nips nips2013 nips2013-167 knowledge-graph by maker-knowledge-mining

167 nips-2013-Learning the Local Statistics of Optical Flow

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Author: Dan Rosenbaum, Daniel Zoran, Yair Weiss

Abstract: Motivated by recent progress in natural image statistics, we use newly available datasets with ground truth optical ﬂow to learn the local statistics of optical ﬂow and compare the learned models to prior models assumed by computer vision researchers. We ﬁnd that a Gaussian mixture model (GMM) with 64 components provides a signiﬁcantly better model for local ﬂow statistics when compared to commonly used models. We investigate the source of the GMM’s success and show it is related to an explicit representation of ﬂow boundaries. We also learn a model that jointly models the local intensity pattern and the local optical ﬂow. In accordance with the assumptions often made in computer vision, the model learns that ﬂow boundaries are more likely at intensity boundaries. However, when evaluated on a large dataset, this dependency is very weak and the beneﬁt of conditioning ﬂow estimation on the local intensity pattern is marginal. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 il 1 Abstract Motivated by recent progress in natural image statistics, we use newly available datasets with ground truth optical ﬂow to learn the local statistics of optical ﬂow and compare the learned models to prior models assumed by computer vision researchers. [sent-4, score-1.213]

2 We ﬁnd that a Gaussian mixture model (GMM) with 64 components provides a signiﬁcantly better model for local ﬂow statistics when compared to commonly used models. [sent-5, score-0.164]

3 We also learn a model that jointly models the local intensity pattern and the local optical ﬂow. [sent-7, score-0.86]

4 In accordance with the assumptions often made in computer vision, the model learns that ﬂow boundaries are more likely at intensity boundaries. [sent-8, score-0.471]

5 However, when evaluated on a large dataset, this dependency is very weak and the beneﬁt of conditioning ﬂow estimation on the local intensity pattern is marginal. [sent-9, score-0.465]

6 We leverage these newly available resources to learn the statistics of optical ﬂow and compare this to assumptions used by computer vision researchers. [sent-11, score-0.496]

7 Interestingly, the best models in terms of log likelihood, when used as priors in image restoration tasks, also yield state-of-the-art performance [14]. [sent-14, score-0.242]

8 A notable example is the computation of optical ﬂow: a vector at every pixel that corresponds to the two dimensional projection of the motion 1 at that pixel. [sent-16, score-0.475]

9 Since local motion information is often ambiguous, nearly all optical ﬂow estimation algorithms work by minimizing a cost function that has two terms: a local data term and a “prior” term (see. [sent-17, score-0.593]

10 Given the success in image restoration tasks, where learned priors give state-of-the-art performance, one might expect a similar story in optical ﬂow estimation. [sent-21, score-0.591]

11 However, with the notable exception of [9] (which served as a motivating example for this work and is discussed below) there have been very few attempts to learn priors for optical ﬂow by modeling local statistics. [sent-22, score-0.51]

12 Instead, the state-ofthe-art methods still use priors that were formulated by computer vision researchers. [sent-23, score-0.177]

13 In fact, two of the top performing methods in modern optical ﬂow benchmarks use a hand-deﬁned smoothness constraint that was suggested over 20 years ago [6, 2]. [sent-24, score-0.495]

14 One big difference between image statistics and ﬂow statistics is the availability of ground truth data. [sent-25, score-0.263]

15 Whereas for modeling image statistics one merely needs a collection of photographs (so that the amount of data is essentially unlimited these days), for modeling ﬂow statistics one needs to obtain the ground truth motion of the points in the scene. [sent-26, score-0.349]

16 In the past, the lack of availability of ground truth data did not allow for learning an optical ﬂow prior from examples. [sent-27, score-0.559]

17 The Sintel dataset (ﬁgure 1) consists of a thousand pairs of frames from a highly realistic computer graphics ﬁlm with a wide variety of locations and motion types. [sent-29, score-0.195]

18 The vehicle was equipped with accurate range ﬁnders as well as accurate localization of its own motion, and the combination of these two sources allow computing optical ﬂow for points that are stationary in the world. [sent-32, score-0.363]

19 Although this is real data, it is sparse (only about 50% of the pixels have ground truth ﬂow). [sent-33, score-0.154]

20 In this paper we leverage the availability of ground truth datasets to learn explicit statistical models of optical ﬂow. [sent-34, score-0.578]

21 We compare our learned model to the assumptions made by computer vision algorithms for estimating ﬂow. [sent-35, score-0.159]

