iccv iccv2013 iccv2013-185 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Jiaolong Yang, Hongdong Li, Yunde Jia
Abstract: Registration is a fundamental task in computer vision. The Iterative Closest Point (ICP) algorithm is one of the widely-used methods for solving the registration problem. Based on local iteration, ICP is however well-known to suffer from local minima. Its performance critically relies on the quality of initialization, and only local optimality is guaranteed. This paper provides the very first globally optimal solution to Euclidean registration of two 3D pointsets or two 3D surfaces under the L2 error. Our method is built upon ICP, but combines it with a branch-and-bound (BnB) scheme which searches the 3D motion space SE(3) efficiently. By exploiting the special structure of the underlying geometry, we derive novel upper and lower bounds for the ICP error function. The integration of local ICP and global BnB enables the new method to run efficiently in practice, and its optimality is exactly guaranteed. We also discuss extensions, addressing the issue of outlier robustness.
Reference: text
sentIndex sentText sentNum sentScore
1 The Iterative Closest Point (ICP) algorithm is one of the widely-used methods for solving the registration problem. [sent-3, score-0.26]
2 Its performance critically relies on the quality of initialization, and only local optimality is guaranteed. [sent-5, score-0.114]
3 This paper provides the very first globally optimal solution to Euclidean registration of two 3D pointsets or two 3D surfaces under the L2 error. [sent-6, score-0.429]
4 By exploiting the special structure of the underlying geometry, we derive novel upper and lower bounds for the ICP error function. [sent-8, score-0.268]
5 The integration of local ICP and global BnB enables the new method to run efficiently in practice, and its optimality is exactly guaranteed. [sent-9, score-0.134]
6 This paper is, to the best of our knowledge, the very first that proposes a truly globally optimal solution to ICP type Euclidean registration in 3D. [sent-27, score-0.378]
7 It provides guaranteed optimality without the need for a good initialization. [sent-28, score-0.112]
8 We have conducted extensive tests on both synthetic data and real data; satisfactory results (both in terms of theoretical optimality and computational efficiency) are obtained for all tests. [sent-34, score-0.11]
9 Although Go-ICP is specifically designed for 3D Euclidean registration since we take advantage ofthe geometry of SE(3), the same techniques used in this paper may be inspiring for other cases as well (e. [sent-35, score-0.278]
10 Below we only list a few most relevant works, that either aimed to achieve optimal ICP or, more generally, addressed optimality in Euclidean registration. [sent-43, score-0.13]
11 To alleviate the local minima issue, previous work has attempted to enlarge the basin of convergence by smoothing out the objective function. [sent-45, score-0.105]
12 particle swarm optimization [34] and particle filtering [32], to help the registration jump out of local minima. [sent-51, score-0.379]
13 Registration methods that come with guaranteed optimality were published in the past, though in a smaller number. [sent-58, score-0.112]
14 While being globally optimal, their method makes unrealistic assumption such as the two pointsets are of equal size and there is only pure rotation. [sent-64, score-0.13]
15 Branch-and-bound based Euclidean registration was investigated in Olsson et al. [sent-65, score-0.26]
16 [12] converted the registration problem to graph vertex cover and provided an optimal solution. [sent-68, score-0.299]
17 spin image [19], shape contexts [4], EGI [24]) are mostly heuristic and do not address the optimality issue. [sent-72, score-0.134]
18 [14] in which they proposed a globally optimal solution on top of the local descriptors. [sent-74, score-0.114]
19 In this paper, we solve the 3D Euclidean registration problem with global optimality guarantee. [sent-78, score-0.371]
20 For the case of optimal registration under L2-norm, few results are available. [sent-83, score-0.299]
