Point clouds registration based on the point-to-plane approach for orthogonal transformations A Makovetskii1, S Voronin1, V Kober2, A Voronin1 and D Tihonkih1 1 Chelyabinsk State University, Bratiev Kashirinykh str. 129, Chelyabinsk, Russia, 454001 2 Department of Computer Science, CICESE, Carretera Ensenada-Tijuana 3918, Ensenada, B.C., Mexico, 22860 Abstract. The most popular algorithm for aligning of 3D point data is the Iterative Closest Point (ICP). This paper proposes a new algorithm for orthogonal registration of point clouds based on the point-to-plane ICP algorithm for affine transformation. At each iterative step of the algorithm, an approximation of the closed-form solution for the orthogonal transformation is derived. 1. Introduction The Iterative Closest Point (ICP) algorithm [1-5] has become the dominant method for aligning three dimensional models based purely on the geometry. For alignment it is necessary to find a geometric transformation that connects two point clouds in ℝ3 by the best way with respect to the 𝐿2 norm. The ICP algorithm consists of two main stages: 1. Searching of corresponding points (pairs) in two clouds; 2. Minimizing the error metric (variational subproblem of the ICP). There are two basic approaches to choosing the error metric for pairs of points. Within the point-to- point approach [1], the distance between the elements of the pair in ℝ3 is used. Within the point-to- plane approach [2] the distance between the point of the first cloud and the tangent plane to the corresponding point of the second cloud is used. The key point [6] of the ICP algorithm is the search of either an orthogonal or affine transformations, best in the sense of a quadratic metric that combines two point clouds with a given correspondence between points (the variational subproblem of the ICP algorithm). For the point-to-point metric in the case of orthogonal transformations, the solution in a closed- form was obtained by Horn [7,8]. The solution [7] is based on the use of quaternions, whereas the solution [8] uses orthogonal matrices. The solutions are linear in time with respect to the number of point pairs. The original ICP algorithm is widely used for the rigid objects registration, but it does not work well for the case of the non-rigid objects. An extension of the ICP algorithm is proposed [9], using scaling in addition to rotation and translation. A generalization of this algorithm to the case of an arbitrary affine transformation was done [10,11]. A closed-form solution to the point-to-point problem was derived [12-14]. The above mentioned approaches for solving the variational subproblem of the ICP algorithm are based on the point-to-point metric. The point-to-plane metric has