nips nips2004 nips2004-44 knowledge-graph by maker-knowledge-mining

44 nips-2004-Conditional Random Fields for Object Recognition

Source: pdf

Author: Ariadna Quattoni, Michael Collins, Trevor Darrell

Abstract: We present a discriminative part-based approach for the recognition of object classes from unsegmented cluttered scenes. Objects are modeled as ﬂexible constellations of parts conditioned on local observations found by an interest operator. For each object class the probability of a given assignment of parts to local features is modeled by a Conditional Random Field (CRF). We propose an extension of the CRF framework that incorporates hidden variables and combines class conditional CRFs into a uniﬁed framework for part-based object recognition. The parameters of the CRF are estimated in a maximum likelihood framework and recognition proceeds by ﬁnding the most likely class under our model. The main advantage of the proposed CRF framework is that it allows us to relax the assumption of conditional independence of the observed data (i.e. local features) often used in generative approaches, an assumption that might be too restrictive for a considerable number of object classes.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We present a discriminative part-based approach for the recognition of object classes from unsegmented cluttered scenes. [sent-3, score-0.383]

2 Objects are modeled as ﬂexible constellations of parts conditioned on local observations found by an interest operator. [sent-4, score-0.248]

3 For each object class the probability of a given assignment of parts to local features is modeled by a Conditional Random Field (CRF). [sent-5, score-0.607]

4 We propose an extension of the CRF framework that incorporates hidden variables and combines class conditional CRFs into a uniﬁed framework for part-based object recognition. [sent-6, score-0.627]

5 The parameters of the CRF are estimated in a maximum likelihood framework and recognition proceeds by ﬁnding the most likely class under our model. [sent-7, score-0.177]

6 local features) often used in generative approaches, an assumption that might be too restrictive for a considerable number of object classes. [sent-10, score-0.333]

7 1 Introduction The problem that we address in this paper is that of learning object categories from supervised data. [sent-11, score-0.206]

8 Given a training set of n pairs (xi , yi ), where xi is the ith image and yi is the category of the object present in xi , we would like to learn a model that maps images to object categories. [sent-12, score-1.101]

9 The part-based models we consider represent images as sets of patches, or local features, which are detected by an interest operator such as that described in [4]. [sent-14, score-0.204]

10 Thus an image xi can be considered to be a vector {xi,1 , . [sent-15, score-0.178]

11 Each patch xi,j has a feature-vector representation φ(xi,j ) ∈ Rd ; the feature vector might capture various features of the appearance of a patch, as well as features of its relative location and scale. [sent-19, score-0.523]

12 Moreover, the local patches underlying the local feature vectors may have complex interdependencies: for example, they may correspond to different parts of an object, whose spatial arrangement is important to the classiﬁcation task. [sent-24, score-0.518]

13 The most widely used approach for part-based object recognition is the generative model proposed in [1]. [sent-25, score-0.325]

14 , local features) given their assignment to parts in the model. [sent-29, score-0.314]

15 This assumption might be too restrictive for a considerable number of object classes made of structured patterns. [sent-30, score-0.276]

16 A second limitation of generative approaches is that they require a model P (xi,j |hi,j ) of patches xi,j given underlying variables hi,j (e. [sent-31, score-0.336]

17 , hi,j may be a hidden variable in the model, or may simply be yi ). [sent-33, score-0.277]

18 Accurately specifying such a generative model may be challenging – in particular in cases where patches overlap one another, or where we wish to allow a hidden variable hi,j to depend on several surrounding patches. [sent-34, score-0.384]

19 A more direct approach may be to use a feature vector representation of patches, and to use a discriminative learning approach. [sent-35, score-0.168]

20 Similar observations concerning the limitations of generative models have been made in the context of natural language processing, in particular in sequence-labeling tasks such as part-of-speech tagging [7, 5, 3] and in previous work on conditional random ﬁelds (CRFs) for vision [2]. [sent-37, score-0.137]

21 In sequence-labeling problems for NLP each observation xi,j is typically the j’th word for some input sentence, and hi,j is a hidden state, for example representing the part-of-speech of that word. [sent-38, score-0.142]

