cvpr cvpr2013 cvpr2013-97 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Zsolt Sánta, Zoltan Kato
Abstract: A novel correspondence-less approach is proposed to find a thin plate spline map between a pair of deformable 3D objects represented by triangular surface meshes. The proposed method works without landmark extraction and feature correspondences. The aligning transformation is found simply by solving a system of nonlinear equations. Each equation is generated by integrating a nonlinear function over the object’s domains. We derive recursive formulas for the efficient computation of these integrals. Based on a series of comparative tests on a large synthetic dataset, our triangular mesh-based algorithm outperforms state of the art methods both in terms of computing time and accuracy. The applicability of the proposed approach has been demonstrated on the registration of 3D lung CT volumes.
Reference: text
sentIndex sentText sentNum sentScore
1 hu Abstract A novel correspondence-less approach is proposed to find a thin plate spline map between a pair of deformable 3D objects represented by triangular surface meshes. [sent-6, score-0.787]
2 The aligning transformation is found simply by solving a system of nonlinear equations. [sent-8, score-0.264]
3 We derive recursive formulas for the efficient computation of these integrals. [sent-10, score-0.227]
4 Based on a series of comparative tests on a large synthetic dataset, our triangular mesh-based algorithm outperforms state of the art methods both in terms of computing time and accuracy. [sent-11, score-0.413]
5 The applicability of the proposed approach has been demonstrated on the registration of 3D lung CT volumes. [sent-12, score-0.436]
6 Introduction A wide range of application areas, such as registration of 3D laser scans [22] or volumetric medical images [16], requires the alignment of deformable objects [11, 19]. [sent-14, score-0.498]
7 From this point of view, registration techniques can be classified into two main categories: physical model-based and parametric or functional representation [10]. [sent-16, score-0.279]
8 A broadly used class of such parametric models are splines, in particular thin plate splines (TPS) [2, 24]. [sent-18, score-0.311]
9 TPS models are quite useful whenever a parametric free-form registration is required but the underlying physical model of the object deformation is unknown or too complex. [sent-19, score-0.337]
10 The aligning transformation is then found by maximizing similarity between the objects, which usually yields a complex non-linear optimization procedure. [sent-24, score-0.165]
11 While point-based registration has the advantage of tolerating rather high occlusions, they are typically inefficient for large point sets. [sent-31, score-0.23]
12 In many cases, however, the goal is the accurate alignment of whole volumetric objects of several megavoxels (e. [sent-32, score-0.178]
13 These approaches could handle large scope of deformations, but usually they do not use classical parametric transformation models. [sent-37, score-0.134]
14 While elastic registration of planar shapes has been addressed by many researchers [1, 8], the extension of 2D methods for 3D object registration is far from trivial. [sent-38, score-0.506]
15 Herein, we propose a general TPS framework for aligning 3D deformable objects. [sent-39, score-0.118]
16 The basic idea of the method is inspired by [8]: an overdetermined system of nonlinear equations is constructed by integrating a set of nonlinear functions over the object volumes, which is then solved in the least squares sense. [sent-40, score-0.321]
17 However, the procedure we develop here to construct the equations is substantially different: in [8], the integrals are evaluated in terms of 2D pix- els, which is relatively straightforward to extend to 3D vox222222777533 els. [sent-41, score-0.362]
18 In this paper, we will show that complexity can be drastically reduced when the input objects are represented as triangular surface meshes. [sent-43, score-0.593]
19 The performance and robustness of the proposed approach has been quantitatively evaluated on a synthetic dataset, and it has also been successfully applied to the alignment of segmented lung CT images. [sent-47, score-0.301]
20 Thin plate spline registration framework Thin plate splines (TPS) [2, 24] are commonly used as parametric models for elastic deformations using radial basis functions. [sent-49, score-0.688]