been shown to perform better than the point-point metric in terms of accuracy and convergence rate [15]. A closed-form solution to the point-to-plane case for orthogonal transformations is an open problem. Instead, iterative methods IV International Conference on "Information Technology and Nanotechnology" (ITNT-2018) Image Processing and Earth Remote Sensing A Makovetskii, S Voronin, V Kober, A Voronin and D Tihonkih based on the linear least-squares optimization or closed-form methods for small angles only are often used [12]. Iterative solutions require an initial approximate estimate of the transformation parameters, and the iterations might converge slowly, converge to a local optimum or not converge at all. In [16,17] a closed-form solution to the point-to-plane problem for an arbitrary affine transformation is proposed. The affine approach works well when the correspondence between point clouds is good. In this case, the affine point-to-plane method precisely reconstructs original geometric transformation for arbitrary affine transformations, in particular for orthogonal transformations [16,17]. When a correspondence between clouds is not sufficiently good, the affine approach cannot reconstructs an original orthogonal transformation. In this paper, we propose an approximation of a closed-form solution to the point-to-plane problem for orthogonal transformation. The method is based on the closed-form solution for the affine point-to- plane problem [16,17], matrix polar decomposition and the Horn’s method for calculating the nearest orthonormal matrix [8]. The proposed method does not require an initial approximate estimate. Computer simulation results are provided to illustrate the performance of the proposed method of solving the minimization problem. 2. Closed-form solution for affine point-to-plane problem Let 𝑃 = {𝑝1 , … , 𝑝𝑛 } be a source point cloud, and 𝑄 = {π‘ž1 , … , π‘žπ‘› } be a destination point cloud in ℝ3 . Suppose that the relationship between points in 𝑃 and 𝑄 is given in such a manner that for each point 𝑝𝑖 exists a corresponding point π‘žπ‘– . The ICP algorithm is commonly considered as a geometrical transformation for rigid objects mapping 𝑃 to 𝑄 𝑅𝑝𝑖 + 𝑑, (1) where 𝑅 is a rotation matrix, 𝑑 is a translation vector, 𝑖 = 1, … , 𝑛. The group of affine transformations in the dimension of three has 12 generators. It means that the affine transformation in the dimension of three is a function of 12 variables. Let us consider the ICP variational problem for an arbitrary affine transformation in the point-to-plane case. Denote by 𝑆(𝑄) a surface constructed from the cloud 𝑄, by 𝑇(π‘žπ‘– ) denote a tangent plane of 𝑆(𝑄) at point π‘žπ‘– . Let 𝐽(𝐴, 𝑇) be the following function: 