nips nips2006 nips2006-160 knowledge-graph by maker-knowledge-mining

160 nips-2006-Part-based Probabilistic Point Matching using Equivalence Constraints

Source: pdf

Author: Graham Mcneill, Sethu Vijayakumar

Abstract: Correspondence algorithms typically struggle with shapes that display part-based variation. We present a probabilistic approach that matches shapes using independent part transformations, where the parts themselves are learnt during matching. Ideas from semi-supervised learning are used to bias the algorithm towards ﬁnding ‘perceptually valid’ part structures. Shapes are represented by unlabeled point sets of arbitrary size and a background component is used to handle occlusion, local dissimilarity and clutter. Thus, unlike many shape matching techniques, our approach can be applied to shapes extracted from real images. Model parameters are estimated using an EM algorithm that alternates between ﬁnding a soft correspondence and computing the optimal part transformations using Procrustes analysis.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk Abstract Correspondence algorithms typically struggle with shapes that display part-based variation. [sent-6, score-0.192]

2 We present a probabilistic approach that matches shapes using independent part transformations, where the parts themselves are learnt during matching. [sent-7, score-0.575]

3 Shapes are represented by unlabeled point sets of arbitrary size and a background component is used to handle occlusion, local dissimilarity and clutter. [sent-9, score-0.119]

4 Thus, unlike many shape matching techniques, our approach can be applied to shapes extracted from real images. [sent-10, score-0.409]

5 Model parameters are estimated using an EM algorithm that alternates between ﬁnding a soft correspondence and computing the optimal part transformations using Procrustes analysis. [sent-11, score-0.25]

6 Over the last decade, numerous shape matching algorithms have been proposed that perform well on benchmark shape retrieval tests. [sent-13, score-0.437]

7 However, many of these techniques share the same limitations: Firstly, they operate on contiguous shape boundaries (i. [sent-14, score-0.181]

8 the ordering of the boundary points matters) and assume that every point on one boundary has a counterpart on the boundary it is being matched to (c. [sent-16, score-0.198]

9 Finally, they struggle to handle shapes that display signiﬁcant part-based variation. [sent-21, score-0.217]

10 The ﬁrst two limitations mean that many algorithms are unsuitable for matching shapes extracted from real images; the latter is important since many common objects (natural and man made) display part-based variation. [sent-22, score-0.271]

11 The methods described in [2, 3, 4] can handle outliers, occlusions and clutter, but are not designed to handle shapes whose parts are independently transformed. [sent-26, score-0.381]

12 In this paper, we introduce a probabilistic model that retains the desirable properties of these techniques but handles parts explicitly by learning the most likely part structure and correspondence simultaneously. [sent-27, score-0.428]

13 In this framework, a part is deﬁned as a set of points that undergo a common transformation. [sent-28, score-0.136]

14 Learning these variation-based parts from scratch is an underconstrained problem. [sent-29, score-0.196]

15 Firstly, the distributions of our hierarchical mixture model are chosen so that the learnt parts are spatially localized. [sent-31, score-0.394]

16 Secondly, ideas from semi-supervised learning [5] are used to encourage a perceptually meaningful part decomposition. [sent-32, score-0.146]

17 Localized dissimilarity Figure 1: Examples of probabilistic point matching (PPM) using the technique described in [4]. [sent-43, score-0.207]

18 In each case, the initial alignment and the ﬁnal match are shown. [sent-44, score-0.199]

19 Natural Part Decomposition (NPD): Most shapes have a natural part decomposition (NPD) (Fig. [sent-47, score-0.253]

20 We note that in tasks such as object recognition and CBIR, the query image is frequently a template shape (e. [sent-51, score-0.283]

21 Throughout this paper, it is assumed that we have obtained a sensible NPD for the query shape only1 – it is not reasonable to assume that an NPD can be computed for each database shape/image. [sent-55, score-0.181]

22 Variation-based Part Decomposition (VPD): A different notion of parts has been used in computer vision [7], where a part is deﬁned as a set of pixels that undergo the same transformations across images. [sent-56, score-0.349]

23 We refer to this type of part decomposition (PD) as a variation-based part decomposition (VPD). [sent-57, score-0.188]

