds, product manifolds, one-shot-learning, kinect data 1. [sent-15, score-0.771]

7 In recent years, many gesture recognition algorithms have been proposed (Mitra and Acharya, 2007; Wang et al. [sent-19, score-0.496]

8 However, reliable gesture recognition remains a challenging area due in part to the complexity of human movements. [sent-21, score-0.542]

9 Consequently, heavy-duty models may not have substantial gains in overall gesture recognition problems. [sent-23, score-0.496]

10 In this paper, we propose a new representation to gesture recognition based upon tensors and the geometry of product manifolds. [sent-24, score-0.757]

11 Since human actions are expressed as a sequence of video frames, an action video may be characterized as a third order data tensor. [sent-25, score-0.652]

12 The goal of this paper is to demonstrate the importance of the intrinsic geometry of tensor space where it provides a very discriminating structure for action recognition. [sent-31, score-0.586]

13 (2005) modeled human shapes from a shape manifold and expressed the dynamics of human silhouettes using an autoregressive (AR) model on the tangent space. [sent-34, score-0.537]

14 The use of tangent bundles on special manifolds was investigated by Lui (2012b) where a set of tangent spaces was exploited for action recognition. [sent-36, score-0.626]

15 Lui and Beveridge (2008) characterized tangent spaces of a registration manifold as elements on a Grassmann manifold for face recognition. [sent-40, score-0.736]

16 The method proposed in this paper characterizes action videos as data tensors and demonstrates their association with a product manifold. [sent-44, score-0.601]

17 We focus attention on the intrinsic geometry of tensor space, and draw upon the fact that the geodesic on a product manifold is equivalent to the Cartesian product of geodesics from multiple factor manifolds. [sent-45, score-0.963]

18 In other words, elements of a product manifold are the set of all elements inherited from factor manifolds. [sent-46, score-0.494]

19 Thus, in our approach, action videos are factorized to three factor elements using Higher Order Singular Value Decomposition (HOSVD) in which the factor elements give rise to three factor manifolds. [sent-47, score-0.636]

20 We further extend the product manifold representation to least squares regression. [sent-48, score-0.464]

21 The least squares ﬁtted elements from a training set can then be exploited for gesture recognition where the similarity is expressed in terms of the geodesic distance on a product manifold associated with ﬁtted elements from factor manifolds. [sent-52, score-1.289]

22 We demonstrate the merits of our method on three gesture recognition problems including hand gestures, body gestures, and gestures collected from the Microsoft KinectTM camera for the oneshot-learning CHALEARN gesture challenge. [sent-53, score-1.063]

23 The key contributions of the proposed work are summarized as follows: • A new way of relating tensors on a product manifold to action recognition. [sent-55, score-0.803]

24 The use of SIFT features with CCA was also considered for gesture recognition by Kim and Cipolla (2007). [sent-110, score-0.496]

25 Recently, nonnegative tensor factorization has been exploited for action recognition by Krausz and Bauckhage (2010) where action videos were factorized using a gradient descent method and represented as the sum of rank-1 tensors associated with a weighting factor. [sent-111, score-1.219]

26 (2009) modeled the motion manifold as a collection of local linear models. [sent-115, score-0.543]

27 This method learned a selection of mappings to encode the motion manifold from a product space. [sent-116, score-0.625]

28 Despite these efforts, the geometry of the product space has not been directly considered and the geodesic nature on the product manifold remains unexamined. [sent-123, score-0.715]

29 The illustration is for a video action sequence with two spatial dimensions X and Y and a temporal dimension T . [sent-137, score-0.477]

30 The Stiefel manifold Vn,p can be considered a quotient space of O (n) so we can identify an isotropy subgroup H of O (n) expressed as Ip 0 0 Qn−p : Qn−p ∈ O (n − p) where the isotropy subgroup leaves the element unchanged. [sent-170, score-0.466]

31 4 Grassmann Manifolds When we impose a group action of O (n) onto the Stiefel manifold, this gives rise to the equivalence relation between orthogonal matrices so that the elements of Stiefel manifolds are rotation and reﬂection invariant. [sent-178, score-0.638]

32 As such, the element of the Grassmann manifold represents the orbit of a Stiefel manifold under the group action of orthogonal groups. [sent-183, score-0.983]

33 Elements of Product Manifolds This section discusses the elements of product manifolds in the context of gesture recognition. [sent-188, score-0.69]

34 We illustrate the essence of product manifolds and the factorization of action videos. [sent-189, score-0.558]

35 Further, we describe the realization of geodesic distance on the product manifold and its use for action classiﬁcation. [sent-190, score-0.894]

36 × Mq is called the product of the manifolds where the manifold topology is equivalent to the product topology. [sent-204, score-0.655]

37 Note that the dimension of a product manifold is the sum of all factor manifolds (Lee, 2003). [sent-212, score-0.599]

38 For action video classiﬁcation, third order data tensors are manifested as elements on three factor manifolds. [sent-215, score-0.594]

