nips nips2012 nips2012-146 knowledge-graph by maker-knowledge-mining

146 nips-2012-Graphical Gaussian Vector for Image Categorization

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Author: Tatsuya Harada, Yasuo Kuniyoshi

Abstract: This paper proposes a novel image representation called a Graphical Gaussian Vector (GGV), which is a counterpart of the codebook and local feature matching approaches. We model the distribution of local features as a Gaussian Markov Random Field (GMRF) which can efﬁciently represent the spatial relationship among local features. Using concepts of information geometry, proper parameters and a metric from the GMRF can be obtained. Then we deﬁne a new image feature by embedding the proper metric into the parameters, which can be directly applied to scalable linear classiﬁers. We show that the GGV obtains better performance over the state-of-the-art methods in the standard object recognition datasets and comparable performance in the scene dataset. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 jp Abstract This paper proposes a novel image representation called a Graphical Gaussian Vector (GGV), which is a counterpart of the codebook and local feature matching approaches. [sent-11, score-0.327]

2 We model the distribution of local features as a Gaussian Markov Random Field (GMRF) which can efﬁciently represent the spatial relationship among local features. [sent-12, score-0.379]

3 Using concepts of information geometry, proper parameters and a metric from the GMRF can be obtained. [sent-13, score-0.111]

4 Then we deﬁne a new image feature by embedding the proper metric into the parameters, which can be directly applied to scalable linear classiﬁers. [sent-14, score-0.292]

5 1 Introduction The Bag of Words (BoW) [7] is the de facto standard image feature for the image categorization. [sent-16, score-0.3]

6 In a BoW, each local feature is assigned to the nearest codeword and an image is represented by a histogram of the quantized features. [sent-17, score-0.293]

7 While it is well established that using a large number of codewords improves classiﬁcation performance, the drawback is that assigning local features to the nearest codeword is computationally expensive. [sent-19, score-0.228]

8 To overcome this problem, some studies have proposed building an efﬁcient image representation with a smaller number of codewords [22], [24]. [sent-20, score-0.155]

9 Finding an explicit correspondence between local features is another way of categorizing images using a BoW [4], [12], [26], and this approach has been improved by representing a spatial layout of local features as a graph [11], [2], [16], [8]. [sent-21, score-0.569]

10 Explicit correspondences between features have an advantage over a BoW as information loss in the vector quantization can be avoided. [sent-22, score-0.077]

11 Therefore, the aim of our research is to build an efﬁcient image representation without using codewords or explicit correspondences between local features, while still achieving high classiﬁcation accuracy. [sent-24, score-0.244]

12 Since having a spatial layout of local features is important for an image to have semantic meaning, it is natural that embedding spatial information into an image feature improves classiﬁcation performance [18], [5], [14], [17]. [sent-25, score-0.742]

13 Several approaches take advantage of this fact, ranging from local (e. [sent-26, score-0.105]

14 Meanwhile, we will focus on the spatial layout of local features, which is the midlevel of the spatial information. [sent-31, score-0.351]

15 In this paper, we model an image as a graph representing the spatial layout of local features and deﬁne a new image feature based on this graph, where a proper metric is embedded into the feature. [sent-32, score-0.735]

16 Speciﬁcally, we model an image as a Gaussian Markov Random Field (GMRF) whose nodes correspond to local features and consider the GMRF parameters as the image feature. [sent-34, score-0.428]

17 Although the GMRF is commonly used for image segmentation, it is rarely used in modern image categorization pipelines despite being an effective way of modeling the spatial layout. [sent-35, score-0.36]

18 In order to extract the repre1 sentative feature vector from the GMRF, the choice of coordinates for the parameters and the metric between them needs to be carefully made. [sent-36, score-0.137]

19 We deﬁne the proper coordinates and the metric from an information geometry standpoint [1] and derive an optimal feature vector. [sent-37, score-0.166]

20 The contributions of this study are summarized as follows: 1) A novel and efﬁcient image feature is developed by utilizing the GMRF as a tool for object categorization. [sent-39, score-0.198]