24 point sets), PBPM matches them by applying a different transformation to each variation-based part of the generating shape. [sent-60, score-0.163]

25 These variation-based parts are learnt during matching, where the known NPD of the data shape is used to bias the algorithm towards choosing a ‘perceptually valid’ VPD. [sent-61, score-0.492]

26 This is achieved using the equivalence constraint Constraint 1 (C1): Points that belong to the same natural part should belong to the same variation-based part. [sent-62, score-0.221]

27 3, this inﬂuences the learnt VPD by changing the generative model from one that generates individual data points to one that generates natural parts (subsets of data points). [sent-64, score-0.397]

28 To further increase the perceptual validity of the learnt VPD, we assume that variation-based parts are composed of spatially localized points of the generating shape. [sent-65, score-0.444]

29 PBPM aims to ﬁnd the correct correspondence at the level of individual points, i. [sent-66, score-0.113]

30 each point of the generating shape should be mapped to the correct position on the data shape despite the lack of an exact point wise correspondence (e. [sent-68, score-0.507]

31 Soft correspondence techniques that achieve this using a single nonlinear transformation [2, 3] perform well on some challenging problems. [sent-72, score-0.146]

32 However, the smoothness constraints used to control the nonlinearity of the transformation will prevent these techniques from selecting the discontinuous transformations associated with part-based movements. [sent-73, score-0.128]

33 PBPM learns an independent linear transformation for each part and hence, can ﬁnd the correct global match. [sent-74, score-0.112]

34 In relation to the point matching literature, PBPM is motivated by the success of the techniques described in [8, 2, 3, 4] on non-part-based problems. [sent-75, score-0.149]

35 [8]) in that we use ‘structural information’ about the point sets to constrain the matching problem. [sent-78, score-0.131]

36 In addition to learning multiple parts and transformations, our work differs in the type of structural information used (the NPD rather then the Delauney triangulation) and the way in which this information is incorporated. [sent-79, score-0.196]

37 With respect to the shape-matching literature, PBPM can be seen as a novel correspondence technique for use with established NPD algorithms. [sent-80, score-0.121]

38 Figure 2: The natural part decomposition (NPD) (b-d) for different representations of a shape (a). [sent-86, score-0.281]

39 Siddiqi and Kimia show that the parts used in their NPD algorithm [6] correspond to speciﬁc types of shocks when shock graph representations are used. [sent-88, score-0.228]

40 The points need not belong to the shape boundary and the ordering of the points is irrelevant. [sent-93, score-0.334]

41 , V, (2) m=1 with y|(m, v) ∼ N (Tv xm , σ 2 I). [sent-108, score-0.121]

42 (3) Here, Tv is the transformation used to match points of part v on X to points of part v on Y. [sent-109, score-0.318]

43 Finally, we deﬁne p(m|v) in such a way that the variation-based parts v are forced to be spatially coherent: µ ) Èexp{−(xm − −vµ exp{−(x Σ−1 (xm − µv )/2} v , T −1 v ) Σv (xm − µv )/2} T p(m|v) = m m (4) where µv ∈ R2 is a mean vector and Σv ∈ R2×2 is a covariance matrix. [sent-110, score-0.222]

44 , M } with the point xm that it indexes and assume that the xm follow a bivariate Gaussian distribution. [sent-114, score-0.283]

45 This assumption means that the xm themselves are essentially generated by a GMM with V components. [sent-122, score-0.121]

46 However, this GMM is embedded in the larger model and maximizing the data likelihood will balance this GMM’s desire for coherent parts against the need for the parts and transformations to explain the actual data (the yn ). [sent-123, score-0.705]

47 3 Parameter Estimation With respect to the model deﬁned in the previous section, C1 states that all yn that belong to the same subset Yl were generated by the same mixture component v. [sent-126, score-0.322]

48 [5] for incorporating equivalence constraints between data points in mixture models. [sent-129, score-0.162]

49 Assuming that subsets and points within subsets are sampled i. [sent-132, score-0.136]

50 p(v|Yl ) log πv + (5) v=0 l=1 yn ∈Yl v=0 l=1 Note that eq. [sent-136, score-0.216]

