=Paper= {{Paper |id=Vol-2210/paper31 |storemode=property |title=Point clouds registration based on the point-to-plane approach for orthogonal transformations |pdfUrl=https://ceur-ws.org/Vol-2210/paper31.pdf |volume=Vol-2210 |authors=Artyom Makovetskii,Sergei Voronin,Vitaly Kober,Aleksei Voronin,Dmitrii Tihonkih }} ==Point clouds registration based on the point-to-plane approach for orthogonal transformations== https://ceur-ws.org/Vol-2210/paper31.pdf 
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

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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)




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                                          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


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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.


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  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




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                              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.



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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. References
[1] Besl P and McKay N 1992 A Method for Registration of 3-D Shapes IEEE Transactions on
      Pattern Analysis and Machine Intelligence 14 239-256
[2] Chen Y and Medioni G 1992 Object Modeling by Registration of Multiple Range Images
      Image and Vision Computing 10 145-155
[3] Protsenko V I, Kazanskiy N L and Serafimovich P G 2015 Real-time analysis of parameters of
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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).




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