𝐽(𝐴) = βˆ‘π‘›π‘–=1(< 𝐴 𝑝𝑖 βˆ’ π‘žπ‘– , 𝑛𝑖 > )2 , (2) where <βˆ™,βˆ™> denotes the inner product, 𝐴 is a matrix of an affine transformation in the homogenous coordinates: π‘Ž11 π‘Ž12 π‘Ž13 𝑑1 π‘Ž π‘Ž22 π‘Ž23 𝑑2 𝐴 = ( 21 ), (3) π‘Ž31 π‘Ž32 π‘Ž33 𝑑3 0 0 0 1 𝑝𝑖 is a point from the cloud 𝑃, 𝑛𝑖 is the unitary normal for 𝑇(π‘žπ‘– ) 𝑝1𝑖 𝑛1𝑖 𝑖 𝑖 𝑝𝑖 = 𝑝2 , 𝑛𝑖 = 𝑛2 . (4) 𝑝3𝑖 𝑛3𝑖 (1) (0) The ICP variational problem can be stated as follows: arg π‘šπ‘–π‘›π΄ 𝐽(𝐴) . (5) The solution of the problem (5) is given by the following way [16,17]: 𝑀𝐴 = 𝐢. (6) 𝑀 is the coefficients matrix 12 Γ— 12 𝑗 𝑗 𝑗 𝑗 𝑖 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 𝑀𝑗 = (π‘š11 π‘š21 π‘š31 π‘š41 π‘š12 π‘š22 π‘š32 π‘š42 π‘š13 π‘š23 π‘š33 π‘š43 ), 𝑗 = 1, … ,3, (7) 𝑗 π‘šπ‘˜π‘™ = βˆ‘π‘›π‘–=1(𝑛𝑗 𝑃𝑁)π‘–π‘˜π‘™ , π‘˜, 𝑙 = 1, … ,4, 𝑗 = 1, … ,3, (8) IV International Conference on "Information Technology and Nanotechnology" (ITNT-2018) 237 Image Processing and Earth Remote Sensing A Makovetskii, S Voronin, V Kober, A Voronin and D Tihonkih 𝑝1𝑖 𝑛1𝑖 𝑛𝑗𝑖 𝑝1𝑖 𝑛2𝑖 𝑛𝑗𝑖 𝑝1𝑖 𝑛3𝑖 𝑛𝑗𝑖 0 𝑝2𝑖 𝑛1𝑖 𝑛𝑗𝑖 𝑝2𝑖 𝑛2𝑖 𝑛𝑗𝑖 𝑝2𝑖 𝑛3𝑖 𝑛𝑗𝑖 0 (𝑛𝑗 𝑃𝑁)𝑖 = , 𝑖 = 1, … , 𝑛, 𝑗 = 1, … ,3, (9) 𝑝3𝑖 𝑛1𝑖 𝑛𝑗𝑖 𝑝3𝑖 𝑛2𝑖 𝑛𝑗𝑖 𝑝3𝑖 𝑛3𝑖 𝑛𝑗𝑖 0 𝑖 𝑖 ( 𝑛1 𝑛𝑗 𝑛2𝑖 𝑛𝑗𝑖 𝑛3𝑖 𝑛𝑗𝑖 0) 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑖 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑀3𝑖+𝑗 = (π‘š11 π‘š21 π‘š31 π‘š41 π‘š12 π‘š22 π‘š32 π‘š42 π‘š13 π‘š23 π‘š33 π‘š43 ), 𝑖, 𝑗 = 1, … ,3, (10) 𝑖𝑗 π‘šπ‘˜π‘™ = βˆ‘π‘›π‘–=1(𝑝𝑗 𝑛𝑖 𝑃𝑁)π‘–π‘˜π‘™ , π‘˜, 𝑙 = 1, … ,4, 𝑖, 𝑗 = 1, … ,3, (11) 𝑝1π‘˜ 𝑛1π‘˜ π‘π‘—π‘˜ π‘›π‘–π‘˜ 𝑝1π‘˜ 𝑛2π‘˜ π‘π‘—π‘˜ π‘›π‘–π‘˜ 𝑝1π‘˜ 𝑛3π‘˜ π‘π‘—π‘˜ π‘›π‘–π‘˜ 0 π‘˜ 𝑝2π‘˜ 𝑛1π‘˜ π‘π‘—π‘˜ π‘›π‘–π‘˜ 𝑝2π‘˜ 𝑛2π‘˜ π‘π‘—π‘˜ π‘›π‘–π‘˜ 𝑝2π‘˜ 𝑛3π‘˜ π‘π‘—π‘˜ π‘›π‘–π‘˜ 0 (𝑝𝑗 𝑛𝑖 𝑃𝑁) = , π‘˜ = 1, … , 𝑛, 𝑖, 𝑗 = 1, … ,3. (12) 𝑝3π‘˜ 𝑛1π‘˜ π‘π‘—π‘˜ π‘›π‘–π‘˜ 𝑝3π‘˜ 𝑛2π‘˜ π‘π‘—π‘˜ π‘›π‘–π‘˜ 𝑝3π‘˜ 𝑛3π‘˜ π‘π‘—π‘˜ π‘›π‘–π‘˜ 0 π‘˜ π‘˜ π‘˜ ( 𝑛1 𝑝𝑗 𝑛𝑖 𝑛2π‘˜ π‘π‘—π‘˜ π‘›π‘–π‘˜ 𝑛3π‘˜ π‘π‘—π‘˜ π‘›π‘–π‘˜ 0) 𝐢 is the coefficients column with 12 elements 𝑐𝑗 = βˆ‘π‘›π‘–=1 𝑛𝑗𝑖 < π‘žπ‘– , 𝑛𝑖 >, 𝑗 = 1, … ,3, (13) 𝑐3𝑖+𝑗 = βˆ‘π‘›π‘˜=1 π‘π‘—π‘˜ π‘›π‘–π‘˜ < π‘žπ‘˜ , π‘›π‘˜ > , 𝑖, 𝑗 = 1, … ,3. (14) 𝐴 is the column of variables with 12 elements 𝐴 = (π‘Ž11 π‘Ž12 π‘Ž14 π‘Ž14 = 𝑑1 π‘Ž21 π‘Ž22 π‘Ž23 π‘Ž24 = 𝑑2 π‘Ž31 π‘Ž32 π‘Ž33 π‘Ž34 = 𝑑3 ) 𝑑 . (15) The reconstructed affine transform is done by the following formula: 𝐴 = 𝑀 βˆ’1 𝐢. (16) 3. Polar decomposition and orthogonal transformations A square matrix M can be decomposed into the product of an orthonormal matrix R and a positive semi-definite matrix S [5]. The matrix S is always uniquely determined. The matrix R is uniquely determined when M is nonsingular. When M is nonsingular, we can actually write directly [8] 𝑀 = 𝑅𝑆, (17) 1 𝑅 = 𝑀(𝑀𝑑 𝑀)βˆ’2. (18) 𝑑 The matrix 𝑀 𝑀 is positive semi-definite