iccv iccv2013 iccv2013-196 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Yuandong Tian, Srinivasa G. Narasimhan
Abstract: Real-world surfaces such as clothing, water and human body deform in complex ways. The image distortions observed are high-dimensional and non-linear, making it hard to estimate these deformations accurately. The recent datadriven descent approach [17] applies Nearest Neighbor estimators iteratively on a particular distribution of training samples to obtain a globally optimal and dense deformation field between a template and a distorted image. In this work, we develop a hierarchical structure for the Nearest Neighbor estimators, each of which can have only a local image support. We demonstrate in both theory and practice that this algorithm has several advantages over the nonhierarchical version: it guarantees global optimality with significantly fewer training samples, is several orders faster, provides a metric to decide whether a given image is “hard” (or “easy ”) requiring more (or less) samples, and can handle more complex scenes that include both global motion and local deformation. The proposed algorithm successfully tracks a broad range of non-rigid scenes including water, clothing, and medical images, and compares favorably against several other deformation estimation and tracking approaches that do not provide optimality guarantees.
Reference: text
sentIndex sentText sentNum sentScore
1 The recent datadriven descent approach [17] applies Nearest Neighbor estimators iteratively on a particular distribution of training samples to obtain a globally optimal and dense deformation field between a template and a distorted image. [sent-5, score-1.261]
2 The proposed algorithm successfully tracks a broad range of non-rigid scenes including water, clothing, and medical images, and compares favorably against several other deformation estimation and tracking approaches that do not provide optimality guarantees. [sent-8, score-0.622]
3 However, estimating the parameters of nonrigid deformation is hard due to its high-dimensionality and strong nonconvexity. [sent-11, score-0.531]
4 They show that if a generative model for deformation is available, then the training samples can be generated by simply deforming the template using parameters from a particular distribuSrinivasa G. [sent-18, score-0.852]
5 Our hierarchical framework for deformation estimation achieves O(C1d + C2 log 1/? [sent-29, score-0.575]
6 A constant number of samples per iteration is needed in [17]. [sent-32, score-0.259]
7 The number of samples needed is a constant for the first few iterations, and then decays double exponentially for our algorithm. [sent-33, score-0.303]
8 Intuitively, this approach captures the group-like structure in deformation and uses the train- O(Cd ing samples which are far away from the test image for prediction. [sent-40, score-0.623]
9 Their approach shows good empirical results for local deformation, but fails to capture general deformation that contains both global and local components (e. [sent-41, score-0.505]
10 In this paper, we develop a top-down hierarchical structure for deformation estimation with global optimality guarantee. [sent-44, score-0.634]
11 First, the deformation field is parameterized so that the deformation happening within a local image patch can be predicted by the content of that patch, reducing the dimensionality. [sent-45, score-1.087]
12 Then, we model the relationship between the image content and the deformation parameters using a novel criterion. [sent-46, score-0.472]
13 With this criterion, all patches at different locations and scales can be regarded as predictors with guaranteed worst-case precisions. [sent-47, score-0.16]
14 Finally, combining these predic22228888 tors together in a top-down hierarchical manner leads to an overall predictor that can handle large and high-dimensional deformation with both local and global components. [sent-48, score-0.572]
15 In particular, the number of samples required in each iteration stays constant for the first few iterations (layers of hierarchy), and then decays double exponentially (Fig. [sent-52, score-0.262]
16 Our work not only has strong theoretical foundations, but also demonstrates good quantitative and qualitative results on real video sequences containing different types of deformation, including clothing and water surface deformations as well as medical images of internal organs. [sent-59, score-0.333]