22 Hidden Markov models (HMMs), a generative approach, require a model of P (xi,j |hi,j ), and this can be a challenging task when features such as word preﬁxes or sufﬁxes are included in the model, or where hi,j is required to depend directly on words other than xi,j . [sent-39, score-0.208]

23 This has led to research on discriminative models for sequence labeling such as MEMM’s [7, 5] and conditional random ﬁelds (CRFs)[3]. [sent-40, score-0.237]

24 We propose a new model for object recognition based on Conditional Random Fields. [sent-42, score-0.272]

25 A key difference of our approach from previous work on CRFs is that we make use of hidden variables in the model. [sent-44, score-0.23]

26 , [2, 3]) each “label” yi is a sequence hi = {hi,1 , hi,2 , . [sent-47, score-0.135]

27 In our case the labels yi are unstructured labels from some ﬁxed set of object categories, and the relationship between yi and each observation xi,j is not clearly deﬁned. [sent-52, score-0.527]

28 Instead, we model intermediate part-labels hi,j as hidden variables in the model. [sent-53, score-0.271]

29 The model deﬁnes conditional probabilities P (y, h | x), and hence indirectly P (y | x) = h P (y, h | x), using a CRF. [sent-54, score-0.125]

30 Dependencies between the hidden variables h are modeled by an undirected graph over these variables. [sent-55, score-0.332]

31 The result is a model where inference and parameter estimation can be carried out using standard graphical model algorithms such as belief propagation [6]. [sent-56, score-0.359]

32 1 Conditional Random Fields with Hidden Variables Our task is to learn a mapping from images x to labels y. [sent-58, score-0.169]

33 1 Each patch xj is represented by a feature vector φ(xj ) ∈ Rd . [sent-64, score-0.224]

34 For example, in our experiments each xj corresponds to a patch that is detected by the feature detector in [4]; section [3] gives details of the feature-vector representation φ(xj ) for each patch. [sent-65, score-0.39]

35 Our training set consists of labeled images (xi , yi ) for i = 1 . [sent-66, score-0.301]

36 n, where each yi ∈ Y, and each xi = {xi,1 , xi,2 , . [sent-69, score-0.251]

37 For any image x we also assume a vector of “parts” variables h = {h1 , h2 , . [sent-73, score-0.15]

38 These variables are not observed on training examples, and will therefore form a set of hidden variables in the 1 Note that the number of patches m can vary across images, and did vary in our experiments. [sent-77, score-0.593]

39 For convenience we use notation where m is ﬁxed across different images; in reality it will vary across images but this leads to minor changes to the model. [sent-78, score-0.234]

40 Each hj is a member of H where H is a ﬁnite set of possible parts in the model. [sent-80, score-0.765]

41 Intuitively, each hj corresponds to a labeling of xj with some member of H. [sent-81, score-0.681]

42 Given these deﬁnitions of image-labels y, images x, and part-labels h, we will deﬁne a conditional probabilistic model: eΨ(y,h,x;θ) . [sent-82, score-0.207]

43 Ψ(y ,h,x;θ) y ,h e h (2) Given a new test image x, and parameter values θ∗ induced from a training example, we will take the label for the image to be arg maxy∈Y P (y | x, θ∗ ). [sent-86, score-0.307]

44 Following previous work on CRFs [2, 3], we use the following objective function in training the parameters: L(θ) = log P (yi | xi , θ) − i 1 ||θ||2 2σ 2 (3) The ﬁrst term in Eq. [sent-87, score-0.159]

45 We assume an undirected graph structure, with the hidden variables {h1 , . [sent-94, score-0.332]

46 We use E to denote the set of edges in the graph, and we will write (j, k) ∈ E to signify that there is an edge in the graph between variables hj and hk . [sent-98, score-0.921]

47 2 We deﬁne Ψ to take the following form: m 2 fl2 (j, k, y, hj , hk , x)θl 1 fl1 (j, y, hj , x)θl + Ψ(y, h, x; θ) = j=1 l (j,k)∈E (4) l 1 2 where fl1 , fl2 are functions deﬁning the features in the model, and θl , θl are the components 1 of θ. [sent-100, score-1.293]

48 The f features depend on single hidden variable values in the model, the f 2 features can depend on pairs of values. [sent-101, score-0.37]