21 , Nets, w Xher=e xi, yj ∈i R=3 are point cooYrd =ina {teys,} , bej t=he d,a. [sent-92, score-0.111]
22 The aim is to estimate a rigid motion with rotation R ∈ SO(3) and translation t ∈ R3, which minimizes the following 3L)2 error Ean:s ? [sent-96, score-0.258]
23 1 where ei (R, t) is the per-point residual error for xi. [sent-106, score-0.184]
24 The point yj∗ ∈ Y is denoted as the optimal correspondence of xi, which ∈in Y Yth ies cdeonntoetxetd o asf ItCheP ips timhea lc cloosrresets point ntoc eth oef transformed xi in Y, i. [sent-107, score-0.118]
25 In order to apply BnB to 3D registration, one must first answer the following questions: (i) how to parameterize and branch the domain of 3D motions, (ii) how to efficiently find an upper bound and lower bound. [sent-129, score-0.188]
26 For ease of manipulation, we use the minimum cube [−π, π]3 that encloses the tπio-bna,ll w as utshee t rhoeta mtiionni mduommai cnu. [sent-140, score-0.123]
27 b eFo [−r tπh,eπ t]ranslation part, we assume the optimal translation must lie within a bounded cube [−ξ, ξ]3 which may be readily set by choosing a big ncuubmebe [−r as ξ]. [sent-141, score-0.253]
28 During BnB searches, initial cubes will be subdivided into smaller sub-cubes Cr, Ct using the octree data-structure and the process is repeated. [sent-142, score-0.127]
29 This way, our method enjoys both the efficiency provided by the local ICP search, and the optimality guaranteed by the BnB search. [sent-147, score-0.135]
30 Derive New Bounding Functions As for any BnB method, finding quality bounds is the key to success. [sent-149, score-0.13]
31 In our method, we need to find the bounds of the particular type of L2-norm error function used in ICP within a domain Cr Ct. [sent-150, score-0.203]
32 Next, we will introduce the concept of uncertainty ×ra Cdius as a mathematical preparation, then derive our new bounds based on it. [sent-151, score-0.266]
33 Uncertainty radius The intuition behind the concept of uncertainty radius is: we want to examine, if we perturb a 3D rigid motion with rotation r ∈ Cr and/or translation t ∈ Ct applied to a 3wDit point x, nw rh ∈at Cthe uncertainty region to f∈ t hCe transformed point will be. [sent-154, score-0.638]
34 We aim to find a ball enclosing such an uncertainty region. [sent-155, score-0.174]
35 maxX (a) Rotation uncertainty radius (b) Translation uncertainty radius Figure 1. [sent-158, score-0.358]
36 Left: rotation uncertainty ball for Cr (in red) with center Rr0 x (blue dot) and radius γr. [sent-160, score-0.331]
37 Right: translation uncertainty ball for Ct (in red) with center x + t0 (blue dot) and radius γt . [sent-161, score-0.322]
38 In both diagrams, the uncertainty balls enclose the range of Rrx or x + t (in green). [sent-162, score-0.121]
39 For a rotation cube Cr of half side-length σr with r0 as the center, examining the maximum distance from Rrx to Rr0x, we have ? [sent-165, score-0.204]
40 Similarly, we can derive a translation uncertainty radius γt, for a translation cube Ct with half side-length σt centered at t0: ? [sent-199, score-0.482]
41 Both uncertainty radii are used in deriving the lower bound for our method. [sent-208, score-0.317]
42 Note that γr is point-dependent, therefore γri refers to the rotation uncertainty radius at xi. [sent-209, score-0.278]
43 It is clear that a ≤ b ≤ c while a = ei and c = ei (Rr , t). [sent-212, score-0.206]
44 Bounding the L2 error Given a 3D data point xi, a rotation cube Cr centered at r0 and a translation cube Ct centered at t0, an upper bound for the per-pixel residual error ei (R, t) within the cubes can be trivially found as ei= ei(Rr0,t0) ? [sent-216, score-0.914]
45 (6) Finding a suitable lower bound for this L2 residual error appears to be a harder task, especially considering the domain is in SE(3). [sent-218, score-0.237]
46 However, below we will show how a lower bound can be found, using the concept of uncertainty radius. [sent-219, score-0.27]
47 (γri +γt) (9) = ei (Rr0 ,t0) −(γri +γt) , − − − where Eq. [sent-240, score-0.103]
48 Now we have reached a lower bound of the per-point residual for Cr Ct as ei=max(ei(Rr0,t0)−(γri+γt),0)? [sent-249, score-0.164]
