nips nips2003 nips2003-54 knowledge-graph by maker-knowledge-mining

54 nips-2003-Discriminative Fields for Modeling Spatial Dependencies in Natural Images

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Author: Sanjiv Kumar, Martial Hebert

Abstract: In this paper we present Discriminative Random Fields (DRF), a discriminative framework for the classiﬁcation of natural image regions by incorporating neighborhood spatial dependencies in the labels as well as the observed data. The proposed model exploits local discriminative models and allows to relax the assumption of conditional independence of the observed data given the labels, commonly used in the Markov Random Field (MRF) framework. The parameters of the DRF model are learned using penalized maximum pseudo-likelihood method. Furthermore, the form of the DRF model allows the MAP inference for binary classiﬁcation problems using the graph min-cut algorithms. The performance of the model was veriﬁed on the synthetic as well as the real-world images. The DRF model outperforms the MRF model in the experiments. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract In this paper we present Discriminative Random Fields (DRF), a discriminative framework for the classiﬁcation of natural image regions by incorporating neighborhood spatial dependencies in the labels as well as the observed data. [sent-3, score-0.383]

2 The proposed model exploits local discriminative models and allows to relax the assumption of conditional independence of the observed data given the labels, commonly used in the Markov Random Field (MRF) framework. [sent-4, score-0.226]

3 The parameters of the DRF model are learned using penalized maximum pseudo-likelihood method. [sent-5, score-0.122]

4 how to model arbitrarily complex dependencies in the observed image data as well as the labels in a principled manner. [sent-13, score-0.201]

5 Let the corresponding labels at the image sites be given by x = {xi }i∈S . [sent-17, score-0.208]

6 In the MRF framework, the posterior over the labels given the data is expressed using the Bayes’ rule as, p(x|y) ∝ p(x, y) = p(x)p(y|x) where the prior over labels, p(x) is modeled as a MRF. [sent-18, score-0.12]

7 The data belonging to such a class is highly dependent on its neighbors since the lines or edges at spatially adjoining sites follow some underlying organization rules rather than being random (See Fig. [sent-26, score-0.167]

8 Now considering a different point of view, for classiﬁcation purposes, we are interested in estimating the posterior over labels given the observations, i. [sent-31, score-0.095]

9 In a generative framework, one expends efforts to model the joint distribution p(x, y), which involves implicit modeling of the observations. [sent-34, score-0.113]

10 In a discriminative framework, one models the distribution p(x|y) directly. [sent-35, score-0.154]

11 As noted in [2], a potential advantage of using the discriminative approach is that the true underlying generative model may be quite complex even though the class posterior is simple. [sent-36, score-0.336]

12 This means that the generative approach may spend a lot of resources on modeling the generative models which are not particularly relevant to the task of inferring the class labels. [sent-37, score-0.144]

13 This approach allows one to capture arbitrary dependencies between the observations without resorting to any model approximations. [sent-42, score-0.102]

14 Our model further enhances the CRFs by proposing the use of local discriminative models to capture the class associations at individual sites as well as the interactions in the neighboring sites on 2-D grid lattices. [sent-43, score-0.394]

15 The proposed model uses local discriminative models to achieve the site classiﬁcation while permitting interactions in both the observed data and the label ﬁeld in a principled manner. [sent-44, score-0.376]

16 With a slight abuse of notations, in the rest of the paper we will call Ai as association potential and Iij as interaction potential. [sent-57, score-0.344]

17 In the DRFs, the association potential is seen as a local decision term which decides the association of a given site to a certain class ignoring its neighbors. [sent-59, score-0.416]

18 The interaction potential is seen as a data dependent smoothing function. [sent-60, score-0.302]

19 1 Association potential In the DRF framework, A(xi , y) is modeled using a local discriminative model that outputs the association of the site i with class xi . [sent-67, score-0.591]

20 Generalized Linear Models (GLM) are used extensively in statistics to model the class posteriors given the observations [8]. [sent-68, score-0.117]