17 Regression-based approaches aim to learn a mapping from the distorted image to the deformation parameters using labeled training samples. [sent-66, score-0.705]
18 (a)-(b) The deformation field is controlled by a set of landmarks on the template image. [sent-115, score-0.777]
19 The Image Deformation Model Denote T as the template image and Ip as the distorted image with deformation parameters p. [sent-121, score-0.794]
20 The deformation field W(x; p) maps the pixel location x on the template to the pixel location W(x; p) on the distorted image Ip: Ip(W(x; p)) = T(x) (1) We locally parameterize the deformation field W(x; p) at any 2D point x by a weighted linear combination of displacements p = [p(1) , p(2) , . [sent-122, score-1.422]
21 , bK (x)] is a Kdimensional row vector of weighting factors on location x from K landmarks, p is a K-by-2 matrix storing 2K displacement components and p(i) is a 2-dimensional deformation vector for landmark i. [sent-129, score-0.613]
22 Due to strong correlations between nearby landmark displacements, the dimensionality d of the warping field could be much lower than 2K. [sent-130, score-0.179]
23 2 is an over-parameterization of the deformation field W(x; p), which leads to further reduction of training samples needed. [sent-140, score-0.71]
24 1, given the parameter p, one can generate the deformed image Ip from the template T. [sent-143, score-0.248]
25 This is done by assigning every pixel y of the deformed image Ip with the pixel value on location x = W−1 (y; p) of the template T. [sent-144, score-0.214]
26 Choosing different parameters {pi} gives many training samples {(pi , Ipi ) }. [sent-145, score-0.244]
27 , sam22228899 ple complexity) to achieve the globally optimal prediction of the unknown parameter for a distorted test image. [sent-149, score-0.386]
28 The Relationship between Image Evidence and Distortion Parameters Suppose we have training samples {p, Ip} and want to predict tohsee parameter rfaoirn a gte ssta image {Ip w,Iith} an dun wkannotw ton true parameter p1. [sent-152, score-0.262]
29 This can be represented by the following Lipchitz conditions proposed in [17]: L1ΔI ≤ Δp ≤ L2ΔI (3) where, L1 and L2 are two constants that are dependent on the template T. [sent-156, score-0.166]
30 [17] shows that the ratio L2/L1 is a charactistic for samples complexity for guaranteed Nearest Neighbor prediction. [sent-157, score-0.268]
31 For simple images that contain one salient object with a clear background, L2/L1 is small and a few samples suffice. [sent-158, score-0.151]
32 For difficult images with repetitive patterns, L2/L1 is large and a lot of samples are needed to distinguish among locally similar-looking structures. [sent-159, score-0.351]
33 Adding noise to a distorted image Ip changes iatsg appearance nb uoti neo tto i tas parameters. [sent-165, score-0.19]
34 In both cases, [17] gives a trivial (infinite) bound on sample complexity and global optimality cannot be guaranteed. [sent-172, score-0.228]
35 I(R) is the patch within R and S = S(x, r) is the subset of landmarks whose displacements p(S) influence the patch content I(R). [sent-175, score-0.395]
36 4 holds only for deformation within the acceptance range r, i. [sent-226, score-0.528]
37 image ciso nate pnrta cIt(icRa)l is no longer related to the local patch deformation p(S). [sent-232, score-0.571]
38 Therefore, we assume: Assumption 2 (Degrees of Freedom for Patches) The local degrees of freedom of a patch (x, r) is min(d, 2|S|). [sent-242, score-0.222]
39 Guaranteed Prediction using Nearest Neighbor Now let us study how the relaxed Lipchitz condition helps Nearest Neighbor prediction. [sent-245, score-0.149]
40 We wish to know how well patch (x, r) can predict the deformation p(S) within its acceptance range r (i. [sent-246, score-0.627]
41 Without any training samples, we can trivially s||et th≤e prediction tˆp a(nSy) =ai 0and get a worst-case guaranteed prediction error of r. [sent-249, score-0.355]
42 Now the problem is: if we want to obtain a slightly better prediction, how many training samples do we need? [sent-250, score-0.194]
43 It shows that if the relaxed Lipchitz condition (Eqn. [sent-252, score-0.149]