21 Thus in classical correspondence based approaches, control points are placed at extracted point matches, and the deformation at other positions is interpolated by the TPS. [sent-66, score-0.137]
22 However, we are interested in solving the TPS registration problem without correspondences. [sent-68, score-0.23]
23 Obviously, a finer grid would allow a more refined approximation of the deformation field at the price of an increased number of free parameters. [sent-73, score-0.115]
24 When objects are represented as a set of foreground voxels, the generation of the transformed volume would be computationally expensive. [sent-75, score-0.125]
25 [8] proposed attoi use integral transformation ∫ℱ표y푑y =∫ℱ푡휑(x)∣퐽휑(x)∣푑x, (5) which involves the Jacobian determinant of 휑, composed of the partial derivatives of the transformation. [sent-77, score-0.12]
26 In the next section, we will show that when 3D objects are represented as triangulated surface meshes, the computational complexity of 휑(ℱ푡) can be drastically recduomcepdu tbayt coonmalp cuotimngpl ethxiet integrals in (4) via recursive formulas. [sent-79, score-0.579]
27 To generate sufficiently many equations, we will extend the 2D solution [8] and apply a set of non-linear functions {휔푖}푖ℓ=1 to both sides of (3) yielding a system eoafr rℓ f unonnctli onneasr { e휔quations: ∫ℱ표휔푖(y)푑y =∫휑(ℱ푡)휔푖(z)푑z 푖 = 1,. [sent-81, score-0.118]
28 The least squares solution of the kabnoovwen system directly provides tlehea parameters oluft tiohen aligning TPS transformation. [sent-86, score-0.152]
29 Computing the integrals over meshes In the following, we will propose an efficient computational scheme for the integrals in (6). [sent-88, score-0.635]
30 For that purpose, let us denote the triangular surface meshes of the template and observation objects by 푇△ and 푂△ and the volumes enclosed by these meshes by and 퐹표△, respectively. [sent-89, score-0.971]
31 Since ℱ푡 ≈ and ℱ표 ≈ 퐹표△, the integrals nfrootmed ( b4y) can . [sent-91, score-0.255]
32 Ttrahedronsgenratedfromatringularmeshofatrus: The blue tetrahedron has positive signed volume and the red one has negative. [sent-94, score-0.292]
33 each triangle 표 = (퐴, 퐵, 퐶), let us create a tetrahedron 풯표 = (퐴, 퐵, 퐶, 0) defined by the vertices ofthe corresponding triangle 표 and the origin 0. [sent-95, score-0.509]
34 Thus for all triangle, we generate a tetrahedron with the fourth vertex shared by all of these tetrahedrons (see Fig. [sent-96, score-0.422]
35 Although the choice of the fourth vertex is arbitrary, setting it to the origin will greatly simplify our computations. [sent-98, score-0.11]
36 The computation of the signed volume of such a tetrahedron 풯표 is straightforward: vol(풯표) =61? [sent-99, score-0.292]
37 1, there is a torus given by its triangular mesh. [sent-114, score-0.372]
38 The volumes of the tetrahedrons generated by the two marked triangles has different signs, caused by the different vertex order. [sent-115, score-0.293]
39 In general, we cannot compute the integrals in a straightforward way for an arbitrary function. [sent-117, score-0.284]
40 However, it is possible to derive a similar closed form formula for the integrals as before, when 휔푖 are polynomials. [sent-118, score-0.255]
41 The integral over a tetrahedron 풯표 can then be written as e[13 i]n ∫풯표푦푚1푖푦2푛푖푦3표푖푑y = 6(∣푚v푖ol+(풯 푛표)푖∣푚+ 표푖! [sent-128, score-0.248]
42 푆푚푖푛푖표푖(풯표) with 푆푚푖푛푖표푖 = ∑ ∑ 푎 1+푎2∑ ∑+푎3 =푚푖 푏1 +푏2∑ ∑+푏3 ∑ =푛푖 (15) 푐1 +푐2∑ ∑+푐3 =표푖 (푎1, 푏1, 푐1)(푎2, 푏2, 푐2)(푎3, 푏3, 푐3) 퐴푎11퐴푏21퐴3푐1퐵푎12퐵2푏2퐵3푐2퐶1푎3퐶푏23퐶3푐3 퐴, 퐵, (16) where 퐶 ∈ ℝ3 are the vertices of the corresponding triangle 표. [sent-131, score-0.159]
43 Hence the complexity of computing all of the integrals for 푖 = 1, . [sent-135, score-0.287]
44 , ℓ over one tetrahedron will be 풪(푀3), where 푀 is the maximal degree of polynomials f풪ro(m푀 the {휔푖}푖ℓ=1 set. [sent-138, score-0.285]
45 Numerical implementation As we presented in the previous section the equations of (6) are only approximately valid, due to the discrete representation of triangular surfaces. [sent-140, score-0.45]
46 Since the system is solved by minimizing the algebraic error, proper normalization is critical for numerical stability. [sent-145, score-0.134]