21 For each site i, let f i (y) be a function that maps the observations y on a feature vector such that f i : y → l . [sent-69, score-0.206]

22 Using a logistic function as the link, the local class posterior can be modeled as, 1 P (xi = 1|y) = = σ(w0 + wT f i (y)) (2) 1 −(w0 +w T f i (y )) 1 1+e where w = {w0 , w1 } are the model parameters. [sent-70, score-0.216]

23 To extend the logistic model to induce a nonlinear decision boundary in the feature space, a transformed feature vector at each site i is deﬁned as, hi (y) = [1, φ1 (f i (y)), . [sent-71, score-0.413]

24 Further, since xi ∈ {−1, 1}, the probability in (2) can be compactly expressed as P (xi |y) = σ(xi wT hi (y)). [sent-77, score-0.135]

25 Finally, the association potential is deﬁned as, A(xi , y) = log(σ(xi wT hi (y)) (3) This transformation makes sure that the DRF yields standard logistic classiﬁer if the interaction potential in (1) is set to zero. [sent-78, score-0.584]

26 Note that the transformed feature vector at each site i, i. [sent-79, score-0.191]

27 hi (y) is a function of whole set of observations y. [sent-81, score-0.109]

28 y i to get the log-likelihood, which acts as the association potential. [sent-84, score-0.115]

29 [2] used the scaled likelihoods to approximate the actual likelihoods at each site required by the generative formulation. [sent-86, score-0.229]

30 These scaled likelihoods were obtained by scaling the local class posteriors learned using a neural network. [sent-87, score-0.116]

31 On the contrary, in the DRF model, the local class posterior is an integral part of the full conditional model in (1). [sent-88, score-0.122]

32 Also, unlike [2], the parameters of the association and interaction potential are learned simultaneously in the DRF framework. [sent-89, score-0.411]

33 2 Interaction potential To model the interaction potential I, we ﬁrst analyze the interaction potential commonly used in the MRF framework. [sent-91, score-0.598]

34 Note that the MRF framework does not permit the use of data in the interaction potential. [sent-92, score-0.229]

35 For a homogeneous and isotropic Ising model, the interaction potential is given as I = βxi xj , which penalizes every dissimilar pair of labels by the cost β [1]. [sent-93, score-0.408]

36 This form of interaction prefers piecewise constant smoothing without explicitly considering discontinuities in the data. [sent-94, score-0.25]

37 In the DRF formulation, the interaction potential is a function of all the observations y. [sent-95, score-0.289]

38 We would like to have similar labels at a pair of sites for which the observed data supports such a hypothesis. [sent-96, score-0.171]

39 In other words, we are interested in learning a pairwise discriminative model as the interaction potential. [sent-97, score-0.417]

40 For a pair of sites (i, j), let µij (ψ i (y), ψ j (y)) be a new feature vector such that µij : γ × γ → q , where ψ k : y → γ . [sent-98, score-0.132]

41 Denoting this feature vector as µij (y) for simpliﬁcation, the interaction potential is modeled as, I(xi , xj , y) = xi xj v T µij (y) (4) where v are the model parameters. [sent-99, score-0.447]

42 This form of interaction potential is much simpler than the one proposed in [7], and makes the parameter learning a convex problem. [sent-101, score-0.252]

43 There are two interesting properties of the interaction potential given in (4). [sent-102, score-0.252]

44 First, if the association potential at each site and the interaction potentials of all the pairwise cliques except the pair (i, j) are set to zero in (1), the DRF acts as a logistic classiﬁer which yields the probability of the site pair to have the same labels given the observed data. [sent-103, score-0.959]

45 Second, the proposed interaction potential is a generalization of the Ising model. [sent-104, score-0.252]

46 Thus, the form in (4) acts as a data-dependent discontinuity adaptive model that will moderate smoothing when the data from the two sites is ’different’. [sent-106, score-0.187]

47 The data-dependent smoothing can especially be useful to absorb the errors in modeling the association potential. [sent-107, score-0.173]