and symmetric. The orthogonal matrix 𝑅 in (18) can be computed by the following way [8]: 1 0 0 βˆšπœ†1 1 𝑅 = 𝑀𝐢 0 0 𝐢𝑑 , (19) βˆšπœ†2 1 0 0 ( βˆšπœ†3 ) where 𝐢 is orthogonal matrix consisting of columns, that are eigenvectors of the matrix 𝑀𝑑 𝑀. Numbers πœ†π‘– , 𝑖 = 1, … ,3, are eigenvalues of the matrix 𝑀𝑑 𝑀. The formula (18) also defines [8] a nearest orthogonal matrix 𝑅 for the nonsingular matrix 𝑀. It means that the formula (18) describes the projection from the group 𝑆𝐿(3) to the subgroup 𝑆𝑂(3). 4. Projection on 𝑺𝑢(πŸ‘) For approximation of the exact solution of the problem (5) we propose the following method. At each step of the ICP algorithm, we project a top-left submatrix 𝐴′ (size of 3 Γ— 3) of a matrix 𝐴 of an affine transform, computed by the formula (16), to 𝑆𝑂(3) by the formula (19). After that it is necessary to recalculate a translation 𝑑 = (𝑑1 , 𝑑2 , 𝑑3 ) 𝑑 . Denote by 𝑅 a result of projection of a top-left submatrix 3 Γ— 3 of a matrix 𝐴 to 𝑆𝑂(3). Denote by 𝑁 the following matrix 𝑛 Γ— 3: 𝑛11 𝑛12 𝑛13 𝑁=( … ), (20) 𝑛 𝑛 𝑛 𝑛1 𝑛2 𝑛1 IV International Conference on "Information Technology and Nanotechnology" (ITNT-2018) 238 Image Processing and Earth Remote Sensing A Makovetskii, S Voronin, V Kober, A Voronin and D Tihonkih denote by 𝑣 the following vector-column 𝑛 Γ— 1: 𝑣𝑖 =< π‘žπ‘– βˆ’ 𝑅𝑝 𝑖 , 𝑛𝑖 >. (21) Then the problem βˆ‘π‘›π‘–=1(< 𝑅 𝑝𝑖 + 𝑑 βˆ’ π‘žπ‘– , 𝑛𝑖 > )2 = βˆ‘π‘›π‘–=1(< 𝑑, 𝑛𝑖 > βˆ’< π‘žπ‘– βˆ’ 𝑅 𝑝𝑖 , 𝑛𝑖 > )2 β†’ min𝑑 , (22) is the least squares problem for the equation 𝑁𝑑 = 𝑣. (23) Thus we have: 𝑑 = (𝑁 𝑑 𝑁)βˆ’1 𝑁 𝑑 𝑣. (24) Figure 1. Illustration of projection of Figure 2. Block-diagram of the proposed the top-left submatrix 𝐴′ onto 𝑆𝑂3 . algorithm. Figure 1 illustrates projection of the top-left submatrix 𝐴′ of the matrix 𝐴 of the affine transform onto submanifold 𝑆𝑂3 . The block-diagram of the proposed algorithm as part of the ICP algorithm is shown in Figure 2. 5. Computer simulation 5.1. We consider two variants of the ICP algorithm here The first is point-to-point ICP based on Horn algorithm. The second is point-to-plane ICP based on the proposed approximation of an exact solution of the variational problem. Other elements of ICP algorithm are same. 5.1.1. Let 𝑃 be the cloud consisting of 34817 points, see figure 1 (blue colour) The cloud 𝑄 (green colour) is obtained from 𝑃 by the orthogonal transformation 𝑄 = 𝑇1 βˆ— 𝑃, where 𝑇1 is given by 1.00000 0.00000 0.00000 3.10000 0.00000 0.83867 βˆ’0.54464 1.13270 𝑇1 = ( ). 0.00000 0.54464 0.83867 1.92795 0.00000 0.00000 0.00000 1.00000 Computed by the proposed method transformation 𝑀1 is given as 1.00000 0.00000 0.00000 3.10000 0.00000 0.83867 βˆ’0.54464 1.13270 𝑀1 = ( ). 