51 (2) and rearranging slightly, we have L V E L p(v=0|Yl ) log{u|Yl | } p(v|Yl ) log πv + = v=0 l=1 V l=1 L + ´M p(yn |m, v)p(m|v) , p(v|Yl ) log v=1 l=1 yn ∈Yl µ (6) m=1 where u is the constant associated with the uniform distribution p(yn |v=0). [sent-139, score-0.243]

52 Here, we deﬁne Tv x ≡ sv Γv x + cv , where sv is a scale parameter, cv ∈ R2 is a translation vector and Γv is a 2D rotation matrix. [sent-158, score-0.136]

53 (12) becomes a weighted Procrustes matching problem between two points sets, each of size N × M – the extent to which xm corresponds to yn in the context of part v is given by p(v|Yl,n )p(m|yn , v). [sent-160, score-0.555]

54 (10-12) are similar to p(v|yn )p(m|yn , v) = p(m, v|yn ), the responsibility of the hidden variables (m, v) for the observed data, yn . [sent-163, score-0.256]

55 4, we initialize πv , µv and Σv by ﬁtting a standard GMM to the xm . [sent-168, score-0.121]

56 5, we describe a sequential algorithm that can be used to select the number of parts V as well as provide initial estimates for all parameters. [sent-170, score-0.333]

57 X and initial Gaussians for p(m|v) Y NPD of Y Initial alignment VPD of Y NPD of X Final match Input Output VPD of X with final Gaussians for p(m|v) Transformed X Figure 3: An example of applying PBPM with V =3. [sent-171, score-0.364]

58 PPM PBPM 2 parts 4 parts 5 parts 6 parts VPD of X VPD of Y Final match Figure 4: Results for the problem in Fig. [sent-172, score-0.856]

59 1 and 2, unsupervised matching of shapes with moving parts is a relatively unexplored area – particularly for shapes not composed of single closed boundaries. [sent-175, score-0.577]

60 Here, we provide illustrative examples which demonstrate the various properties of PBPM and then consider more challenging problems involving shapes extracted from real images. [sent-177, score-0.151]

61 The number of parts, V , is ﬁxed prior to matching in these examples; a technique for estimating V is described in Sec. [sent-178, score-0.136]

62 To visualize the matches found by PBPM, each point yn is assigned to a part v using maxv p(v|yn ). [sent-180, score-0.36]

63 Those xm not assigned to any parts are removed from the ﬁgure. [sent-186, score-0.381]

64 The means and the ellipses of constant probability density associated with the distributions N (µv , Σv ) are plotted on the original shape X. [sent-187, score-0.163]

65 We also assign the xm to natural parts using the known natural part label of the yn that they are assigned to. [sent-188, score-0.685]

66 3 shows an example of matching two human body shapes using PBPM with V =3. [sent-190, score-0.246]

67 The learnt VPD is intuitive and the match is better than that found using PPM (Fig. [sent-191, score-0.205]

68 When V =5, one of the parts is effectively repeated, suggesting that four parts is sufﬁcient to cover all the interesting variation. [sent-196, score-0.392]

69 However, when V =6 all parts are used and the VPD looks very similar to the NPD – only the lower leg and foot on each side are grouped together. [sent-197, score-0.244]

70 5, there are two genuine variation-based parts and X contains additional features. [sent-199, score-0.196]

71 PBPM effectively ignores the extra points of X and ﬁnds the correct parts and matches. [sent-200, score-0.257]

72 We ﬁnd that deletion from the generating shape tends to be very precise (e. [sent-203, score-0.191]

73 5), whereas PBPM is less inclined to delete points from the data shape when it involves breaking up natural parts (e. [sent-206, score-0.427]

74 This is X and initial Gaussians for p(m|v) Y NPD of Y Initial alignment VPD of Y NPD of X Final match Input Output VPD of X with final Gaussians for p(m|v) Transformed X Figure 5: Some features of X are not present on Y; the main building of X is smaller and the tower is more central. [sent-210, score-0.364]

75 X and initial Gaussians for p(m|v) Y NPD of Y Initial alignment NPD of X Final match Input Output VPD of X with final Gaussians for p(m|v) Transformed X VPD of Y Figure 6: The left legs do not match and most of the right leg of X is missing. [sent-211, score-0.484]

76 largely due to the equivalence constraints trying to keep natural parts intact, though the value of the uniform density, u, and the way in which points are assigned to parts is also important. [sent-212, score-0.607]