22 We ﬁnd that a Gaussian mixture model with 64 components provides a signiﬁcantly better model for local ﬂow statistics when compared to commonly used models. [sent-36, score-0.164]

23 We also learn a model that jointly models the local intensity pattern and the local optical ﬂow. [sent-38, score-0.86]

24 In accordance with the assumptions often made in computer vision, the model learns that ﬂow boundaries are more likely at intensity boundaries. [sent-39, score-0.471]

25 However, when evaluated on a large dataset, this dependency is very weak and the beneﬁt of conditioning ﬂow estimation on the local intensity pattern is marginal. [sent-40, score-0.465]

26 1 Priors for optical ﬂow One of the earliest methods for optical ﬂow that is still used in applications is the celebrated LucasKanade algorithm [7]. [sent-42, score-0.684]

27 It overcomes the local ambiguity of motion analysis by assuming that the optical ﬂow is constant within a small image patch and ﬁnds this constant motion by least-squares estimation. [sent-43, score-0.788]

28 It ﬁnds the optical ﬂow by minimizing a cost function that has a data term and a “smoothness” term. [sent-45, score-0.342]

29 Denoting by u the horizontal ﬂow and v the vertical ﬂow, the smoothness term is of the form: 2 2 u2 + u2 + vx + vy x y JHS = x,y where ux , uy are the spatial derivatives of the horizontal ﬂow u and vx , vy are the spatial derivatives of the vertical ﬂow v. [sent-46, score-0.582]

30 Rather than using a quadratic smoothness term, many authors have advocated using more robust terms that would be less sensitive to outliers in smoothness. [sent-48, score-0.144]

31 Both the Lorentzian and the absolute value robust smoothness 2 terms were shown to outperform quadratic smoothness in [11] and the absolute value was better among the two robust terms. [sent-51, score-0.288]

32 Several authors have also suggested that the smoothness term be based on the local intensity pattern, since motion discontinuities are more likely to occur at intensity boundaries. [sent-52, score-1.08]

33 Ren [8] modiﬁed the weights in the Lucas and Kanade least-squares estimation so that pixels that are on different sides of an intensity boundary will get lower weights. [sent-53, score-0.411]

34 Perhaps the simplest such regularizer is of the form: 2 2 w(Ix )(u2 + vx ) + w(Iy )(u2 + vy ) x y JHSI = (1) x,y As we discuss below, this prior can be seen as a Gaussian prior on the ﬂow that is conditioned on the intensity. [sent-55, score-0.23]

35 They used a training set of optical ﬂow obtained by simulating the motion of a camera in natural range images. [sent-57, score-0.475]

36 The prior learned by their system was similar to a robust smoothness prior, but the ﬁlters are not local derivatives but rather more random-looking high pass ﬁlters. [sent-58, score-0.347]

37 One way to answer this question is to use these priors as a basis for an optical ﬂow estimation algorithm and see which algorithm gives the best performance. [sent-61, score-0.428]

38 [11] reported that adding a non-local smoothness term to a robust smoothness prior signiﬁcantly improved results on the Middlebury benchmark, while Geiger et al. [sent-64, score-0.299]

39 Perhaps the main difﬁculty with this approach is that the prior is only one part of an optical ﬂow estimation algorithm. [sent-66, score-0.382]

40 Motivated by recent advances in natural image statistics and the availability of new datasets, we compare different priors in terms of (1) log likelihood on held-out data and (2) inference performance with tractable posteriors. [sent-70, score-0.263]

41 2 Comparing priors as density models In order to compare different prior models as density models, we generate a training set and test set of optical ﬂow patches from the ground truth databases. [sent-72, score-0.91]

42 Denoting by f a single vector that concatenates all the optical ﬂow in a patch (e. [sent-73, score-0.419]

43 Given a prior probability model Pr(f ; θ) we use the training set to estimate the free parameters of the model θ and then we measure the log likelihood of held out patches from the test set. [sent-76, score-0.396]

44 From Sintel, we divided the pairs of frames for which ground truth is available into 708 pairs which we used for training and 333 pairs which we used for testing. [sent-77, score-0.169]

45 We created a second test set from the KITTI dataset by choosing a subset of patches for which full ground truth ﬂow was available. [sent-79, score-0.37]