0.00000 0.54464 0.83867 1.92795 0.00000 0.00000 0.00000 1.00000 The reconstructed by the point-to-point ICP geometrical transformation has the same matrix. The point-to-point ICP method converges in 31 iterations, processing time 1745 milliseconds. The proposed ICP method converges in 10 iterations, processing time 913 milliseconds. Figure 3 shows the clouds 𝑃 (blue) and 𝑄 (green), figure 4 shows the clouds 𝑃′ = 𝑀1 βˆ™ 𝑃 (blue) and 𝑄 (green) together. IV International Conference on "Information Technology and Nanotechnology" (ITNT-2018) 239 Image Processing and Earth Remote Sensing A Makovetskii, S Voronin, V Kober, A Voronin and D Tihonkih Figure 3. Cloud 𝑃 (blue), cloud 𝑄 (green). Figure 4. Cloud 𝑃′ = 𝑀1 βˆ™ 𝑃 (blue), cloud 𝑄 (green). 5.1.2. Let 𝑃 be the cloud consisting of 34817 points, see figure 3 (blue colour) The cloud 𝑄 (green colour) is obtained from 𝑃 by the orthogonal transformation 𝑄 = 𝑇2 βˆ— 𝑃, where 𝑇1 is given by 0.91015 βˆ’0.36772 0.19081 βˆ’0.79646 0.21782 0.81653 0.53463 2.18083 ). 𝑇2 = ( βˆ’0.35240 βˆ’0.44503 0.82326 2.41239 0.00000 0.00000 0.00000 1.00000 Computed by the proposed method transformation 𝑀2 is given as 0.91015 βˆ’0.36772 0.19081 βˆ’0.79646 0.21782 0.81653 0.53463 2.18083 ). 𝑀2 = ( βˆ’0.35240 βˆ’0.44503 0.82326 2.41239 0.00000 0.00000 0.00000 1.00000 The reconstructed by the point-to-point ICP geometrical transformation has the same matrix. The point-to-point ICP method converges in 41 iterations, processing time 2458 milliseconds. The proposed ICP method converges in 16 iterations, processing time 1491 milliseconds. Figure 5. Cloud 𝑃 (blue), cloud 𝑄 (green). Figure 6. Cloud 𝑃′ = 𝑀2 βˆ™ 𝑃 (blue), cloud 𝑄 (green). Figure 5 shows the clouds 𝑃 (blue) and 𝑄 (green), figure 6 shows the clouds 𝑃′ = 𝑀2 βˆ™ 𝑃 (blue) and 𝑄 (green) together. 5.1.3. Let 𝑃 be the cloud consisting of 34817 points, see figure 5 (blue colour) The cloud 𝑄 (green colour) is obtained from 𝑃 by the orthogonal transformation 𝑄 = 𝑇3 βˆ— 𝑃, where 𝑇3 is given by 0.98163 0.00000 βˆ’0.19081 βˆ’0.64070 0.03641 0.98163 0.18730 0.03261 𝑇3 = ( ). 0.18730 βˆ’0.19081 0.96359 1.21591 0.00000 0.00000 0.00000 1.00000 Computed by the proposed method transformation 𝑀1 is given as IV International Conference on "Information Technology and Nanotechnology" (ITNT-2018) 240 Image Processing and Earth Remote Sensing A Makovetskii, S Voronin, V Kober, A Voronin and D Tihonkih 0.98163 0.00000 βˆ’0.19081 βˆ’0.64070 0.03641 0.98163 0.18730 0.03261 𝑀3 = ( ). 0.18730 βˆ’0.19081 0.96359 1.21591 0.00000 0.00000 0.00000 1.00000 The reconstructed by the point-to-point ICP geometrical transformation has the same matrix. The point-to-point ICP method converges in 19 iterations, processing time 984 milliseconds. The proposed ICP method converges in 9 iterations, processing time 747 milliseconds. Figure 7. Cloud 𝑃 (blue), cloud 𝑄 (green). Figure 8. Cloud 𝑃′ = 𝑀3 βˆ™ 𝑃 (blue), cloud 𝑄 (green). Figure 7 shows the clouds 𝑃 (blue) and 𝑄 (green), figure 8 shows the clouds 𝑃′ = 𝑀3 βˆ™ 𝑃 (blue) and 𝑄 (green) together. 