44 4) holds, then a Nearest Neighbor prediction with 1/α samples per dimension will always reduce the error by a factor of γ < 1: Theorem 1(Guaranteed Nearest Neighbor) Given a distorted image Ip with |p||∞ ≤ r, then with N(x,r) = min ? [sent-253, score-0.462]
45 (5) number of samples uniformly distributed in the hypercube [−r, r]2|S|, we can compute a prediction ˆ p(S) so that | pˆ(S) − p(S)|| ≤ γr (6) using Nearest Neighbor in the region R with image metric. [sent-259, score-0.321]
46 Proof Sketch We first fill the 2|S|-dimensional hypercPurobeo f[− Srk,e rt]c2h|S| W weith fi r(s1t/α fil)l2|S th| training samples uniformly. [sent-260, score-0.194]
47 If the local deformation is d-dimensional with d < 2|S| , thenIf i t h etu lrnosc olu dte tfhoartm only a ssm d-aldli mfreacntsiioonn aolf wthiteh hypercube are sampled and css ? [sent-272, score-0.58]
48 6, now both α and γ have their physical meanings: α is the inverse of sample complexity per dimension, while γ is the inverse of prediction accuracy. [sent-281, score-0.204]
49 3, this trade-off reflects the difficulty level of images for deformation prediction (See Sec. [sent-285, score-0.593]
50 Within the same layer t, scale of patches is fixed and denoted as rt. [sent-291, score-0.156]
51 1:INPUT Training samples Tr(x,r) ≡ {(pi,Ii)} for 2: 3: 4: 5: eIaNcPhU UimTa gTera pinaitcnhg ( sxa,m mrp). [sent-294, score-0.151]
52 6: 7: 8: 9: 10: 11: 12: 13: 14: for Patch (xj , rt) within layer t do Sj = S(xj , rt), Rj = R(xj , rt) Find the Nearest Neighbor i∗ for patch I(Rj) : i∗ = arg mini∈Tr(x,r) |Ic(Rj ) − Ii (Rj ) | Set the estimation ˜ pj→i (Sj) =) pi∗ (Sj). [sent-298, score-0.21]
53 For the first iteration, the test image Itest is directly compared with the training samples generated from the entire image with scale r1 to obtain the Nearest Neighbor prediction pˆ1 . [sent-305, score-0.315]
54 Then for the second iteration, we have a slightly less distorted image Itest (W(x, pˆ1 )), from which we estimate p − pˆ1 . [sent-306, score-0.19]
55 1will converge to the globally optimal solution (Theorem 2), while the required number of samples is O(C1d + C2 log 1/? [sent-310, score-0.228]
56 Note that a less distorted image Itest (W(x; pˆt−1)), as the input of layer t, is not necessarily the same as a distorted image Ip− pˆt−1 generated directly from the template image. [sent-312, score-0.623]
57 Work flow of our hierarchical algorithm for deformation estimation. [sent-367, score-0.539]
58 On Layer 1, a global prediction is made and the estimation is updated. [sent-368, score-0.154]
59 On Layer 2, local deformation is estimated and aggregated. [sent-369, score-0.472]
60 givenby[Tian dNar simhan,IJCV2012] [N1=16, N2=30] [N1=15, N2=24] [N1=14, N2=41 ] [N1=14, N2=40] [N1=13, N2=41 ] [N1=13, N2=45] [N1=12, N2=40] (b) Hard Images for deformation estimation. [sent-376, score-0.472]
61 Exemplar images and the theoretical bounds for the number of samples needed per dimension. [sent-378, score-0.259]
62 Top Row: Images with a salient object and clean background require only a few samples per dimension. [sent-382, score-0.151]
63 Bottom Row: Images with repetitive patterns require more samples per dimension. [sent-383, score-0.323]
64 Proof Sketch From Theorem 1, from the top layer, after each layer the residue is contracted by at least γ¯. [sent-385, score-0.155]
65 Theorem 3 (The Number of Samples Needed) The total number N of samples needed is bounded by: N ≤ C3C1d + C2 log1/ ¯γ 1/? [sent-387, score-0.225]
66 Twheellre afsor |Se,| ,t dhee nrueamsbee bry o af samples ¯γ needed stays the same until 2|S| ≈ d, and then goes down double-exponentially. [sent-392, score-0.297]
67 uTnhteilor 2e|Sm| |1 gives nthde tnheumngb oere osf d samples batl any pleonveeln otiaf tlhlye. [sent-393, score-0.201]
68 The supplementary report [16] shows that the summation of the samples at all levels is a fast decaying series bounded by Eqn. [sent-395, score-0.151]
69 Empirical Upper Bounds For Images Given a spectific template and a specific family of deformation, we can generate many deformed images and their parameters (pi, Ipi ), compute all-pair image/parameter distances {Δpi , ΔIi} and estimate the monotonous curve γ = tγa(nαce) sli {keΔ Fig. [sent-398, score-0.308]
70 T}h aisn curve can help predict tohues t chuerovree tγic=a l difficulties of images for deformation estimation. [sent-400, score-0.506]
71 We randomly generate 1000 deformed samples and compute all-pair distances. [sent-408, score-0.233]
