nips nips2008 nips2008-208 knowledge-graph by maker-knowledge-mining

208 nips-2008-Shared Segmentation of Natural Scenes Using Dependent Pitman-Yor Processes

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Author: Erik B. Sudderth, Michael I. Jordan

Abstract: We develop a statistical framework for the simultaneous, unsupervised segmentation and discovery of visual object categories from image databases. Examining a large set of manually segmented scenes, we show that object frequencies and segment sizes both follow power law distributions, which are well modeled by the Pitman–Yor (PY) process. This nonparametric prior distribution leads to learning algorithms which discover an unknown set of objects, and segmentation methods which automatically adapt their resolution to each image. Generalizing previous applications of PY processes, we use Gaussian processes to discover spatially contiguous segments which respect image boundaries. Using a novel family of variational approximations, our approach produces segmentations which compare favorably to state-of-the-art methods, while simultaneously discovering categories shared among natural scenes. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We develop a statistical framework for the simultaneous, unsupervised segmentation and discovery of visual object categories from image databases. [sent-7, score-0.6]

2 Examining a large set of manually segmented scenes, we show that object frequencies and segment sizes both follow power law distributions, which are well modeled by the Pitman–Yor (PY) process. [sent-8, score-0.444]

3 This nonparametric prior distribution leads to learning algorithms which discover an unknown set of objects, and segmentation methods which automatically adapt their resolution to each image. [sent-9, score-0.265]

4 Generalizing previous applications of PY processes, we use Gaussian processes to discover spatially contiguous segments which respect image boundaries. [sent-10, score-0.547]

5 Using a novel family of variational approximations, our approach produces segmentations which compare favorably to state-of-the-art methods, while simultaneously discovering categories shared among natural scenes. [sent-11, score-0.399]

6 We would like to build systems which can automatically discover the visual categories (e. [sent-13, score-0.186]

7 In simple cases, topic models can be used to cluster local textural elements, coarsely representing categories via a bag of visual features [1, 2]. [sent-17, score-0.259]

8 One approach to modeling additional spatial dependence begins by precomputing one, or several, segmentations of each input image [4–6]. [sent-19, score-0.308]

9 Markov random ﬁelds (MRFs) have been used to segment images into one of several known object classes [7, 8], but these approaches require manual segmentations to train category-speciﬁc appearance models. [sent-21, score-0.508]

10 In this paper, we instead develop a statistical framework for the unsupervised discovery and segmentation of visual object categories. [sent-22, score-0.34]

11 Using color and texture cues, our method simultaneously groups dense features into spatially coherent segments, and reﬁnes these partitions using shared appearance models. [sent-25, score-0.514]

12 This extends the cosegmentation framework [9], which matches two views of a single object instance, to simultaneously segment multiple object categories across a large image database. [sent-26, score-0.627]

13 This generalization of the Dirichlet process (DP) leads to heavier-tailed, power law distributions for the frequencies of observed objects or topics. [sent-29, score-0.229]

14 2 demonstrates that PY priors closely match the true distributions of natural segment sizes, and frequencies with which object categories are observed. [sent-31, score-0.545]

15 Importantly, this approach coherently models uncertainty in the number of object categories and instances. [sent-34, score-0.192]

16 90) 100 80 60 40 20 0 1 2 3 4 5 6 7 8 Number of Segments per Image (a) (b) (c) (d) Figure 1: Validation of stick-breaking priors for the statistics of human segmentations of the forest (top) and insidecity (bottom) scene categories. [sent-53, score-0.368]

17 (c) Number of segments occupying varying proportions of the image area, on a log-log scale. [sent-59, score-0.437]

18 (d) Counts of segments of size at least 5,000 pixels in 256 × 256 images of natural scenes. [sent-60, score-0.29]

19 4, we use thresholded Gaussian processes to link assignments of features to regions, and thereby produce smooth, coherent segments. [sent-62, score-0.309]

20 Simulations show that our use of continuous latent variables captures long-range dependencies neglected by MRFs, including intervening contour cues derived from image boundaries [13]. [sent-63, score-0.229]

21 Furthermore, our formulation naturally leads to an efﬁcient variational learning algorithm, which automatically searches over segmentations of varying resolution. [sent-64, score-0.23]

22 5 concludes by demonstrating accurate segmentation of complex images, and discovery of appearance patterns shared across natural scenes. [sent-66, score-0.393]

23 2 Statistics of Natural Scene Categories To better understand the statistical relationships underlying natural scenes, we analyze manual segmentations of Oliva and Torralba’s eight categories [3]. [sent-67, score-0.258]

