Dmitrii Tihonkih - The Iterative Closest Points Algorithm and Affine Transformations
Download as PPTX, PDF1 like566 views
This document describes modifications made to the iterative closest point (ICP) algorithm. The authors propose a new matching procedure that uses the angles between line segments connecting points to find initial correspondences. They also formulate the ICP variational problem for an arbitrary affine transformation. A computer simulation applies the standard ICP approach and the authors' algorithm to two point sets related by a known transformation, finding the latter estimates the transformation more accurately.
![Computer simulation
Let 𝑋 be the set consists of 80 points. The
coordinates of points are randomly generated
(by the uniform distribution). The values of all
coordinates belong to the range [0, . . . , 100].
The set 𝑌 is obtained from the set 𝑋 by the
geometrical transformation 𝑌 = 𝑅 ∗ 𝑋 + 𝑡,
where 𝑅 and 𝑡 are described below:
𝑅 =
1 0 0
0 0.5 −0.866025
0 0.866025 0.5
,](https://image.slidesharecdn.com/tihonkihtheiterativeclosestpointsalgorithm-160427212240/85/Dmitrii-Tihonkih-The-Iterative-Closest-Points-Algorithm-and-Affine-Transformations-16-320.jpg)
Dmitrii Tihonkih - The Iterative Closest Points Algorithm and Affine Transformations
- 1. Dmitrii Tihonkih Department of Mathematics, Chelyabinsk State University, Russian Federation Artyom Makovetskii Department of Mathematics, Chelyabinsk State University, Russian Federation Vladislav Kuznetsov Department of Mathematics, Chelyabinsk State University, Russian Federation E-mails: sparsemind@gmail.com, artemmac@csu.ru, k.v.net@rambler.ru The iterative closest points algorithm and affine transformations AIST'2016
- 2. Introduction The standard ICP starts with two point clouds for their relative rigid-body transform, and iteratively refines the transform by repeatedly generating pairs of corresponding points in the clouds and minimizing an error metric.
- 3. The ICP stages: 1. Selection of some set of points in one clouds. 2. Matching these points to samples in the other cloud. 3. Rejecting certain pairs based on looking at each pair individually or considering the entire set of pairs. 4. Assigning an error metric based on the point pairs. 5. Minimizing the error metric (variational subproblem of the ICP).
- 4. Our main focus is on the accuracy of the final answer and the ability of ICP to reach the correct solution for a given difficult geometry. We consider transformation that hold the angles between lines in the cloud of points. Also we consider the ICP minimizing the error metric subproblem for the case of an arbitrary affine transformation.
- 5. The matching procedure for sets 𝐗 and 𝐘 Let 𝑋 = {𝑥0, … , 𝑥 𝑘−1} be an set consist of 𝑘 points in ℝ3 and 𝑌 = {𝑦0, … , 𝑦 𝑛−1} be an set consist of 𝑛 points in ℝ3 . Denote by (𝑥𝑖, 𝑦𝑗), 𝑥𝑖 ∈ 𝑋, 𝑦𝑗 ∈ 𝑌 the pair of corresponding points. Note, that each point from 𝑋 and 𝑌 can be included to the set of pairs just one time.
- 6. At the beginning the set of pairs is empty. Let 𝑚 ∈ ℕ be a number such that: 3 ≤ 𝑚 ≤ min(𝑛, 𝑘). 1. Consider the following subset 𝑋𝑖 of the 𝑋: 𝑋𝑖 = {𝑥 𝑚∗ 𝑖−1 , … , 𝑥 𝑚∗ 𝑖−1 +𝑚−1}. 2. Let 𝐶 be a closed piecewise linear curve in ℝ3 that consist of 𝑚 line segments. The 𝑗-th segment connects points 𝑥 𝑚∗ 𝑖−1 +𝑗 and 𝑥 𝑚∗ 𝑖−1 +𝑗+1.
- 7. Denote by 𝛼𝑗 a minimal flat angle that is constructed by 𝑗-th and (𝑗 + 1)-th segments. Let 𝑉𝑋 be a vector 𝑉𝑋 = {𝛼0, … , 𝛼 𝑚−1}, where elements αj, j = 0, … , m − 1 are respective angles.
- 8. 3. Consider all possible combinations of m points in the set Y besides the points that already included to the set of pairs. For an each combination we construct the vector 𝑉 by the same way as in step 2. 4. We choose a vector from the set of vectors of the step 3 such that distance between them and 𝑉𝑋 is minimal relatively the norm 𝐿1. Denote this vector as 𝑉𝑌.
