cvpr cvpr2013 cvpr2013-341 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Minsik Lee, Jungchan Cho, Chong-Ho Choi, Songhwai Oh
Abstract: Non-rigid structure from motion is a fundamental problem in computer vision, which is yet to be solved satisfactorily. The main difficulty of the problem lies in choosing the right constraints for the solution. In this paper, we propose new constraints that are more effective for non-rigid shape recovery. Unlike the other proposals which have mainly focused on restricting the deformation space using rank constraints, our proposal constrains the motion parameters so that the 3D shapes are most closely aligned to each other, which makes the rank constraints unnecessary. Based on these constraints, we define a new class ofprobability distribution called the Procrustean normal distribution and propose a new NRSfM algorithm, EM-PND. The experimental results show that the proposed method outperforms the existing methods, and it works well even if there is no temporal dependence between the observed samples.
Reference: text
sentIndex sentText sentNum sentScore
1 In this paper, we propose new constraints that are more effective for non-rigid shape recovery. [sent-5, score-0.142]
2 Unlike the other proposals which have mainly focused on restricting the deformation space using rank constraints, our proposal constrains the motion parameters so that the 3D shapes are most closely aligned to each other, which makes the rank constraints unnecessary. [sent-6, score-0.425]
3 Based on these constraints, we define a new class ofprobability distribution called the Procrustean normal distribution and propose a new NRSfM algorithm, EM-PND. [sent-7, score-0.152]
4 The experimental results show that the proposed method outperforms the existing methods, and it works well even if there is no temporal dependence between the observed samples. [sent-8, score-0.108]
5 Structure from motion (SfM) [10], which estimates the 3D shape and pose of a rigid object from 2D-point tracks, is the most simple form of this problem. [sent-12, score-0.237]
6 Majority of the approaches [4, 8, 11, 13] fix the number of shape bases to restrict the ‘degree’ of de{c j c 8 3 , chcho i , s onghwai }@ s nu . [sent-19, score-0.14]
7 [1] showed that only an orthogonality constraint on the rotations is sufficient to find a unique solution. [sent-23, score-0.249]
8 However, the choice on the number of shape bases greatly affects the reconstruction performance and it is difficult to know the right number. [sent-24, score-0.178]
9 These approaches assume temporal dependence between frames, and incorporate the discrete cosine transform (DCT) bases in the model. [sent-26, score-0.173]
10 Most importantly, this number changes the solution of rotations, because the rotations are found based on the factorization results. [sent-29, score-0.244]
11 This means that finding the correct rotations are vital for NRSFM. [sent-31, score-0.173]
12 Because finding the correct rotations is important, we consider NRSfM as an alignment problem and introduce an additional constraint to each rotation matrix. [sent-33, score-0.294]
13 This constraint is derived from the generalized Procrustes analysis (GPA) [6, 14], which aligns a set of shapes most closely to each other. [sent-34, score-0.18]
14 We also modify the scale constraint in GPA to make the aligned shapes lie in a linear subspace, which makes the problem tractable. [sent-36, score-0.284]
15 This subspace includes all possible deformation of shapes, and moreover, the null space of this subspace is 7-dimensional that is related to the variations due to rigid transformations. [sent-37, score-0.177]
16 In other words, rigid and nonrigid shape variations are strictly separated under these con- straints. [sent-38, score-0.316]
17 This leads us to define a new class of probability distribution called the Procrustean normal distribution (PND), which is a special case of the normal distribution. [sent-39, score-0.202]
18 Since there is no constraint on the deformation space, EM-PND loses less detail in the reconstructed shapes. [sent-41, score-0.139]
19 Moreover, it does 1 1 12 2 27 78 80 8 not require any temporal dependence between the observed 2D tracks. [sent-42, score-0.108]