77 7 and 8, a template shape is matched to the edge detector output from two real images. [sent-214, score-0.317]

78 We have not focused on optimizing the parameters of the edge detector since the aim is to demonstrate the ability of PBPM to handle suboptimal shape representations. [sent-215, score-0.242]

79 The correct correspondence and PDs is estimated in all cases, though the results are less precise for these difﬁcult problems. [sent-216, score-0.113]

80 Note that the left shoulder is not assigned to the same variation-based part as the other points of the torso, i. [sent-220, score-0.148]

81 4 (with V =5) and 8 indicate that it may be possible to start with more parts than are required and either allow extraneous parts to go unused or perhaps prune parts during matching. [sent-225, score-0.588]

82 Finally, one could attempt to learn the parts in a sequential fashion. [sent-229, score-0.279]

83 5 Sequential Algorithm for Initialization When part variation is present, one would expect PBPM with V =1 to ﬁnd the most signiﬁcant part and allow the background to explain the remaining parts. [sent-231, score-0.163]

84 This suggests a sequential approach whereby a single part is learnt and removed from further consideration at each stage. [sent-232, score-0.302]

85 Speciﬁcally, assume that the ﬁrst part (v=1) has been learnt and now learn the second part using the Y X = edge detector output. [sent-235, score-0.313]

86 NPD of Y Initial alignment Input Output VPD of X with final Gaussians for p(m|v) Transformed X VPD of Y NPD of X Final match Figure 7: Matching a template shape to an object in a cluttered scene. [sent-236, score-0.543]

87 NPD of Y Initial alignment Input Output VPD of X with final Gaussians for p(m|v) Transformed X VPD of Y NPD of X Final match Figure 8: Matching a template shape to a real image. [sent-238, score-0.52]

88 J2 = (13) l=1 Here, π1 is known and 1 zl ≡ u|Yl | (1 − π1 ) p(Yl |v=1)π1 + u|Yl | (1 − π1 ) (14) is the responsibility of the background component for the subset Yl after learning the ﬁrst part – the superscript of z indicates the number of components that have already been learnt. [sent-240, score-0.183]

89 Note that we use the responsibilities for the subsets Yl rather than the individual yn [7], in line with the assumption that complete subsets belong to the same part. [sent-243, score-0.387]

90 Having learnt the second part, additional components v = 3, 4, . [sent-252, score-0.133]

91 are learnt in the same way except for minor adjustments to eqs. [sent-255, score-0.133]

92 The sequential algorithm terminates when the uniform component is not signiﬁcantly responsible for any data or the most recently learnt component is not signiﬁcantly responsible for any data. [sent-257, score-0.343]

93 As discussed in [7], the sequential algorithm is expected to have fewer problems with local minima since the objective function will be smoother (a single component competes against a uniform component at each stage) and the search space smaller (fewer parameters are learnt at each stage). [sent-258, score-0.305]

94 Preliminary experiments suggest that the sequential algorithm is capable of solving the model selection problem (choosing the number of parts) and providing good initial parameter values for the full model described in Sec. [sent-259, score-0.137]

95 9 and 10 – the initial transformations for each part are not shown. [sent-262, score-0.178]

96 We are currently investigating how the model can be made more robust to this value and also how the used xm should be subtracted (in a probabilistic sense) at each step. [sent-264, score-0.165]

97 X and initial Gaussians for p(m|v) Y NPD of Y Initial alignment VPD of Y NPD of X Final match Input Output VPD of X with final Gaussians for p(m|v) Transformed X Figure 9: Results for PBPM; V and initial parameters were found using the sequential approach. [sent-265, score-0.501]

98 X and initial Gaussians for p(m|v) Y NPD of Y Initial alignment VPD of Y NPD of X Final match Input Output VPD of X with final Gaussians for p(m|v) Transformed X Figure 10: Results for PBPM; V and initial parameters were found using the sequential approach. [sent-266, score-0.501]

99 In this paper, we have presented a probabilistic correspondence algorithm that handles part-based variation by learning the parts and correspondence simultaneously. [sent-271, score-0.443]

100 Shape matching and recognition using generative models and informative features. [sent-292, score-0.127]

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