- 9. 5. We construct 𝑚 pairs of the points by 𝑉𝑋 and 𝑉𝑌. Add this m pairs to the set of pairs. 6. If the number of remaining points in 𝑋 or 𝑌 less that 𝑚 then procedure terminates. Else 𝑖 ≔ 𝑖 + 1 and go to step 1.
- 10. We use this procedure only as first iteration on the ICP algorithm. Obtained after the first iteration the transformation matrix and the translation vector are used for a second iteration. In the next iterations we use the standard nearest neighbor approach. The described above approach can good work not for rigid transformation only but for sufficiently wide subset of the affine transformations.
- 11. The ICP variational subproblem for an arbitrary affine transformation Suppose that the relationship between points in 𝑋 and 𝑌 is done by such a way that for each point 𝑥𝑖 is calculated corresponding point 𝑦𝑖. The ICP algorithm is offten considered as a geometrical transformation for rigid objects mapping 𝑋 to 𝑌:
- 12. 𝑅𝑥𝑖 + 𝑡, where 𝑅 is a rotation matrix, 𝑇 is a translation vector, 𝑖 = 0, … , 𝑛 − 1. The S-ICP algorithm is given by 𝑅𝑆𝑥𝑖 + 𝑡, where 𝑆 is a scaling matrix.
- 13. ICP variational problem for the case of an arbitrary affine transformation. Let 𝐽(𝐴, 𝑇) be the following function: 𝐽 𝐴, 𝑇 = 𝑖=0 𝑛−1 ∥ 𝐴 𝑥𝑖 + 𝑡 − 𝑦𝑖 ∥2 . The ICP variational problem can be stated as follows: arg 𝑚𝑖𝑛 𝐽 𝐴, 𝑡 , 𝐴, 𝑡
- 14. where 𝐴 = 𝑎11 𝑎12 𝑎13 𝑎21 𝑎22 𝑎23 𝑎31 𝑎32 𝑎33 , 𝑡 = 𝑡1 𝑡2 𝑡3 , 𝑥𝑖 = 𝑥1𝑖 𝑥2𝑖 𝑥3𝑖 , 𝑦𝑖 = 𝑦1𝑖 𝑦2𝑖 𝑦3𝑖 .
- 15. The elements of the first row of the matrix 𝐴∗ that minimizes 𝐽 are computed as 𝑎11 = 𝑖=0 𝑛−1 𝑦1𝑖− 𝑎12 𝑥2𝑖− 𝑎13 𝑥3𝑖 𝑥1𝑖 𝑖=0 𝑛−1 𝑥1𝑖 2 , 𝑎12 = 𝑗=0 𝑛−1 𝛾 𝑗 𝛼 𝑗−𝑎13 𝑗=0 𝑛−1 𝛽 𝑗 𝛼 𝑗 𝑗=0 𝑛−1 𝛼 𝑗 2 , 𝑎13 = 𝑘=0 𝑛−1 𝜑 𝑘 𝜓 𝑘 𝑘=0 𝑛−1 𝜑 𝑘 2 .
- 16. Computer simulation Let 𝑋 be the set consists of 80 points. The coordinates of points are randomly generated (by the uniform distribution). The values of all coordinates belong to the range [0, . . . , 100]. The set 𝑌 is obtained from the set 𝑋 by the geometrical transformation 𝑌 = 𝑅 ∗ 𝑋 + 𝑡, where 𝑅 and 𝑡 are described below: 𝑅 = 1 0 0 0 0.5 −0.866025 0 0.866025 0.5 ,
- 17. 𝑡T = 5 6 7 . The standard approach based on nearest neighbor method gives the following results (open source, C++): 𝑅 = 0.45 0.64 0.61 0.89 −0.36 −0.27 −0.05 −0.67 0.73 , 𝑡T = −7.56 11.22 −11.42 .
- 18. Estimated matrix 𝑅 and vector 𝑡 (our algorithm): 𝑅 = 0.99 9.78e−06 9.41e−05 −0.000100597 0.499911 −0.866098 −2.66211e−05 0.865921 0.499979 , 𝑡T = 5.0028 6.0127 7.00388 .
- 19. Conclusion In this work we considered matching and error minimizing steps of the ICP algorithm. On the base of the obtained results, a new efficient algorithm for the sets alignment was designed. The obtained results are illustrated with the help of computer simulation.