72 The deformation is 2D translation and in-plane rotation (d = 3) up to ±π/8. [sent-409, score-0.472]
73 Note that images with a salient object and uniform background requires fewer samples, while images with repetitive patterns and cluttered backgrounds require more. [sent-415, score-0.172]
74 To obtain the same level of accuracy of our approach with 400 samples, [17] requires 10000 samples or more. [sent-463, score-0.151]
75 For all the experiments, our approach adopts a hierarchical structure using a grid of 256 landmarks with ¯γ = 0. [sent-468, score-0.196]
76 While our theory gives an upper bound of the sample complexity, practically 350 training samples over all layers suffice for good performance. [sent-471, score-0.428]
77 Convergence Behavior We artificially distorted 100 images with a 20dimensional global warping field specified in [17]. [sent-474, score-0.302]
78 For each image, its 10 distorted versions are generated with random parameters, which are estimated using Data-driven Descent (TN) [17] and using our approach. [sent-475, score-0.19]
79 Note that the strong drop in error shows that our method achieves very high accuracy by adding very few samples once it starts to work. [sent-479, score-0.151]
80 Deformation Estimation on Repetitive Patterns We further test our approach on synthetic data containing distorted repetitive patterns, and compare it with previ$? [sent-484, score-0.316]
81 Starting from initialization, the algorithm applies predictors of different layers to estimate the landmark locations. [sent-528, score-0.238]
82 From an undistorted template (240-by-240), we generate a dataset of 200 distorted images, each with labeled 49 points. [sent-531, score-0.322]
83 The deformation field is created by random Gaussian noise without temporal continuity. [sent-532, score-0.516]
84 The overall degree of freedom for this dataset is very high (50 dimensions are needed to achieve < 1pixel reconstruction error). [sent-533, score-0.149]
85 It is in general impossible to have sufficient number of samples for global optimality conditions to be satisfied. [sent-534, score-0.246]
86 LK and TN use a local parametric deformation model. [sent-537, score-0.472]
87 LK, FF and TN compute dense deformations and our hierarchy outputs 256 predicted landmarks, from which 49 landmark locations are interpolated. [sent-539, score-0.194]
88 Due to repetitive patterns, previous approaches fail to estimate the landmarks correctly. [sent-546, score-0.255]
89 The prediction of ESR is restricted to be on the linear shape subspace spanned by the training samples. [sent-548, score-0.164]
90 Thus, it is insufficient to use the template to capture the subspace of a complex deformation field. [sent-549, score-0.604]
91 When more layers are switched off, the algorithm is unable to identify global deformation and is essentially the same as local template matching at each landmark. [sent-611, score-0.771]
92 Performance on synthetic data if the first L layer of predictors are switched off, showing the bottom layers play a critical role for performance. [sent-632, score-0.29]
93 8 demonstrates how prediction from coarse layers (large patch) help the lower layer (small patch) find correct correspondences in repetitive patterns, justifying the hierarchy. [sent-635, score-0.451]
94 Real Experiments We also apply our framework to real world scenarios such as water distortion, cloth deformation and registration of medical images. [sent-637, score-0.851]
95 We captured the cloth sequence in the 5th row of Fig. [sent-650, score-0.199]
96 10, we use temporal information by adding training samples generated from perturbing the final estimation of the previous frame. [sent-654, score-0.194]
97 In comparison, SIFT+RANSAC only obtains a sparse set of distinctive matches, not enough for estimating a nonrigid deformation (even if we are using Thin-Plate Spline). [sent-660, score-0.531]
98 TN can capture detailed local deformations but not global shifts of the cloth without modeling the relationship between local patches. [sent-661, score-0.246]
99 Each row is a video sequence, two from underwater imaging, two from cloth deformation and the final one is from medical imaging. [sent-826, score-0.803]
100 Eachrowisa video, two from cloth deformation and one from underwater imaging. [sent-832, score-0.674]
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