49 Moreover the features respect the structure of the graph, in that no feature depends on more than two hidden variables hj , hk , and if a feature does depend on variables hj and hk there must be an edge (j, k) in the graph E. [sent-104, score-2.062]

50 4, then exact methods exist for inference and parameter estimation in the model. [sent-106, score-0.127]

51 This follows because belief propagation [6] can be used to calculate the following quantities in O(|E||Y|) time: ∀y ∈ Y, Z(y | x, θ) = exp{Ψ(y, h, x; θ)} h ∀y ∈ Y, j ∈ 1 . [sent-107, score-0.178]

52 m, a ∈ H, P (hj = a | y, x, θ) = P (h | y, x, θ) h:hj =a ∀y ∈ Y, (j, k) ∈ E, a, b ∈ H, P (h | y, x, θ) P (hj = a, hk = b | y, x, θ) = h:hj =a,hk =b 2 This will allow exact methods for inference and parameter estimation in the model, for example using belief propagation. [sent-110, score-0.39]

53 If E contains cycles then approximate methods, such as loopy beliefpropagation, may be necessary for inference and parameter estimation. [sent-111, score-0.157]

54 The second and third terms are marginal distributions over individual variables hj or pairs of variables hj , hk corresponding to edges in the graph. [sent-115, score-1.421]

55 2 Parameter Estimation Using Belief Propagation This section considers estimation of the parameters θ∗ = arg max L(θ) from a training sample, where L(θ) is deﬁned in Eq. [sent-118, score-0.131]

56 In our work we used a conjugate-gradient method to optimize L(θ) (note that due to the use of hidden variables, L(θ) has multiple local minima, and our method is therefore not guaranteed to reach the globally optimal point). [sent-120, score-0.195]

57 Ψ(y ,h,xi ;θ) y ,h e h Li (θ) = log P (yi | xi , θ) = log (5) 1 We ﬁrst consider derivatives with respect to the parameters θl corresponding to features fl1 (j, y, hj , x) that depend on single hidden variables. [sent-123, score-0.922]

58 We deﬁne Ψ(y, h, x; θ) = φ(xj ) · θ(hj ) + j θ(y, hj ) + j θ(y, hj , hk ) (6) (j,k)∈E Here θ(k) ∈ Rd for k ∈ H is a parameter vector corresponding to the k’th part label. [sent-128, score-1.338]

59 The inner-product φ(xj ) · θ(hj ) can be interpreted as a measure of the compatibility between patch xj and part-label hj . [sent-129, score-0.74]

60 Each parameter θ(y, k) ∈ R for k ∈ H, y ∈ Y can be interpreted as a measure of the compatibility between part k and label y. [sent-130, score-0.233]

61 Finally, each parameter θ(y, k, l) ∈ R for y ∈ Y, and k, l ∈ H measures the compatibility between an edge with labels k and l and the label y. [sent-131, score-0.234]

62 Hence belief propagation can be used for inference and parameter estimation in the model. [sent-135, score-0.277]

63 The patches xi,j in each image are obtained using the SIFT detector [4]. [sent-136, score-0.312]

64 Each patch xi,j is then represented by a feature vector φ(xi,j ) that incorporates a combination of SIFT and relative location and scale features. [sent-137, score-0.254]

65 The tree E is formed by running a minimum spanning tree algorithm over the parts hi,j , where the cost of an edge in the graph between hi,j and hi,k is taken to be the distance between xi,j and xi,k in the image. [sent-138, score-0.466]

66 Our choice of E encodes our assumption that parts conditioned on features that are spatially close are more likely to be dependent. [sent-140, score-0.301]

67 We also plan to investigate more complex graph structures that involve cycles, which may require approximate methods such as loopy belief propagation for parameter estimation and inference. [sent-142, score-0.346]

68 3 The ﬁrst two experiments consisted of training a two class model (object vs. [sent-144, score-0.134]

69 The third experiment consisted of training a multi-class model to distinguish between n classes. [sent-146, score-0.126]

70 The only parameter that was adjusted in the experiments was the scale of the images upon which the interest point detector was run. [sent-147, score-0.335]

71 In particular, we adjusted the scale on the car side data set: in this data set the images were too small and without this adjustment the detector would fail to ﬁnd a signiﬁcant amount of features. [sent-148, score-0.768]