24 A non-expert user partitioned each image into a variable number of polygonal segments corresponding to distinctive objects or scene elements (see Fig. [sent-68, score-0.474]

25 Each segment has a semantic text label, allowing study of object co-occurrence frequencies across related scenes. [sent-70, score-0.346]

26 There are over 29,000 segments in the collection of 2,688 images. [sent-71, score-0.203]

27 1 Stick Breaking and Pitman–Yor Processes The relative frequencies of different object categories, as well as the image areas they occupy, can be statistically modeled via distributions on potentially inﬁnite partitions. [sent-73, score-0.36]

28 When γa > 0, E[wk ] decreases with k, and the resulting partition frequencies follow heavier-tailed, power law distributions. [sent-81, score-0.191]

29 While the sequences of beta variables underlying PY processes lead to inﬁnite partitions, only a random, ﬁnite subset of size Kε = {k | ϕk > ε} will have probability greater than any threshold ε. [sent-82, score-0.209]

30 2 Object Label Frequencies Pitman–Yor processes have been previously used to model the well-known power law behavior of text sequences [15, 16]. [sent-89, score-0.183]

31 Intuitively, the labels assigned to segments in the natural scene database have similar properties: some (like sky, trees, and building) occur frequently, while others (rainbow, lichen, scaffolding, obelisk, etc. [sent-90, score-0.321]

32 1(b) plots the observed frequencies with which unique text labels, sorted from most to least frequent, occur in two scene categories. [sent-93, score-0.21]

33 By varying PY hyperparameters, we also capture interesting differences among scene types: urban, man-made environments have many more unique objects than natural ones. [sent-99, score-0.225]

34 3 Segment Counts and Size Distributions We have also used the natural scene database to quantitatively validate PY priors for image partitions [17]. [sent-101, score-0.4]

35 1(d), PY priors also model uncertainty in the number of segments at various resolutions. [sent-106, score-0.257]

36 While power laws are often used simply as a descriptive summary of observed statistics, PY processes provide a consistent generative model which we use to develop effective segmentation algorithms. [sent-107, score-0.305]

37 We do not claim that PY processes are the only valid prior for image areas; for example, log-normal distributions have similar properties, and may also provide a good model [18]. [sent-108, score-0.233]

38 However, PY priors lead to efﬁcient variational inference algorithms, avoiding the costly MCMC search required by other segmentation methods with region size priors [18, 19]. [sent-109, score-0.417]

39 We ﬁrst describe a “bag of features” model [1, 2] capturing prior knowledge about region counts and sizes, and then extend it to model spatially coherent shapes in Sec. [sent-111, score-0.196]

40 1 Hierarchical Pitman–Yor Processes Each image is ﬁrst divided into roughly 1,000 superpixels [18] using a variant of the normalized cuts spectral clustering algorithm [13]. [sent-117, score-0.264]

41 Superpixel i in image j is then represented by histograms xji = (xt , xc ) indicating its texture xt and color xc . [sent-120, score-0.413]

42 ji ji ji ji Figure 2 contains a directed graphical model summarizing our HPY model for collections of local image features. [sent-121, score-0.56]

43 Each of the potentially inﬁnite set of global object categories occurs with frequency ϕk , where ϕ ∼ GEM(γa , γb ) as motivated in Sec. [sent-122, score-0.226]

44 Each category k also has an assot c t c ciated appearance model θk = (θk , θk ), where θk and θk parameterize multinomial distributions on the Wt texture and Wc color bins, respectively. [sent-125, score-0.22]

45 Consider a dataset containing J images of related scenes, each of which is allocated an inﬁnite set of potential segments or regions. [sent-127, score-0.257]

46 3, region t occupies a random proportion πjt of the area in image j, where π j ∼ GEM(αa , αb ). [sent-130, score-0.285]

47 Each region is also associated with a particular global object category kjt ∼ ϕ. [sent-131, score-0.221]

48 kjt wk f 6 5 5 4 3 2 1 Probability Density vjt Probability Density J D 6 5 Probability Density 6 4 3 2 1 0 0 0. [sent-133, score-0.24]

49 Left: Directed graphical model in which global category frequencies ϕ ∼ GEM(γ) are constructed from stickbreaking proportions wk ∼ Beta(1 − γa , γb + kγa ), as in Eq. [sent-166, score-0.269]