39 However, the traditional deﬁnition of HOSVD does not lead to a well-deﬁned product manifold in the context of action recognition. [sent-219, score-0.691]

40 Because we (1) (2) are performing action recognition on videos, the orthogonal matrices, Vhorizontal-motion , Vvertical-motion , (3) and Vappearance , correspond to horizontal motion, vertical motion, and appearance. [sent-228, score-0.541]

41 When we impose a group action of the orthogonal group, elements on the Stiefel manifold become rotation and reﬂection invariant. [sent-231, score-0.734]

42 As such, the action data are represented as the orbit of 3304 H UMAN G ESTURE R ECOGNITION ON P RODUCT M ANIFOLDS elements on the Stiefel manifold under the rotation and reﬂection actions with respect to appearance and dynamics. [sent-233, score-0.839]

43 3 Geodesic Distance on Product Manifolds The geodesic in a product manifold M is the product of geodesics in M1 , M2 , . [sent-236, score-0.648]

44 Hence, for any differentiable curve γ parametrized by t, we have γ(t) = (γi (t), γ j (t)) where γ is the geodesic on the product manifold M , and γi and γ j are the geodesics on the factor manifold Mi and M j respectively. [sent-241, score-0.904]

45 From this observation, the geodesic distance on a product manifold may be expressed as a Cartesian product of canonical angles computed by factor manifolds. [sent-242, score-0.912]

46 , 1998) using canonical angles, the geodesic distance on a product manifold could also be deﬁned in different ways. [sent-244, score-0.713]

47 Consequently, we deﬁne the geodesic distance on a product manifold as: dM (A , B ) = sin Θ 2 (3) where A and B are the N order data tensors, Θ = (θ1 , θ2 , . [sent-247, score-0.597]

48 o This development of geodesic distance on the product manifold can be related back to our cylinder example where a circle in R2 and a line in R1 form a cylinder in R3 where R3 is the product space. [sent-251, score-0.777]

49 (k) (k) Note that canonical angles θk are measured between VA and VB where each is an orthogonal matrix spanning the row space associated with nonzero singular values from a mode-k unfolded matrix. [sent-262, score-0.477]

50 The Product Manifold Representation The tensor representation on a product manifold models the variations in both space and time for action videos. [sent-271, score-0.885]

51 Speciﬁcally, the product manifold captures the individual characteristics of spatial and temporal evolution through three factor manifolds. [sent-272, score-0.478]

52 As such, one factor manifold is acquiring the change in time, resulting in the appearance (XY) component, while the other two capture the variations in horizontal and vertical directions, demonstrating the horizontal motion (YT) and vertical motion (XT). [sent-273, score-1.058]

53 Putting all these representations together, geodesic distance on the product manifold measures the changes in both appearance and dynamics. [sent-274, score-0.699]

54 The aim of this section is to illustrate how the product manifold characterizes appearance and dynamics from action videos. [sent-275, score-0.825]

55 On the contrary, the second column shows the same action performed by different actors and the canonical variates are much more similar than the ﬁrst column, resulting in smaller canonical angles overall. [sent-287, score-0.678]

56 One of the advantages of the product manifold representation is that actions do not need to be aligned in temporal space. [sent-288, score-0.517]

57 To demonstrate this merit, we permute the frame order from action 3 denoted as action 4 and match it to action 1. [sent-289, score-0.891]

58 We should ﬁrst note that the appearance (XY) of action 3 and action 4 span the same space despite the visual differences resulting in the identical sum of canonical angles 38. [sent-291, score-0.929]

59 This important concept is illustrated in Figure 5 where the exchange matrix O (p) maps the appearance of action 4 to the appearance of action 3. [sent-294, score-0.798]

60 The canonical angles for the appearance indicates that the action is not affected by the frame order. [sent-302, score-0.606]

61 Figure 5: The characterization of the Grassmann manifold where a point is mapped to another point on the Stiefel manifold via an exchanged matrix. [sent-303, score-0.624]

62 The group action is (X, Q) −→ XQ where X ∈ Vn,p and Q ∈ O (p) so that elements on the Grassmann manifold are closed under the orthogonal matrix multiplication. [sent-304, score-0.708]

63 In the example given in Figure 4, the most prominent change is related to the motion in vertical directions (XT) between action 3 and action 4. [sent-305, score-0.859]

64 This arises from the fact that the change of motion mostly occurs in the vertical direction when we permute the order of the video frames from action 3. [sent-306, score-0.716]

65 As a result, the product manifold representation is resilient to misregistration in the temporal space for appearance while keeping the dynamics intact. [sent-317, score-0.586]

66 While the structure of horizontal motion between walking and running is similar exhibiting a line-like pattern, they have very distinct slopes shown in the horizontal motion column of Figure 6. [sent-323, score-0.595]

67 In general, it is possible to see the rate of motion through both motion representations depending on the type of actions. [sent-326, score-0.462]