5.1.4. Let 𝑃 be the cloud consisting of 106289 points, see figure 7 (blue colour) The cloud 𝑄 (green colour) is obtained from 𝑃 by the orthogonal transformation 𝑄 = 𝑇4 βˆ— 𝑃, where 𝑇4 is given by 0.83867 0.54464 βˆ’0.00000 1.38331 βˆ’0.45677 0.70337 βˆ’0.54464 βˆ’0.29804 𝑇4 = ( ). βˆ’0.29663 0.45677 0.83867 0.99881 0.00000 0.00000 0.00000 1.00000 Computed by the proposed method transformation 𝑀4 is given as 0.83867 0.54464 βˆ’0.00000 1.38331 βˆ’0.45677 0.70337 βˆ’0.54464 βˆ’0.29804 ). 𝑀4 = ( βˆ’0.29663 0.45677 0.83867 0.99881 0.00000 0.00000 0.00000 1.00000 The reconstructed by the point-to-point ICP geometrical transformation has the same matrix. The point-to-point ICP method converges in 24 iterations, processing time 6316 milliseconds. The proposed ICP method converges in 16 iterations, processing time 5792 milliseconds. Figure 9. Cloud 𝑃 (blue), cloud 𝑄 (green). Figure 10. Cloud 𝑃′ = 𝑀4 βˆ™ 𝑃 (blue), cloud 𝑄 (green). Figure 9 shows the clouds 𝑃 (blue) and 𝑄 (green), figure 10 shows the clouds 𝑃′ = 𝑀4 βˆ™ 𝑃 (blue) and 𝑄 (green) together. IV International Conference on "Information Technology and Nanotechnology" (ITNT-2018) 241 Image Processing and Earth Remote Sensing A Makovetskii, S Voronin, V Kober, A Voronin and D Tihonkih 6. Conclusion In this paper, we revised error minimizing steps of the ICP algorithm. A new algorithm for orthogonal registration of point clouds based on the point-to-plane ICP algorithm for affine transformation is proposed. At each iterative step of the algorithm, an approximation of the closed-form solution for the orthogonal transformation is derived. 7. 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SPIE's 60 Annual Meeting: Applications of Digital Image Processing XXXVIII 9599 959929-1 [15] Rusinkiewicz S and Levoy M 2001 Efficient Variants of the ICP Algorithm Proceedings of the International Conference on 3-D Digital Imaging and Modeling 145-152 [16] Makovetskii A, Voronin S, Kober V and Tihonkih D 2017 An efficient point-to-plane registration algorithm for affine transformations Applications of Digital Image Processing 10396 103962J [17] Makovetskii A, Voronin S, Kober V and Tihonkih D 2017 Affine registration of point clouds based on point-to-plane approach Procedia Engineering 201 322-330 Acknowledgments The work was supported by the Ministry of Education and Science of Russian Federation (grant β„– 2.1743.2017) and by the RFBR (grant β„– 18-07-00963). IV International Conference on "Information Technology and Nanotechnology" (ITNT-2018) 242