50 Similarly, vjt ∼ Beta(1 − αa , αb + tαa ) deﬁne region areas π j ∼ GEM(α) for image j. [sent-168, score-0.311]

51 Upper right: Beta distributions from which stick proportions wk are sampled for three different PY processes: k = 1 (blue), k = 10 (red), k = 20 (green). [sent-171, score-0.252]

52 2 Variational Learning for HPY Mixture Models To allow efﬁcient learning of HPY model parameters from large image databases, we have developed a mean ﬁeld variational method which combines and extends previous approaches for DP mixtures [21, 22] and ﬁnite topic models. [sent-177, score-0.234]

53 We truncate the variational posterior [21] by setting q(vjT = 1) = 1 for each image or group, and q(wK = 1) = 1 for the shared global clusters. [sent-180, score-0.323]

54 Multinomial assignments q(kjt | κjt ), q(tji | τji ), and beta stick proportions q(wk | ωk ), q(vjt | νjt ), then have closed form update equations. [sent-181, score-0.34]

55 To avoid bias, we sort the current sets of image segments, and global categories, in order of decreasing aggregate assignment probability after each iteration [22]. [sent-182, score-0.182]

56 4 Segmentation with Spatially Dependent Pitman–Yor Processes We now generalize the HPY image segmentation model of Fig. [sent-183, score-0.332]

57 For simplicity, we consider a single-image model in which features xi are assigned to regions by indicator variables zi , and each segment k has its own appearance parameters θk (see Fig. [sent-185, score-0.365]

58 1 Coupling Assignments using Thresholded Gaussian Processes Consider a generative model which partitions data into two clusters via assignments zi ∈ {0, 1} sampled such that P[zi = 1] = v. [sent-191, score-0.229]

59 zi = 1 0 if ui < Φ−1 (v) otherwise Φ(u) We adapt this idea to PY processes using the stick-breaking representation of Eq. [sent-194, score-0.186]

60 In particuk−1 lar, we note that if zi ∼ π where πk = vk ℓ=1 (1 − vℓ ), a simple induction argument shows that vk = P[zi = k | zi = k − 1, . [sent-196, score-0.362]

61 The stick-breaking proportion vk is thus the conditional probability of choosing cluster k, given that clusters with indexes ℓ < k have been rejected. [sent-200, score-0.232]

62 Combining uk2 uk1 51 52 53 54 uk4 uk3 51 52 53 54 u3 B z2 z1 51 52 53 54 , u2 vk z4 z3 x1 x2 u1 6k B x3 x4 7 Figure 3: A nonparametric Bayesian approach to image segmentation in which thresholded Gaussian processes generate spatially dependent Pitman–Yor processes. [sent-201, score-0.765]

63 Left: Directed graphical model in which expected segment areas π ∼ GEM(α) are constructed from stick-breaking proportions vk ∼ Beta(1 − αa , αb + kαa ). [sent-202, score-0.38]

64 Zero mean Gaussian processes (uki ∼ N (0, 1)) are cut by thresholds Φ−1 (vk ) to produce segment assignments zi , and thereby features xi . [sent-203, score-0.489]

65 Right: Three randomly sampled image partitions (columns), where assignments (bottom, color-coded) are determined by the ﬁrst of the ordered Gaussian processes uk to cross Φ−1 (vk ). [sent-204, score-0.394]

66 (4), we can generate samples zi ∼ π as follows: zi = min k | uki < Φ−1 (vk ) where uki ∼ N (0, 1) and uki ⊥ uℓi , k = ℓ (5) As illustrated in Fig. [sent-206, score-0.493]

67 Intuitively, the ordering of segments underlying this dependent PY model is analogous to layered appearance models [23], in which foreground layers occlude those that are farther from the camera. [sent-213, score-0.332]

68 To retain the power law prior on segment sizes justiﬁed in Sec. [sent-214, score-0.271]

69 3, we transform priors on stick proportions vk ∼ Beta(1 − αa , αb + kαa ) into corresponding random thresholds: p(¯k | α) = N (¯k | 0, 1) · Beta(Φ(¯k ) | 1 − αa , αb + kαa ) v v v vk ¯ Φ−1 (vk ) (6) Fig. [sent-216, score-0.445]

70 As the number of features N becomes large relative to the GP covariance length-scale, the proportion assigned to segment k approaches πk , where π ∼ GEM(αa , αb ) as desired. [sent-218, score-0.273]

71 Figure 4: Five samples from each of four prior models for image partitions (color coded). [sent-230, score-0.228]

72 [24] proposed a generalized spatial Dirichlet process which links assignments via thresholded GPs, as in Sec. [sent-239, score-0.192]