68 The least squares ﬁtted elements from a training set can then be exploited for gesture recognition. [sent-334, score-0.525]

69 To make it speciﬁc for gesture recognition, we impose rotation and reﬂection invariance to the factorized element V (k) such that they are elements on a Grassmann manifold and the computation of the weighted Karcher mean can be realized. [sent-364, score-0.844]

70 From a geometric point of view, the logarithmic operator maps a point on a manifold to a tangent space whereas the exponential map projects a point in the tangent space back to the manifold. [sent-373, score-0.462]

71 To perform gesture recognition, a set of training videos is collected. [sent-379, score-0.528]

72 Because the query gesture Y and the regression instance are realized as elements on a product manifold, we employ the chordal distance given in (3) for gesture classiﬁcation. [sent-382, score-0.964]

73 In summary, the least squares regression model applies HOSVD on a query gesture Y and fac(1) (2) (3) torizes it to three sub-regression models (Ψ j , Ψ j , Ψ j ) on three Grassmann manifolds where regressions are performed. [sent-383, score-0.671]

74 The distance between the regression output and query is then characterized on a product manifold; gesture recognition is achieved using the chordal distance. [sent-384, score-0.639]

75 1 Cambridge Hand-Gesture Data Set Our ﬁrst experiment is conducted using the Cambridge hand-gesture data set which has 900 video sequences with nine different hand gestures (100 video sequences per gesture class). [sent-399, score-0.811]

76 As for the dynamic environment, the gestures acquired from the static scene are used for training while the gestures collected from the dynamic environment are the test videos. [sent-422, score-0.473]

77 3 One-Shot-Learning Gesture Challenge Microsoft KinectTM has recently revolutionized gesture recognition by providing both RGB and depth images. [sent-459, score-0.496]

78 Consequently, we apply the same regression framework on the product manifold to the one-shot-learning gesture challenge. [sent-467, score-0.816]

79 One of the gesture variations is performing gesture positions. [sent-468, score-0.784]

80 Since gesture positions are the key source of variations, we synthesize training examples for translational instances on both RGB and depth images. [sent-470, score-0.46]

81 Each batch is made of 47 gesture videos and split into a training set and a test set. [sent-479, score-0.528]

82 Since the number of gestures varies for test videos, we perform temporal segmentation to localize each gesture segment. [sent-484, score-0.652]

83 We can then localize the gesture segments by identifying the peak locations from the correlations; the number of gestures is the number of peaks + 1. [sent-487, score-0.567]

84 Since we do not have a hand detector, the gross motion dominates the whole action causing it to be confused with other similar gestures. [sent-587, score-0.528]

85 Once we have a reliable hand detector, we expect to further improve gesture recognition from a single training example. [sent-595, score-0.522]

86 It decomposes a video tensor to three Stiefel manifolds via HOSVD where the orthogonal elements are imposed to Grassmannian spaces. [sent-601, score-0.594]

87 A V-shape rightward gesture and a ﬂat leftward gesture are shown in the ﬁrst row and second row. [sent-607, score-0.863]

88 We superpose a cluttered background on every frame of the ﬂat leftward gesture exhibited in the third row. [sent-608, score-0.511]

89 While the appearances between the uniform ﬂat gesture and the cluttered ﬂat gesture emerge differently, the deterioration on the dynamics is quite minimal. [sent-609, score-0.868]

90 Numerically, the sum of the canonical angles between the uniform (second row) and the cluttered background (third row) gestures is (56. [sent-611, score-0.462]

91 In addition, when the V-shape gesture (ﬁrst row) matches against the cluttered ﬂat gesture (third row), the sum of the canonical angles is (76. [sent-621, score-1.043]

92 This ﬁnding reveals that the geodesic distance between the uniform and cluttered background gestures are quite similar against inter-class gestures, while the geodesic distance is signiﬁcantly smaller for the intra-class gestures. [sent-626, score-0.661]

93 We have presented a geometric framework for least squares regression and applied it to gesture recognition. [sent-633, score-0.492]

94 We view action videos as third order tensors and impose them on a product manifold where each factor is 3317 L UI (a) V-shape rightward gesture. [sent-634, score-0.939]

95 The realization of points on these Grassmannians is achieved by applying HOSVD to a tensor representation of the action video. [sent-640, score-0.491]

96 A natural metric is inherited from the factor manifolds since the geodesic on the product manifold is given by the product of the geodesic on the Grassmann manifolds. [sent-641, score-1.025]

97 Future work will focus on developing more sophisticated models for gesture recognition and other regression techniques on matrix manifolds for visual applications. [sent-647, score-0.731]

98 Gesture and action recognition via modeling trajectories on riemannian manifolds. [sent-654, score-0.46]

99 Riemannian geometry of grassmann manifolds with a view on algorithmic computation. [sent-661, score-0.478]

100 Canonical correlation analysis of video volume tensors for action categorization and detection. [sent-796, score-0.531]

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