72 a we show how the number of parts in the model affects performance. [sent-153, score-0.236]

73 In the case of the car side data set, the ten-part model shows a signiﬁcant improvement compared to the ﬁve parts model while for the car rear data set the performance improvement obtained by increasing the number of parts is not as signiﬁcant. [sent-154, score-1.484]

74 The model exhibits best performance for the Leopard data set, for which the presence of part 1 alone is a clear predictor of the class. [sent-160, score-0.118]

75 This shows again that our model can learn discriminative part distributions for each class. [sent-161, score-0.221]

76 Figure 3 shows results for a multi-view experiment where the task is two distinguish between two different views of a car and background. [sent-162, score-0.439]

77 Notice, that since our algorithm does not currently allow for the recognition of multiple instances of an object we test it on a partition of the the training set in http://l2r. [sent-171, score-0.311]

78 Figure 1: Examples of the most likely assignment of parts to features for the two class experiments (car data set). [sent-176, score-0.417]

79 Data set (a) Car Side Car Rear 5 parts 94 % 91 % 10 parts 99 % 91. [sent-177, score-0.39]

80 5 % Figure 2: (a) Equal Error Rates for the car side and car rear experiments with different number of parts. [sent-183, score-1.012]

81 Figure 1 displays the Viterbi labeling4 for a set of example images showing the most likely assignment of local features to parts in the model. [sent-185, score-0.543]

82 Figure 6 shows the mean and variance of each part’s location for car side images and background images. [sent-186, score-0.76]

83 The mean and variance of each part’s location for the car side images were calculated in the following manner: First we ﬁnd for every image classiﬁed as class a the most likely part assignment under our model. [sent-187, score-1.028]

84 Second, we calculate the mean and variance of positions of all local features that were assigned to the same part. [sent-188, score-0.231]

85 As can be seen in Figure 6 , while the mean location of a given part in the background images and the mean location of the same part in the car images are very similar, the parts in the car have a much tighter distribution which seems to suggest that the model is learning the shape of the object. [sent-190, score-1.639]

86 As shown in Figure 5 the model has also learnt discriminative part distributions for each class, for example the presence of part 1 seems to be a clear predictor for the car class. [sent-191, score-0.739]

87 In general part assignments seem to rely on a combination of appearance and relative location. [sent-192, score-0.193]

88 4 This is the labeling h∗ = arg maxh P (h | y, x, θ) where x is an image and y is the label for the image under the model. [sent-194, score-0.268]

89 It appears that the model has learnt a vocabulary of very general parts with signiﬁcant variability in appearance and learns to discriminate between classes by capturing the most likely arrangement of these parts for each class. [sent-201, score-0.739]

90 In some cases the model relies more heavily on relative location than appearance because the appearance information might not be very useful for discriminating between the two classes. [sent-202, score-0.401]

91 One of the reasons for this is that the detector produces a large number of false detections, making the appearance data too noisy for discrimination. [sent-203, score-0.254]

92 The fact that the model is able to cope with this lack of discriminating appearance information illustrates its ﬂexible data-driven nature. [sent-204, score-0.197]

93 This can be a desirable model property of a general object recognition system, because for some object classes appearance is the important discriminant (i. [sent-205, score-0.602]

94 One noticeable difference between our model and similar part-based models is that our model learns large parts composed of small local features. [sent-210, score-0.33]

95 , through their position in minimum spanning tree): the potential functions deﬁned on pairs of hidden variables tend to smooth the allocation of parts to patches. [sent-213, score-0.474]

96 Similarly to CRFs and other maximum entropy models our approach allows us to combine arbitrary observation features for training discriminative classiﬁers with hidden variables. [sent-215, score-0.395]

97 Furthermore, by making some assumptions about the joint distribution of hidden variables one can derive efﬁcient training algorithms based on dynamic programming. [sent-216, score-0.273]

98 The main limitation of our model is that it is dependent on the feature detector picking up discriminative features of the object. [sent-218, score-0.467]

99 Furthermore, our model might learn to discriminate between classes based on the statistics of the feature detector and not the true underlying data, to which it has no access. [sent-219, score-0.353]

100 This is not a desirable property since it assumes the feature detector to be consistent. [sent-220, score-0.203]

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