21 3) Using standard image categorization benchmarks, we demonstrate that the proposed feature has better performance over the state-of-the-art methods, even though it is not based on mainstream modules (such as codebooks and correspondence between local features). [sent-41, score-0.331]

22 To the best of our knowledge, this is the ﬁrst image feature for the object categorization that utilizes the expectation parameters of the GMRF with its Fisher information metric, and achieves a level of accuracy comparable to that of the codebook and local feature matching approaches. [sent-42, score-0.537]

23 ξ2 Figure 1: Overview of image feature extraction based on a multivariate GMRF. [sent-45, score-0.192]

24 Initially, local features {xi ∈ Rd }M are i=1 extracted using a dense sampling strategy (Fig. [sent-47, score-0.182]

25 We then use a multivariate GMRF to model the spatial relationships among local features (Fig. [sent-49, score-0.298]

26 The GMRF is represented as a graph G(V, E), whose vertices V and edges E correspond to local features and the dependent relationships between those features, respectively. [sent-51, score-0.241]

27 Let the vector x be a concatenation of local features in V and let ξ j be a parameter of the GMRF of an image Ij , the image Ij can be represented by a probability distribution p(x; ξ j ) of the GMRF (Fig. [sent-52, score-0.408]

28 We consider the parameter ξj of the GMRF to be a feature vector of the image Ij (Fig. [sent-54, score-0.168]

29 However, because the space spanned by parameters of a probability distribution is not a Euclidean space, we have to be very careful when choosing parameters for the probability distribution and the metric among them. [sent-58, score-0.123]

30 We make use of concepts from the information geometry [1] and extract proper parameters and a metric from the GMRF. [sent-59, score-0.131]

31 Finally, we deﬁne the new image feature by embedding the metric into the extracted parameters to build an image categorization system with a scalable linear classiﬁer. [sent-60, score-0.438]

32 2 Image Model and Parameters Given M local features {xi ∈ Rd }M , the aim is to model a probability distribution of the local i=1 features representing the spatial layout of the image using the multivariate GMRF G = (V, E). [sent-63, score-0.655]

33 First, a vector x is built by concatenating the local features corresponding to the vertices V of the GMRF. [sent-64, score-0.235]

34 Let {xi }n are local features that we are focusing on, we obtain the concatenated vector as i=1 x = (x1 · · · xn ) (e. [sent-65, score-0.182]

35 Note that the dimensionality of x is nd and does not depend on the number of local features M , the image size, or the aspect ratio. [sent-69, score-0.378]

36 However, since all results valid for a scalar local feature are also valid for a multivariate local feature, in this section we consider the dimensionality of local features is 1 (d = 1) for simplicity. [sent-70, score-0.554]

37 The expectation parameters are obtained as [15]: ηi = μi , ηii = Pii + μ2 , ηjk = Pjk + μj μk , (i ∈ V, {j, k} ∈ E). [sent-77, score-0.083]

38 (4) i The natural and expectation parameters can be transformed into each other [1]. [sent-78, score-0.083]

39 If we take the natural parameters or the expectation parameters as a coordinate system for an exponential family, a ﬂat structure can be realized [1]. [sent-81, score-0.124]

40 Those spaces are similar to a Euclidean space, but we need to be careful that the spaces spanned by the natural or expectation parameters are different from a Euclidean space, as the metrics vary for different parameters. [sent-84, score-0.104]

41 To summarize this section, the natural and expectation parameters are similar and interchangeable through the FIMs. [sent-89, score-0.083]

42 Although it does not matter whether we choose natural or expectation parameters, we use expectation parameters (Eq. [sent-91, score-0.146]

43 (4)) as feature vectors because they can be calculated directly from the mean and covariance of local features. [sent-92, score-0.183]

44 3 Calculation of Expectation Parameters In this section, we describe the calculations of the expectation parameters of the multivariate GMRF. [sent-95, score-0.124]

45 While a graph having more neighbors is obviously able to represent richer spatial information, the compact structure is preferable for efﬁciency. [sent-101, score-0.119]

46 2(c), which represents the vertical and horizontal relationships among local features, and Fig. [sent-103, score-0.138]

47 rk + a 2 rk + a 3 rk + a 4 rk + a 2 rk + a1 rk (a) (b) rk + a1 rk (c) (d) Figure 2: Structures of the GMRF. [sent-105, score-0.432]