49 (10) The geometric explanation of this lower bound is as follows. [sent-251, score-0.149]
50 Since yj∗0 is the closest point to the center Rr0xi + t0 of the uncertainty ball with radius γ = γri +γt, it is also the closest point to (the surface of) the ball and ei is the closest distance between pointset Y and the ball. [sent-252, score-0.605]
51 (Bounds of the L2 error) For a 3D motion domain Cr Ct centered at (r0, t0) with uncertainty radii γris and γt, ×theC upperboundE andthe lowerboundE ofthe optimal L2 registration error E∗ can be chosen as ? [sent-261, score-0.573]
52 1 This result gives both the upper bound and lower bound of the registration error, based on which we developed our Go-ICP algorithm. [sent-279, score-0.505]
53 An outer BnB searches the rotation space of SO(3), while solving the bounds and corresponding optimal translation by calling an inner translational BnB. [sent-284, score-0.46]
54 The bounds for both BnB algorithms can be readily derived according to Sec. [sent-285, score-0.149]
55 In the outer rotation BnB, for a cube Cr the bounds of the registration error can be chosen ? [sent-288, score-0.693]
56 he bounds for a translation cube Ct can be chosen as Et = ? [sent-297, score-0.325]
57 The nested BnB structure can be clearly seen: the outer BnB in Algorithm 1 calls the inner BnB in Algorithm 2. [sent-307, score-0.113]
58 Once the difference between so-far-the-best error E∗ and the lower bound E of current cube is less than a threshold ? [sent-317, score-0.305]
59 All we need to do is to show that, as the algorithm iterates, the gap between the lower bound and the upper bound converges uniformly to zero. [sent-321, score-0.245]
60 This is easy to see, as when the side-lengths of all cubes asymptotically diminish to zero, the gap between the two bounds, i. [sent-322, score-0.104]
61 To speed up the computation of inner BnB, we set the initial Et∗ to be E∗ without loss of globally optimal registration based on the following insight. [sent-329, score-0.407]
62 In other words, ≥if Ethe error Et∗ returned by the inner BnB is greater than or equal to so-far-the-best error E∗ in the outer BnB, it makes no contribution. [sent-332, score-0.183]
63 Whenever the outer BnB finds a cube Cr which has an upper-bound lower than the so-farthe-best function value, it will then call the conventional ICP to start over, from the center of Cr and corresponding t∗ as the new initial transformation. [sent-337, score-0.231]
64 Algorithm 2: BnB search for optimal translation given Algorithm 2: BnB search for optimal translation given rotation Input: Data and model points; threshold ? [sent-341, score-0.43]
65 ; initial cube Ct ; rotation r0 ; rotation uncertainty radii γr, so far the best error E∗ . [sent-342, score-0.56]
66 LBenft:ICPBn anBdICP colabrtivelyupdateh uper bounds during the search process. [sent-347, score-0.159]
67 Right: with the guidance of BnB, ICP only explores un-discarded promising cubes with small lower bounds marked up by BnB. [sent-348, score-0.312]
68 Instead of exploring the domain blindly, ICP con- verges into local minima one by one with each local minimum having lower error than the previous one, and reaches the global minimum in the end. [sent-351, score-0.273]
69 [5]), all points (transformation parameter r, t) in its search path should have error lower than E∗, which means that lower bounds of the cubes containing these points should be lower than E∗ . [sent-353, score-0.539]
70 Thus the search path of the local ICP is entirely confined to the undiscarded, promising cubes with small lower bounds, as illustrated in Fig. [sent-354, score-0.211]
71 This way, both the global BnB search and the local ICP search are intimately integrated in our method. [sent-356, score-0.101]
72 The former not only helps the latter to jump out of local minima, but also provides a guidance for the latter’s next search; the latter accelerates the former’s convergence by refining the upper bound, hence improves the overall efficiency. [sent-357, score-0.139]
73 Comparison of the registration error (Left) and rotation error (Right) on random points, by our Go-ICP method, versus ICP initialized with Identity rotation and zero translation. [sent-362, score-0.583]