48 Anisotropy can be easily included in the DRF model by parametrizing the interaction potentials of different directional pairwise cliques with different sets of parameters v. [sent-108, score-0.346]

49 The form of the DRF model resembles the posterior of the MRF framework assuming conditionally independent data. [sent-110, score-0.084]

50 However, in the MRF framework, the parameters of the class generative models, p(y i |xi ) and the parameters of the prior random ﬁeld on labels, p(x) are generally assumed to be independent and learned separately [1]. [sent-111, score-0.196]

51 However, for the Ising model in MRFs, pseudo-likelihood tends to overestimate the interaction parameter β, causing the MAP estimates of the ﬁeld to be very poor solutions [9]. [sent-119, score-0.245]

52 Our experiments in the previous work [7] and Section 4 of this paper verify these observations for the interaction parameters in DRFs too. [sent-120, score-0.267]

53 Similar to the concept of weight decay in neural learning literature, we assume a Gaussian prior over the interaction parameters v such that p(v|τ ) = N (v; 0, τ 2 I) where I is the identity matrix. [sent-122, score-0.23]

54 The problem of inference is to ﬁnd the optimal label conﬁguration x given an image y, where optimality is deﬁned with respect to a cost function. [sent-133, score-0.091]

55 However, since these algorithms do not allow negative interaction between the sites, the data-dependent smoothing for each clique is set to be v Tµij (y) = max{0, v Tµij (y)}, yielding an approximate MAP estimate. [sent-137, score-0.267]

56 This is equivalent to switching the smoothing off at the image discontinuities. [sent-138, score-0.104]

57 4 Experiments and discussion For comparison, a MRF framework was also learned assuming a conditionally independent likelihood and a homogeneous and isotropic Ising interaction model. [sent-139, score-0.312]

58 So, the MRF −1 posterior is p(x|y) = Zm exp i∈S log p(si (y i )|xi ) + i∈S j∈Ni βxi xj where β is the interaction parameter and si (y i ) is a single-site feature vector at ith site such that si : y i → d . [sent-140, score-0.553]

59 Note that si (y i ) does not take into account inﬂuence of the data in the neighborhood of ith site. [sent-141, score-0.113]

60 A ﬁrst order neighborhood (4 nearest neighbors) was used for label interaction in all the experiments. [sent-142, score-0.278]

61 1 Synthetic images The aim of these experiments was to obtain correct labels from corrupted binary images. [sent-144, score-0.167]

62 For each noise model, 50 images were generated from each base image. [sent-148, score-0.131]

63 Each pixel was considered as an image site and the feature vector si (y i ) was simply chosen to be a scalar representing the intensity at ith site. [sent-149, score-0.295]

64 In experiments with the synthetic data, no neighborhood data interaction was used for the DRFs (i. [sent-150, score-0.272]

65 f i (y) = si (y i )) to observe the gains only due to the use of discriminative models in the association and interaction potentials. [sent-152, score-0.517]

66 A linear discriminant was implemented in the association potential such that hi (y) = [1, f i (y)]T . [sent-153, score-0.216]

67 The pairwise data vector µij (y) was obtained by taking the absolute difference of si (y i ) and sj (y j ). [sent-154, score-0.092]

68 1 was used for training while 150 noisy images from the rest of the three base images were used for testing. [sent-157, score-0.162]

69 (i) The interaction parameters for the DRF (v) as well as for the MRF (β) were set to zero. [sent-159, score-0.23]

70 This reduces the DRF model to a logistic classiﬁer and MRF to a maximum likelihood (ML) classiﬁer. [sent-160, score-0.124]

71 (iii) Finally, the DRF parameters were learned using penalized pseudo-likelihood and the best β for the MRF was chosen from cross-validation. [sent-168, score-0.097]

72 The MAP estimates of the labels were obtained using graph-cuts for both the models. [sent-169, score-0.085]

73 Under the ﬁrst noise model, each image pixel was corrupted with independent Gaussian noise of standard deviation 0. [sent-170, score-0.161]