46 By exponentiating JHS we see that Pr(f ; θ) is a multidimensional Gaussian with covariance matrix λDDT where D is a 256 × 128 derivative matrix that computes the derivatives of the ﬂow ﬁeld at each pixel and λ is the weight given to the prior relative to the data term. [sent-85, score-0.152]

47 Motivated by the success of GMMs in modeling natural image statistics [14] we use the training set to estimate GMM priors for optical ﬂow. [sent-93, score-0.517]

48 Each mixture component is a multidimensional Gaussian with full covariance matrix and zero mean and we vary the number of components between 1 and 64. [sent-94, score-0.168]

49 Even with a few mixture components, the GMM has far more free parameters than the previous models but note that we are measuring success on held out patches so that models that overﬁt should be penalized. [sent-96, score-0.411]

50 One interesting thing that can be seen is that the local statistics validate some assumptions commonly used by computer vision researchers. [sent-98, score-0.232]

51 For example, the Horn and Shunck smoothness prior is as good as the optimal Gaussian prior (GMM1) even though it uses local ﬁrst derivatives. [sent-99, score-0.254]

52 It can be seen that the GMM is indeed much better than the other priors including cases where the test set is taken from KITTI (rather than Sintel) and when the patch size is 12 × 12 rather than 8 × 8. [sent-104, score-0.183]

53 5 log-likelihood 4 3 2 1 0 LK HS L1 GMM1 GMM2 GMM4 Models GMM8 GMM16 GMM64 Figure 2: mean log likelihood of the different models for 8 × 8 patches extracted from held out data from Sintel. [sent-105, score-0.392]

54 1 Comparing models using tractable inference A second way of comparing the models is by their ability to restore corrupted patches of optical ﬂow. [sent-108, score-0.654]

55 We are not claiming that optical ﬂow restoration is a real-world application (although using priors to “ﬁll in” holes in optical ﬂow is quite common, e. [sent-109, score-0.816]

56 3 The secret of the GMM We now take a deeper look at how the GMM models optical ﬂow patches. [sent-126, score-0.378]

57 The ﬁrst (and not surprising) thing we found is that the covariance matrices learned by the model are block diagonal (so that the u and v components are independent given the assignment to a particular component). [sent-127, score-0.153]

58 More insight can be gained by considering the GMM as a local subspace model: a patch which is generated by component k is generated as a linear combination of the eigenvectors of the kth covariance. [sent-128, score-0.264]

59 The coefﬁcients of the linear combination have energy that decays with the eigenvalue: so each patch can be well approximated by the leading eigenvectors of the corresponding covariance. [sent-129, score-0.138]

60 Unlike global subspace models, different subspace models can be used for different patches, and during inference with the model one can infer which local subspace is most likely to have generated the patch. [sent-130, score-0.253]

61 Figure 6 shows the dominant leading eigenvectors of all 32 covariance matrices in the GMM32 model: the eigenvectors of u are followed by the eigenvectors of v. [sent-131, score-0.205]

62 The right hand half of each row shows (u,v) patches that are sampled from that Gaussian. [sent-134, score-0.24]

63 It can be seen that the ﬁrst 10 components or so model very smooth components (in fact the samples appear to be completely ﬂat). [sent-137, score-0.14]

64 This can also be seen by comparing the v samples on the top row which are close to gray with those in the next two rows which are much closer to black or white (since the models are zero mean, black and white are equally likely for any component). [sent-139, score-0.18]

65 Thus these components are very similar to a non-local smoothness assumption similar to the one suggested in [11]): they not only assume that derivatives are small but they assume that all the 8 × 8 patch is constant. [sent-141, score-0.345]

66 However, unlike the suggestion in [11] to enforce non-local smoothness by applying a median ﬁlter at all pixels, the GMM only applies non-local smoothness at a subset of patches that are inferred to be generated by such components. [sent-142, score-0.47]

67 we see that the components no longer model ﬂat components but rather motion boundaries. [sent-144, score-0.253]

68 For example, the bottom row of the ﬁgure illustrates a component that seems to generate primarily diagonal motion boundaries. [sent-146, score-0.16]

69 Interestingly, such local subspace models of optical ﬂow have also been suggested by Fleet et al. [sent-147, score-0.515]

70 They used synthetic models of moving occlusion boundaries and bars to learn linear subspace models of the ﬂow. [sent-149, score-0.186]

71 The GMM seems to support their intuition that learning separate linear subspace models for ﬂat vs motion boundary is a good idea. [sent-150, score-0.278]