48 Next, we show a method for estimating the expectation parameters of each image. [sent-106, score-0.083]

49 (4) in a multivariate case can be determined by calculating the local auto-correlations of local 3 features. [sent-108, score-0.234]

50 Let x(r k ) ∈ Rd be the local feature at a reference point rk and let ai and aj be the displacement vectors, which are deﬁned by the structure of the GMRF. [sent-112, score-0.252]

51 Then, the local auto-correlation matrices 1 are obtained as: Ci,j = NJ k∈J x(r k +ai )x(r k +aj ) , where NJ is the number of local features 1 in the image region J. [sent-113, score-0.4]

52 Let a vector concatenating local features in the vertices at the reference point rk be xk = (x(r k ) x(r k + a1 ) x(r k + a2 ) ), P + μμ is calculated to be: P + μμ = 1 NJ xk xk = k∈J C0,0 C1,0 C2,0 C0,1 C1,1 C2,1 C0,2 C1,2 C2,2 . [sent-115, score-0.312]

53 (5) The expectation parameters of the GMRF depicted in Fig. [sent-116, score-0.083]

54 Note that C1,2 is omitted, because there is no edge between the vertices at rk + a1 and rk + a2 . [sent-118, score-0.14]

55 The dimensionality of η is: nd + n(d + 1)d/2 + (n − 1)d2 , where d is the dimensionality of the local feature. [sent-121, score-0.271]

56 2(e)), if we have enough local features, the means {μi }n−1 i=0 and covariance matrices {Ci,i }n−1 of local features in the region J come to the vector μ0 and matrix i=0 C0,0 , respectively. [sent-124, score-0.287]

57 We now derive a metric between the expectation parameters [1]. [sent-138, score-0.145]

58 (8) Thus, the FIM is a proper metric for the feature vectors (the expectation parameters) obtained from the GMRF. [sent-144, score-0.209]

59 5 Implementation of Graphical Gaussian Vector At ﬁrst, we build the concatenated vector as x = (x1 · · · xn ) , where each xi corresponds to the local feature of the vertex i. [sent-149, score-0.16]

60 The dimensionality of the local features is 2 (x = (x1 , x2 ) , y = (y1 , y2 ) , z = (z1 , z2 ) ). [sent-151, score-0.265]

61 A vector concatenating the local features in V is v = (x1 , x2 , y1 , y2 , z1 , z2 ) . [sent-152, score-0.203]

62 Using μ and J, the Fisher information matrix of the full Gaussian family can be calculated as in (b), whose rows and columns correspond to the elements of the expectation parameters. [sent-154, score-0.086]

63 In order to embed the proper metric into the expectation parameters, we multiply G∗ (η c )1/2 by η: ζ= ∗ ∗ ∗ FG,G (η c ) − FG,\G (η c ) F\G,\G (η c ) −1 ∗ F\G,G (η c ) 1/2 η. [sent-169, score-0.174]

64 In practice, since using all training data is infeasible to estimate the FIM, we use a subset of local features randomly sampled from training data. [sent-176, score-0.182]

65 Input: An image region J, and the Fisher information matrix of the GMRF G∗ (η c ) Output: GGV ζ 1. [sent-179, score-0.113]

66 Calculate local auto-correlations of local features: 1 1 μi = NJ k∈J x(r k + ai ), Ci,j = NJ k∈J x(r k + ai )x(r k + aj ) 2. [sent-180, score-0.269]

67 Embed the Fisher information metric into the expectation parameters: 1/2 ζ = (G∗ (η c )) η 3 Experiment We tested our method on the standard object and scene datasets (Caltech101, Caltech256, and 15-Scenes). [sent-182, score-0.178]

68 2(c) (GGV, n = 3), which models a horizontal and vertical spatial layout of the local features. [sent-190, score-0.292]

69 2(d) (GGV, n = 5), which adds diagonal spatial layouts of the features ˆ to Fig. [sent-192, score-0.169]

70 To embed the global spatial information, we used the spatial pyramid representation with a 1 × 1 + 2 × 2 + 3 × 3 pyramid structure. [sent-197, score-0.293]