74 The pixelwise classiﬁcation error for this noise model is given in the top row of Table 1. [sent-174, score-0.119]

75 Since the form of noise is the same as the likelihood model in the MRF, MRF is Table 1: Pixelwise classiﬁcation errors (%) on 150 synthetic test images. [sent-175, score-0.092]

76 From top, ﬁrst row:original images, second row: images corrupted with ’bimodal’ noise, third row: MRF results, fourth row: DRF results. [sent-193, score-0.102]

77 The DRF model is affected more because all the parameters in DRFs are learned simultaneously unlike MRFs. [sent-198, score-0.092]

78 In the second noise model each pixel was corrupted with independent mixture of Gaussian noise. [sent-199, score-0.114]

79 An interesting point to note is that DRF yields lower error than MRF even when the logistic classiﬁer has higher error than the ML classiﬁer on the test data. [sent-211, score-0.116]

80 FP for logistic classiﬁer were kept to be the same as for DRF for DR comparison. [sent-215, score-0.099]

81 Superscript − indicates no neighborhood data interaction was used. [sent-216, score-0.241]

82 2 Real-World images The proposed DRF model was applied to the task of detecting man-made structures in natural scenes. [sent-228, score-0.092]

83 The aim was to label each image site as structured or nonstructured. [sent-229, score-0.251]

84 The training and the test set contained 108 and 129 images respectively, each of size 256×384 pixels, from the Corel image database. [sent-230, score-0.121]

85 For each image site i, a 5-dim single-site feature vector si (y i ) and a 14-dim multiscale feature vector f i (y) is computed using orientation and magnitude based features as described in [16]. [sent-232, score-0.334]

86 Note that f i (y) incorporates data interaction from neighboring sites. [sent-233, score-0.2]

87 For the association potentials, a transformed feature vector hi (y) was computed at each site i using quadratic transforms of vector f i (y). [sent-234, score-0.355]

88 For the MRF, each class conditional density was modeled as a mixture of ﬁve Gaussians. [sent-238, score-0.11]

89 For two typical images from the test set, the detection results for the MRF and the DRF models are given in Fig. [sent-240, score-0.111]

90 For a quantitative evaluation, we compared the detection rates and the number of false positives per image for different techniques. [sent-244, score-0.176]

91 For the comparison of detection rates, in all the experiments, the decision threshold of the logistic classiﬁer was ﬁxed such that it yields the same false positive rate as the DRF. [sent-245, score-0.2]

92 Thus, no neighborhood data interaction was used for both the logistic classiﬁer and the DRF, i. [sent-247, score-0.34]

93 The detection rates of the MRF and the DRF are higher than the logistic classiﬁer due to the label interaction. [sent-251, score-0.18]

94 However, higher detection rate and lower false positives for the DRF in comparison to the MRF indicate the gains due to the use of discriminative models in the association and interaction potentials in the DRF. [sent-252, score-0.621]

95 In the next experiment, to take advantage of the power of the DRF framework, data interaction was allowed for both the logistic classiﬁer as well as the DRF (’Logistic’ and ’DRF’ in Table 2). [sent-253, score-0.299]

96 The DRF detection rate increases substantially and the false positives decrease further indicating the importance of allowing the data interaction in addition to the label interaction. [sent-254, score-0.359]

97 5 Conclusion and future work We have presented discriminative random ﬁelds which provide a principled approach for combining local discriminative classiﬁers that allow the use of arbitrary overlapping features, with adaptive data-dependent smoothing over the label ﬁeld. [sent-255, score-0.436]

98 A class of discrete multiresolution random ﬁelds and its application to image segmentation. [sent-293, score-0.115]

99 Discriminative random ﬁelds: A discriminative framework for contextual interaction in classiﬁcation. [sent-319, score-0.437]

100 Man-made structure detection in natural images using a causal multiscale random ﬁeld. [sent-374, score-0.166]

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