72 6 leading eigenvectors u patch samples v u v Figure 6: The eigenvectors and samples of the GMM components. [sent-154, score-0.199]

73 GMM is better because it explicitly models edges and ﬂat patches separately. [sent-155, score-0.276]

74 4 A joint model for optical ﬂow and intensity As mentioned in the introduction, many authors have suggested modifying the smoothness assumption by conditioning it on the local intensity pattern and giving a higher penalty for motion discontinuities in the absence of intensity discontinuities. [sent-156, score-1.771]

75 We therefore ask, does conditioning on the local intensity give better log likelihood on held out ﬂow patches? [sent-157, score-0.539]

76 We evaluated two ﬂow models that are conditioned on the local intensity pattern. [sent-159, score-0.433]

77 The second one is a Gaussian mixture model that simultaneously models both intensity and ﬂow. [sent-164, score-0.378]

78 The simultaneous GMM we use includes a 200 component GMM to model the intensity together with a 64 dimensional GMM to model the ﬂow. [sent-165, score-0.345]

79 We allow a dependence between the hidden variable of the intensity GMM and that of the ﬂow GMM. [sent-166, score-0.337]

80 This is equivalent to a hidden Markov model (HMM) with 2 hidden variables: one represents the intensity component and one represents the ﬂow component (ﬁgure 8). [sent-167, score-0.41]

81 Initialization is given by independent GMMs learned for the intensity (we actually use the one learned by [14] which is available on their website) and for the ﬂow. [sent-169, score-0.416]

82 The intensity GMM is not changed during the learning. [sent-170, score-0.318]

83 Conditioned on the intensity pattern, the ﬂow distribution is still a GMM with 64 components (as in the previous section) but the mixing weights depend on the intensity. [sent-171, score-0.378]

84 Conditioning on the intensity gives basically zero improvement in log likelihood and a slight improvement in ﬂow denoising only for very large amounts of noise. [sent-174, score-0.428]

85 To investigate this effect, we examine the transition matrix between the intensity components and the ﬂow components (ﬁgure 8). [sent-176, score-0.457]

86 If intensity and ﬂow were independent, we would expect all rows of the transition matrix to be the same. [sent-177, score-0.365]

87 If an intensity boundary always lead to a ﬂow boundary, we would expect the bottom rows of the matrix to have only one nonzero element. [sent-178, score-0.415]

88 7 Regardless of whether the intensity component corresponds to a boundary or not, the most likely ﬂow components are ﬂat. [sent-180, score-0.512]

89 When there is an intensity boundary, the ﬂow boundary in the same orientation becomes more likely. [sent-181, score-0.387]

90 To rule out that this effect is due to a local optimum found by EM, we conducted additional experiments whereby the emission probabilities were held ﬁxed to the GMMs learned independently for ﬂow and motion and each patch in the training set was assigned one intensity and one ﬂow component. [sent-183, score-0.687]

91 We then estimated the joint distribution over ﬂow and motion components by simply counting the relative frequency in the training set. [sent-184, score-0.193]

92 In summary, while our learned model supports the standard intuition that motion boundaries are more likely at intensity boundaries, it suggests that when dealing with a large dataset with high variability, there is very little beneﬁt (if any) in conditioning ﬂow models on the local intensity. [sent-186, score-0.73]

93 un-conditional mixing-weights h intensity 50 intensity conditional mixing-weights 100 150 200 10 20 30 40 h ﬂow 50 60 Figure 8: Left: the transition matrix learned by the HMM. [sent-189, score-0.704]

94 Conditioned on an intensity boundary, motion boundaries become more likely but are still less likely than a ﬂat motion. [sent-191, score-0.578]

95 In this paper, we have leveraged the availability of large ground truth databases to learn priors from data and compare our learned models to the assumptions typically made by computer vision researchers. [sent-193, score-0.481]

96 the Horn and Schunck model is close to the optimal Gaussian model, robust models are better, intensity discontinuities make motion discontinuities more likely). [sent-196, score-0.638]

97 However, a learned GMM model with 64 components signiﬁcantly outperforms the standard models used in computer vision, primarily because it explicitly distinguishes between ﬂat patches and boundary patches and then uses a different form of nonlocal smoothness for the different cases. [sent-197, score-0.832]

98 A framework for the robust estimation of optical ﬂow. [sent-206, score-0.371]

99 A naturalistic open source movie for optical ﬂow evaluation. [sent-212, score-0.342]

100 Design and use of linear models for image motion analysis. [sent-218, score-0.213]

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