47 Furthermore, after normalization the origin becomes the centroid of the objects, therefore the fourth point of each tetrahedron, described in the previous section, will be the centroid of the objects, which also increases numerical stability. [sent-151, score-0.126]
48 The range of the 휔푖 functions should also be normalized into [−1, 1] in order to ensure a balanced contribution of the equations t]o i nth oer algebraic error. [sent-152, score-0.117]
49 Following [t8ri]b, uthtiiosn can h bee achieved by dividing the integrals with the maximal magnitude of the integral over the unit sphere containing the objects: 푁푖=∫∥x∥≤√23∣휔푖(x)∣푑x (23) We thus obtain the following system of equations 푁1푖∫푁표(ℱ표)휔푖(y)푑y =푁1푖∫푁푡(휑(ℱ푡))휔(z)푑z, (24) ℓ. [sent-153, score-0.478]
50 , Using the triangular surface mesh representations, we get the following approximation of our system: 푁1푖표∈푁∑표(푂△)sgn(vol(푇표))∫풯표푦1푚푖푦2푛푖푦3표푖푑y ≈ 푁1푖휋∈휑(∑푁푡(푇△))sgn(vol(풯휋))∫풯휋푧1푚푖푧2푛푖푧3표푖푑z, ℓ (25) where 푖 = 1. [sent-157, score-0.753]
51 and the integrals over tetrahedrons can be computed via (15) using the recursive formulas (20)– (22). [sent-160, score-0.624]
52 Experimental results In our experiments we used a TPS model with 64 control points placed on a uniform grid, yielding a total of 2b0y4 th pear saemte 휔te푖r(sx. [sent-165, score-0.125]
53 2: Compute the normalizing transformations 푁푡 and 푁표 which maps vertex coordinates into [−0. [sent-171, score-0.109]
54 3: Cwohincshtru mcat pthse v system oofr equations o(2 [5−) using 5t]h. [sent-174, score-0.118]
55 The registration error has been quantitatively evaluated based on the Dice coefficient of the aligned objects: 훿 =∣∣퐹퐹푟푟∣△ + ∣ 퐹퐹표표∣∣⋅ 100%, (26) where 퐹표 and 퐹푟 denote the set of foreground voxels of the observation and registered objects respectively. [sent-182, score-0.373]
56 In the last row, registered objects are overlayed, overlapping voxels shown in yellow and non-overlapping ones in red and green. [sent-185, score-0.143]
57 Deformjecattiso onfs were generated ebagasevdo on sth hians plate splines . [sent-189, score-0.203]
58 In order to preserve the topology of the input object, a diffeomorphic transformation were generated. [sent-196, score-0.194]
59 Some examples of our registration results on this dataset can be seen in Fig. [sent-198, score-0.23]
60 The synthetic observations as well as registered objects were generated by the following procedure using Matlab: First, a smooth triangular mesh has been created from the object’s surface using Matlab’s internal i sosurface function. [sent-200, score-0.842]
61 Then the transformation was applied to the vertices yielding the transformed triangular mesh, from which the final voxelized object was obtained by the binvox program available from http : / /www . [sent-201, score-0.614]
62 In particular, the triangular mesh used for the transformation had much higher resolution than what was used in the registration process. [sent-206, score-0.889]
63 Comparison of registration error and computing time with various 푟 values for the Delaunay sphere in surface mesh ex- traction. [sent-208, score-0.616]
64 In the first experiment, we evaluated the sensitivity of the mesh-based algorithm (see Algorithm 1) with respect to the mesh resolution. [sent-209, score-0.202]
65 For that purpose, the triangular mesh extractor algorithms from CGAL library [21] were used where the resolution of the triangular mesh can be controlled by the maximal radius 푟 ofthe corresponding Delaunay sphere. [sent-210, score-1.184]
66 Using our synthetic dataset, triangular meshes has been created for 푟 ∈ {1, 3, 5, 6, 10}. [sent-211, score-0.538]
67 Obviously, trheeresolution of the triangular mesh affects the computation 222222777977 time. [sent-215, score-0.574]
68 On the other hand, the computational time can be significantly reduced by decreasing 푟 at the price of a slightly lower registration accuracy. [sent-216, score-0.258]
69 In Table 1, we compared the registration quality and computing times of the naive voxel-based extension of [8] and the proposed mesh-based algorithm for 푟 = 5. [sent-226, score-0.259]
70 While the registration errors are of similar magnitude, the computing times show an almost 20 times speed-up for the meshbased algorithm. [sent-227, score-0.23]
71 This is not surprising, as the mesh-based algorithm works only with triangle vertices whose number is less than 9000, whereas the voxel-based method has to deal with several hundred thousand voxels. [sent-228, score-0.159]