74 Ground- × truth rotation and translation lie randomly within ±100 degrees tarnudth h± ro0t. [sent-363, score-0.189]
75 In all the expericmliednetas reported t bhel 3o0w0,× we pre-normalized Ithne pointsets psuercihthat all the points are within the domain of [−1, 1]3 and tthhea i aniltlia thl etr panoisnftosrm araetiwo ni dhionm thaien dtoo explore i−s s,e1t to be [−π, π]3 [−0. [sent-371, score-0.13]
76 Optimality The purpose ofthis first experiment is to verify the global optimality of our new Go-ICP algorithm and compare that with standard ICP. [sent-382, score-0.127]
77 In each test, 100 model points are randomly drawn from the uniform distribution in [−1, 1]3; rotation and translation are randomly idsrtariwbunt iwonith inin [ −±11,001] degrees and ±0. [sent-384, score-0.219]
78 5 respectively adnodm applied tno w wthiteh imno ±d1e0l points etos generate 5d raetasp points; zero mean Gaussian noise is added to the points; ICP is initialized with Identity rotation and zero translation. [sent-385, score-0.175]
79 Figure 4 shows the final reported registration error and rotation errors for the 100 runs. [sent-386, score-0.41]
80 Note that our goal is to minimize the L2 error while root-mean-square (RMS) error is reported for better comprehension. [sent-387, score-0.102]
81 Visually inspected, our method yields a more satisfactory registration too. [sent-392, score-0.26]
82 In our experiment, for example, to match 1000 data points to about 30,000–40,000 model points took about 30 seconds for bunny and 15 seconds for the hand. [sent-410, score-0.141]
83 The global lower bound, which is the smallest lower bound of the cubes in the queue, is always close to zero because the globally optimal error is close to zero, which is true for the registration problem. [sent-416, score-0.753]
84 Typical convergence curves ofthe upper bounds and lower bounds in the outer BnB of our method on bunny and hand (M = 1000). [sent-464, score-0.515]
85 ICP refines the better upper-bound found by BnB with local search, and BnB guides ICP to converge into multiple local minima with lower and lower registration errors until the global minimum is reached. [sent-466, score-0.497]
86 In this test, we use ICP with trimming [11] and the same trimming strategy in BnB to handle outliers. [sent-472, score-0.102]
87 Our goal is to register the points of a baseball cap to the point cloud of the scene as shown in Fig. [sent-483, score-0.105]
88 (Better viewed in color) are numerous local minima (with very low registration error) arising when the cap is nestled into the table, the wall, etc. [sent-487, score-0.372]
89 We focus on a few examples, showing mainly how the new lower bound (for Cr Ct) may be derived for these extended cases. [sent-499, score-0.134]
90 Closing Remarks We have described a truly globally-optimal solution to Euclidean registration in 3D, under the L2-norm error 11446633 Figure9. [sent-521, score-0.338]
91 scene; Note that registration result showing 3D points and 2D image points. [sent-524, score-0.29]
92 For certain applications where real-time performance is not critical, our algorithm can be readily applied, or used as an optimality benchmark. [sent-531, score-0.11]
93 Model-based multiple rigid object detection and registration in unstructured range data. [sent-569, score-0.278]
94 Global optimization through searching rotation space and optimal estimation of the essential matrix. [sent-623, score-0.138]
95 A robust algorithm for point set registration using mixture of gaussians. [sent-643, score-0.289]
96 Consensus set maximization with guaranteed global optimality for robust geometry estimation. [sent-667, score-0.15]
97 A robust method for registration and segmentation ofmultiple range images. [sent-683, score-0.26]
98 Point set registration via particle filtering and stochastic dynamics. [sent-725, score-0.286]
99 An approach to multimodal biomedical image registration utilizing particle swarm optimization. [sent-741, score-0.308]
100 Iterative point matching for registration of free-form curves and surfaces. [sent-745, score-0.289]
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