20 EM-PND also works well when there are some missing data in the observation. [sent-43, score-0.082]
21 Procrustean Normal Distribution Estimating rotations based on rank and orthonormality constraints as in other NRSfM algorithms can be troublesome, because of two reasons. [sent-49, score-0.362]
22 First, knowing the correct rank is not easy and wrongly chosen rank can ruin the estimation. [sent-50, score-0.136]
23 Second, how orthonormal the estimated rotations are is not directly connected to the accuracy ofthe rotations. [sent-51, score-0.173]
24 Therefore, we propose another way to model the rotations by incorporating GPA. [sent-52, score-0.173]
25 GPA finds the relative motions (including rotations) between similar shapes by aligning them as closely as possible. [sent-53, score-0.139]
26 This principle determines rigid motions by minimizing non-rigid variations, which can improve the accuracy of the estimated rotations in NRSfM. [sent-54, score-0.324]
27 ) aligned shapes, and propose a new distribution based on the conditions. [sent-58, score-0.123]
28 This distribution can be effectively used to separate rigid and non-rigid shape variations, and serves as a core component of the proposed algorithm. [sent-59, score-0.242]
29 GPA and its modification GPA superimposes multiple landmark shapes to a common reference using rigid transformations. [sent-62, score-0.22]
30 Let Xi ∈ Rnd si ∈ R, Ri ∈ Rnd ×nd, and ti ∈ Rnd be the 3D shape, scale, rotation, and translation, respectively, for the ith sample, 1 ≤ i≤ ns, where nd, np, and ns are the dimension of the coordinate system, the number of landmarks in a frame, and the number of frames, respectively. [sent-63, score-0.15]
31 es shape space if the first constraint is used, and otherwi? [sent-96, score-0.151]
32 Here, the translation component is removed from each shape in Step 1 and does not appear in the iterative procedure, because the optimal ti is in fact given as ti = −n1psiRiXi1 [6]. [sent-119, score-0.122]
33 However, both the Procrustes and Kendall shape spaces are nonlinear manifolds, which make it hard to handle the distribution of the shape. [sent-124, score-0.126]
34 We may drop the scale constraint to resolve this issue, but it is not a good idea because there can be scale changes due to camera motion that may significantly affect the reconstruction performance (as shown in the supplementary material). [sent-125, score-0.16]
35 Therefore, we need to find a new scale constraint that makes the aligned shapes lie in a linear subspace. [sent-126, score-0.284]
36 To do this, we propose another scale constraint so that each shape variation from the mean shape is orthogonal to the mean shape, i. [sent-127, score-0.275]
37 si, and it makes the aligned shapes lie in the stereographic projection [14], which is a mapping from an n-dimensional sphere to a hyperplane, of a Procrustes shape space. [sent-150, score-0.283]
38 This let us describe the distribution of the aligned shapes more easily using typ- ical probability distributions. [sent-151, score-0.227]
39 an aligned shape siRiXi and the last condition forces the set of the aligned shapes to be convex, which are relatively easy to handle. [sent-170, score-0.323]
40 Hence, if we solve NRSfM based on a distribution constrained by these conditions, then we may enforce the reconstructed shapes to be a possible optimal solution for problem (1). [sent-171, score-0.218]
41 Because there are three more conditions in (4) besides the first equation, which is the orthonormality constraint in other NRSfM algorithms, the solution may be quite different from the other NRSfM algorithms. [sent-173, score-0.163]
42 Definition and Properties of PND Let Yi be the ith aligned shape expressed as Yi = siRiXi and Y = X, then (4) can be written as ? [sent-177, score-0.194]
43 e second and last conditions are linear equality and convex constraints w. [sent-188, score-0.1]
44 However, if we relax the PSD constraint to a symmetric constraint, then the last one can be also expressed as a linear equality constraint w. [sent-192, score-0.152]
45 ) To set the centroid of an aligned shape at the origin, an additional constraint is introduced as Yi1 = 0. [sent-199, score-0.223]
46 (7) These constraints can be simplified using the vectorization operator as ? [sent-200, score-0.102]