72 This is due to the way these numerical schemes approximate the continuous integrals in (24). [sent-231, score-0.31]
73 In the voxelbased case, approximation error is due to 1) the discretization error on the object’s surface (inner voxels are not producing such errors) and 2) the Jacobian is implicitly assumed to be constant within each voxel (including the inner ones! [sent-232, score-0.32]
74 However, the mesh-based algorithm computes the exact (continuous) integrals over each tetrahedron, therefore the only source of the approximation error is the difference between the true object surface and its approximating triangular mesh. [sent-234, score-0.806]
75 Considering that a 훿 < 10% corresponds to a visually good alignment, our approach is quite robust up to as high as 20% surface error. [sent-239, score-0.15]
76 We have also compared our results to the point-based registration frameworks in [11] (GMMREG) and [17] (CPD). [sent-242, score-0.23]
77 Robustness test results for various degree of synthetically generated surface errors. [sent-246, score-0.15]
78 For each test, samples of surface errors on a voxel slice are shown. [sent-247, score-0.215]
79 The input of these algorithms were the vertices of the extracted triangular surface meshes, using the same extraction technique as in the previous tests. [sent-249, score-0.587]
80 The mesh resolution was controlled by the maximal radius 푟 of the corresponding Delaunay sphere. [sent-250, score-0.238]
81 For the best GMMREG set both the average and median runtimes were 30 min, for the lower resolutions both the average and median runtimes were 3 min. [sent-251, score-0.122]
82 Medical application Lung alignment is a crucial task in lung cancer diagnosis [5]. [sent-259, score-0.26]
83 The Delaunay sphere in surface mesh extraction was 푟 = 5. [sent-263, score-0.386]
84 Due to respiratory motion, the lung is subject to a nonlinear deformation between suchfollow-ups, hence it is hard to automatically find correspondences. [sent-266, score-0.323]
85 We successfully applied the proposed approach to align 3D lung CT scans. [sent-268, score-0.206]
86 Since the triangular mesh basedapproach is more accurate and faster, the input voxel volumes were first transformed into a triangular mesh using 푟 = 5, and then the TPS parameters were recovered by Algorithm 1. [sent-269, score-1.302]
87 × In medical applications, however, it is necessary to align the inner part of the objects too. [sent-274, score-0.152]
88 6, where we also show the achieved inner alignment on grayscale slices of the original lung CT images. [sent-278, score-0.342]
89 In many medical image registration problem, one is looking for a smooth deformation, thus a diffeomorphic solution is needed. [sent-280, score-0.415]
90 It is a well known fact, that thin plate splines may not be diffeomorphic without further regularization. [sent-281, score-0.371]
91 In out approach, using the bending energy minimization provided sufficient constraint to obtain diffeomorphic solutions. [sent-283, score-0.189]
92 This finding has been experimentally confirmed on the lung dataset by computing the Jacobian of the obtained transformations for all of the input voxels: the values were positive in every cases. [sent-284, score-0.242]
93 Conclusion We have proposed a novel TPS registration method which works without established correspondences to register two objects represented by their triangular surface meshes. [sent-286, score-0.791]
94 The basic idea is to set-up a system of nonlinear equations whose solution directly provides the parameters of the aligning transformation. [sent-287, score-0.257]
95 An efficient numerical scheme were proposed for triangular mesh representation. [sent-288, score-0.629]
96 Finally, the algorithm achieved promising results in aligning lung CT images, which demonstrates the usefulness of the method in real life applications. [sent-291, score-0.286]
97 Generalized multidimensional scaling: A framework for isometryinvariant partial surface matching. [sent-329, score-0.15]
98 3d surface reconstruction and registration for [13] [14] [15] [16] [17] [18] image guided medialization laryngoplasty. [sent-399, score-0.38]
99 Deformable 3D shape registration based on local similarity transforms. [sent-457, score-0.23]
100 Elastic convolved ICP for the registration of deformable objects. [sent-487, score-0.268]
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