73 However, their focus is on modeling spatial random effects for prediction tasks, as opposed to the segmentation tasks which motivate our generalization to PY processes. [sent-242, score-0.231]

74 Moreover, their basic Gibbs sampler takes 12 hours on a toy dataset with 2,000 observations; our variational method jointly segments 200 scenes in comparable time. [sent-244, score-0.362]

75 This produces a ﬁeld of smoothly varying multinomial ˇ ˇ distributions π i , from which segment assignments are independently sampled as zi ∼ π i . [sent-246, score-0.353]

76 Moreover, its bias towards partitions with K segments of similar size is a poor ﬁt for natural scenes. [sent-249, score-0.316]

77 A previous nonparametric image segmentation method deﬁned its prior as a normalized product of a DP sample π ∼ GEM(0, α) and a nearest neighbor MRF with Potts potentials [28]. [sent-250, score-0.418]

78 Due to the phase transition which occurs with increasing potential strength, Potts models assign low probability to realistic image partitions [29]. [sent-252, score-0.228]

79 5 Results Figure 5 shows segmentation results for images from the scene categories considered in Sec. [sent-255, score-0.435]

80 We compare the bag of features PY model (PY-BOF), dependent PY with distance-based squared exponential covariance (PY-Dist), and dependent PY with covariance that incorporates intervening contour cues (PY-Edge) based on the Pb detector [20]. [sent-257, score-0.267]

81 5 and 6, we independently segment each image, without sharing appearance models or supervised training. [sent-261, score-0.261]

82 We compare our results to the normalized cuts spectral clustering method with varying numbers of segments (NCut(K)), and a high-quality afﬁnity function based on color, texture, and intervening contour cues [13]. [sent-262, score-0.431]

83 To quantitatively evaluate results, we measure overlap with held-out human segments via the Rand index [30]. [sent-265, score-0.203]

84 We have also experimented with our hierarchical PY extension, in which color and texture distributions are shared between images. [sent-268, score-0.241]

85 7, many of the inferred global visual categories align reasonably with semantic categories (e. [sent-270, score-0.301]

86 6 Discussion We have developed a nonparametric framework for image segmentation which uses thresholded Gaussian processes to produce spatially coupled Pitman–Yor processes. [sent-273, score-0.611]

87 This approach produces empirically justiﬁed power law priors for region areas and object frequencies, allows visual appear- Figure 5: Segmentation results for two images (rows) from each of the coast, mountain, and tallbuilding scene categories. [sent-274, score-0.56]

88 From left to right, columns show LabelMe human segments, image with boundaries inferred by PY-Edge, and segments for PY-Edge, PY-Dist, PY-BOF, NCut(3), NCut(4), and NCut(6). [sent-275, score-0.351]

89 5 2 4 6 8 10 Number of Normalized Cuts Regions (d) Figure 6: Quantitative comparison of segmentation results to human segments, using the Rand index. [sent-311, score-0.184]

90 We plot the performance of NCut(K) versus the number of segments K, compared to the variable resolution segmentations of PY-Edge, PY-Dist, and PY-BOF. [sent-314, score-0.316]

91 ance models to be ﬂexibly shared among natural scenes, and leads to efﬁcient variational inference algorithms which automatically search over segmentations of varying resolution. [sent-317, score-0.318]

92 We believe this provides a promising starting point for discovery of shape-based visual appearance models, as well as weakly supervised nonparametric learning in other, non-visual application domains. [sent-318, score-0.214]

93 Acknowledgments We thank Charless Fowlkes and David Martin for the Pb boundary estimation and segmentation code, Antonio Torralba for helpful conversations, and Sra. [sent-319, score-0.184]

94 Figure 7: Most signiﬁcant segments associated with each of three shared, global visual categories (rows) for hierarchical PY-Edge models trained with 200 images of mountain (left) or tallbuilding (right) scenes. [sent-343, score-0.611]

95 Spatially coherent latent topic model for concurrent object segmentation and classiﬁcation. [sent-347, score-0.307]

96 Using multiple segmentations to discover objects and their extent in image collections. [sent-358, score-0.33]

97 Cosegmentation of image pairs by histogram matching: Incorporating a global constraint into MRFs. [sent-383, score-0.182]

98 Shared segmentation of natural scenes using dependent Pitman-Yor processes. [sent-447, score-0.331]

99 Learning to detect natural image boundaries using local brightness, color, and texture cues. [sent-469, score-0.258]

100 The Ising/Potts model is not well suited to segmentation tasks. [sent-540, score-0.184]

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