47 T, (9) where yi is the ith column vector of Y and ⊗ is the Kro- R? [sent-235, score-0.11]
48 (10) Note that nN, the number of columns of PN, corresponds to the degree of freedom (DOF) of a rigid transform. [sent-248, score-0.116]
49 For example, nN = 7 for nd = 3, which corresponds to the DOF of a rigid transform (1 for scale, 3 for rotation, and 3 for translation) in a? [sent-249, score-0.172]
50 , perpendicular to the subspace of rigid shape variations, the variations will ? [sent-269, score-0.252]
51 e sum of the exterior products of yi with themselves. [sent-325, score-0.107]
52 This proposition states that each constraint specifies a different orthogonal subspace. [sent-329, score-0.181]
53 In other words, variations due to scaling, rotation and translation are mutually orthogonal to each other under the constraints in (8). [sent-330, score-0.269]
54 Based on these constraints, we can define a new distribution that only includes non-rigid shape variations, eliminating rigid variations. [sent-331, score-0.242]
55 Because PND does not include any rigid shape variations, it is possible to find relative motions between sample shapes by fitting them to a PND. [sent-375, score-0.33]
56 First, because no low-rank constraint is needed for PND, there is no need to adjust the rank and less details will be lost in the fitting process. [sent-378, score-0.144]
57 Second, PND strictly separates rigid and non-rigid variations in the fitting process, which will lead to more accurate motion parameters and reconstructed shapes. [sent-379, score-0.286]
58 The idea ofruling out rigid variations is somewhat similar to that in [12], however, PND does not require accurate motion information in advance, unlike the approach. [sent-380, score-0.223]
59 For PND, estimating the distribution of non-rigid variations determines the rigid motions as a by-product, because QN depends on the mean shape. [sent-381, score-0.263]
60 form of a normal distribution is also Gaussian, hence Y? [sent-405, score-0.101]
61 the Note that (14) can be applied to distributions other than the normal distribution for the analysis of NRSfM, but we use the normal distribution for its simplicity. [sent-471, score-0.202]
62 The proposed algorithm: EM-PND ×np Let Di ∈ Rnd be the input landmark data, observed by an orthographic camera, of the ith sample, and Wi ∈ Rnd be the weight matrix filled with ones and zeros that indicates whether the corresponding elements are observed or missing. [sent-475, score-0.116]
63 lwliwjiljdlijl oifth weirjwkis=e 1, (19) where dijk and wijk are the (j, k)th elements of Di and Wi, respectively. [sent-480, score-0.132]
64 Here, Xi is a hidden variable representing the true 3D shape of the ith sample. [sent-519, score-0.122]
65 In order to represent the prior distribution of Xi, we assume that the aligned shapes Yi = siRiXi are independently and ident? [sent-551, score-0.227]
66 All the constraints in this problem are the same as the constraints in (4) except that Xi is replaced with its expectation Mi. [sent-637, score-0.134]
67 In this framework, the most difficult parameter to update is X, on which Q as well as the last three constraints in (29) depend. [sent-642, score-0.136]
68 To resolve this, we regard Q as an independent parameter and simply ignore the constraints in the update of X. [sent-643, score-0.136]
69 According to the constraints in (29), the feasible si and Ri are unique if the other parameters are fixed and all the samples are non-degenerate. [sent-662, score-0.139]
70 The proposed method works well for random initial rotations when the shape variations are moderate, but the performance may deteriorate for large variations. [sent-688, score-0.309]
71 Hence, we adopt the initialization method used in [5] for the rotations, which calculates rotations based on the factorization results for all possible numbers of shape bases and then automatically chooses the most ‘orthonormal’ ones. [sent-689, score-0.384]
72 = 1, si where zi is the vector of the missing elements for the ith sample and B = I −n1p11T. [sent-716, score-0.285]
73 Li(zi) ∈ Rnd×np is a mapping that places each element of zi to the corresponding location in a shape matrix. [sent-717, score-0.159]