71 Table 1: The relationships between GLC, LAC, GG, and GGV in terms of spatial information and Fisher information metrics. [sent-198, score-0.092]

72 Method GLC LAC GG GGV (proposed) Spatial information √ √ Fisher information metric √ √ In the second experiment, we compared GGVs with the Improved Fisher kernel (IFK) [24], [25], which is the best image representation available at the time of writing. [sent-199, score-0.175]

73 In this experiment, we used the spatial pyramid representation with a 1 × 1 + 2 × 2 + 3 × 1 structure. [sent-200, score-0.128]

74 For all datasets, SIFT features were densely sampled and were described for 16 × 16 patches. [sent-203, score-0.077]

75 As the aforementioned features depend on the dimensionality of the local feature, we reduced its dimensionality using PCA and compared performance as a function of its new dimensionality. [sent-205, score-0.348]

76 Before comparison between GGVs and the baselines, we evaluate the sensitivities of the sampling step of local features. [sent-212, score-0.105]

77 The sampling step is one of the important parameters of GGV, because GGV calculates auto-correlations of the neighboring local features. [sent-213, score-0.125]

78 In this preliminary experiment, we ﬁx the number of vertices is 5 (n = 5) and the dimensionality of local feature is 32. [sent-214, score-0.275]

79 Therefore in the following experiments, we use 6 pixels sampling step for local feature extraction. [sent-224, score-0.186]

80 Figure 4 (left) shows the classiﬁcation accuracies as a function of the dimensionality of the local features. [sent-225, score-0.225]

81 A large dimensionality yielded better performance, and the proposed method (GGV) outperformed the other methods (GLC, LAC, and GG). [sent-226, score-0.099]

82 By comparing GGV with LAC, and GG with GLC, it is clear that embedding the Fisher information metric improved the classiﬁcation accuracy signiﬁcantly. [sent-227, score-0.095]

83 By comparing GGV with GG, as well as LAC with GLC, it can also be seen that embedding the spatial layout of local features also improved the accuracy. [sent-228, score-0.369]

84 4 (right) shows the classiﬁcation accuracy as a function of the dimensionality of the image features which are converted from the results shown in Fig. [sent-238, score-0.273]

85 We see that GGVs achieved higher accuracy for a lower dimensionality of image features. [sent-240, score-0.196]

86 3 % when the dimensionality of the local feature is 32 and the number of vertices is 5. [sent-246, score-0.275]

87 5 (center) and (right) show comparisons of the L2 normalized GGVs and IFKs using the Caltech256 dataset with respect to the dimensionality of local features and image features, respectively. [sent-256, score-0.378]

88 It is known that using multi-scale local features improves classiﬁcation accuracies (e. [sent-264, score-0.219]

89 In the second experiment, the results with respect to the dimensionality of local features and image features are shown in Figs. [sent-280, score-0.455]

90 As the leading method, the spatial Fisher kernel [17] reported the highest score (88. [sent-285, score-0.092]

91 4 Conclusion In this paper, we proposed an efﬁcient image feature called a Graphical Gaussian Vector, which uses neither codebook nor local feature matching. [sent-288, score-0.377]

92 In the proposed method, spatial information about local features and the Fisher information metric are embedded into a feature by modeling the image as the Gaussian Markov Random Field (GMRF). [sent-289, score-0.52]

93 The proposed image feature calculates the expectation parameters of the GMRF simply and effectively while maintaining a high classiﬁcation rate. [sent-291, score-0.267]

94 Improving local descriptors by embedding global and local spatial information. [sent-385, score-0.373]

95 Asymmetric region-to-image matching for comparing images with generic object categories. [sent-394, score-0.075]

96 Modeling spatial layout with ﬁsher vectors for image categorization. [sent-400, score-0.267]

97 Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories. [sent-406, score-0.096]

98 Global gaussian approach for scene categorization using information geometry. [sent-418, score-0.091]

99 Linear spatial pyramid matching using sparse coding for image classiﬁcation. [sent-469, score-0.262]

100 Image classiﬁcation using super-vector coding of local image descriptors. [sent-477, score-0.218]

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