74 Average reconstruction errors without noise and missing data data \ methodEM-PPCAMPCSF2SPMEM-PND TablewpFsdiytrhRa2folcnkieg. [sent-768, score-0.12]
75 48 3706 943 missing data data \ methodEM-PPCAMPCSF2SPMEM-PND 3. [sent-780, score-0.082]
76 If a shape model rather than a reconstructed shape is needed, then X and Σ can be used instead. [sent-797, score-0.213]
77 Similarly, si and Ri can be used to represent a rigid motion. [sent-798, score-0.188]
78 Random scaling and rotation were applied to each of these samples to form a new 3D facial-landmark data with no temporal dependence. [sent-804, score-0.09]
79 For the missing data, we randomly set 30 percents of the landmarks as missing. [sent-808, score-0.149]
80 Average reconstruction errors with missing data and without noise data \ methodEM-PPCAMPCSF2EM-PND wspFdtyfiarhRcolkneGrtgciukCapenhg0 . [sent-812, score-0.12]
81 Average reconstruction errors with noise and missing data data \ methodEM-PPCAMPCSF2EM-PND error, i. [sent-821, score-0.12]
82 There are no experimental results for missing data using SPM because SPM can not handle the case of missing data. [sent-836, score-0.164]
83 Among the 32 cases, excluding four cases ofthe dance sequence, EM-PND gives the best performance except for two cases, and even for these two cases, EM-PND gives the second best performance. [sent-838, score-0.104]
84 For the case in the pickup sequence without noise and missing data, the error difference between EM-PND and the best method is about five percent. [sent-839, score-0.14]
85 For the shark sequence, EM-PPCA gives a smaller error than EM-PND for the case ofno missing data but with noise. [sent-840, score-0.176]
86 This seems to be attributed due to the nature of the shark sequence, which was artificially generated by superposing two basis shapes [11]. [sent-841, score-0.166]
87 Because of this, EMPPCA, which explicitly limits the number of shape bases in the reconstruction process, gives better performance for the shark data. [sent-842, score-0.272]
88 The dance sequence includes a large deformation, and CSF2, which enforces temporal dependence between frames, gives better results than the other schemes. [sent-844, score-0.18]
89 We expect that the performance of EM-PND for the dance sequence can also be improved by enforcing temporal dependence. [sent-845, score-0.085]
90 Reconstructed results (top row: EM-PND, bottom row: the second best method, ◦: observed ground truth, : missing ground truth, reconstructed points). [sent-847, score-0.145]
91 Note that CSF2 gives relatively good performance, but not for the FRGC data because CSF2 assumes the existence of temporal dependence between frames. [sent-853, score-0.14]
92 EM-PND shows a better fit between the reconstructed points and the corresponding ground truth than the second best method, as can be seen from the reconstruction results in Fig. [sent-854, score-0.101]
93 The videos of reconstructed shapes are also provided in the supplementary material to confirm the performance of EM-PND. [sent-856, score-0.167]
94 Instead of rank constraints employed in the other methods, EM-PND imposes constraints on the motion parameters, following the practices in GPA, which makes the 3D shapes most closely aligned in a linear subspace. [sent-859, score-0.424]
95 EM-PND gives state-of-the-art performance, as validated in the experimental results, by separating rigid and non-rigid variations and not using any rank constraint. [sent-861, score-0.277]
96 Future work will consider the problems of adding temporal dependence into the model and designing a new factorization algorithm based on PND to reduce the computation time. [sent-862, score-0.179]
97 In defense of orthonormality constraints for nonrigid structure from motion. [sent-867, score-0.185]
98 Shape and motion from image streams under orthography: a factorization method. [sent-945, score-0.117]
99 Nonrigid structure-from-motion: Estimating shape and motion with hierarchical priors. [sent-952, score-0.121]
100 A closed-form solution to non-rigid shape and motion recovery. [sent-966, score-0.121]
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topicId topicWeight
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simIndex simValue paperId paperTitle
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