Softassign and EM-ICP on GPU
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The document discusses GPU implementations of 3D point set registration algorithms including Softassign, EM-ICP, and weighted Horn's method. It finds that EM-ICP runs 60 times faster on a GPU compared to CPU and can align 5000 points within 7 seconds, making it more robust than CPU-based ICP. The implementations allow registration of up to 8000 points depending on GPU memory. Limitations include inefficient stopping conditions and a fixed number of iterations.
![Contributionofthistalk
Fast GPU implementations of registration
algorithms for 3D point sets.
Softassign [Gold et al., 1998]
EM-ICP [Granger et al., 2002]
(Weighted) Horn’s method [Horn, 1987]
So, what is “registartion” ?](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Softassign and EM-ICP on GPU
- 1. ToruTamaki, MihoAbe, Bisser Raytchev, Kazufumi Kaneda 19th Nov. 2010
- 2. Contributionofthistalk Fast GPU implementations of registration algorithms for 3D point sets. Softassign [Gold et al., 1998] EM-ICP [Granger et al., 2002] (Weighted) Horn’s method [Horn, 1987] So, what is “registartion” ?
- 3. Whatis“Registration”or“Alignment”? A set of images Image registration
- 4. Registrationof3Dpointsets 大石岳史,増田智仁,倉爪亮,池内克史,創建期奈良大仏及び大仏殿のデ ジタル復元,日本バーチャルリアリティ学会論文誌, Vol. 10, No. 3, pp.429-436, 2005.10. A statue Range data from one view Range data from another view Aligned (registered) 3d point cloud An example of rendered CG image of the statue
- 5. 3Dregistrationalgorithm Input Two point sets: 𝑋 and 𝑌 Output Rotation matrix 𝑅 Translation vector 𝒕 X Y 𝑅 and 𝒕
- 6. Algorithmsforregistration Horn’s method • Corresponding point sets are given. • Estimate R and t. ICP (Iterative closest point) • Unknown correspondence. • Fast, standard. • Easily fail due to local minimum. • A lot of variants follow. Softassign • Unknown correspondence. • Robust. • Very slow because of iterations. EM-ICP • Unknown correspondence. • Robust. • Very slow because of iterations. Registration algorithm
- 7. Algorithmsforregistration Horn’s method • Corresponding point sets are given. • Estimate R and t. ICP (Iterative closest point) • Unknown correspondence. • Fast, standard. • Easily fail due to local minimum. • A lot of variants follow. Softassign • Unknown correspondence. • Robust. • Very slow because of iterations. EM-ICP • Unknown correspondence. • Robust. • Very slow because of iterations. Registration algorithm
- 8. Horn’smethod:correspondenceisknown. 𝑋 𝑌 X Y ? Unknown correspondence X Y Known correspondence 𝒙1 𝒚1 𝒙2 𝒚2 ⋮⋮ 𝑇 𝑇 𝑇 𝑇 𝒙1 = (𝑥1𝑥, 𝑥1𝑦, 𝑥1𝑧) 𝑇
- 9. Horn’smethod:correspondenceisknown. 𝑋 𝑌 𝒙1 𝒚1 𝒙2 𝒚2 ⋮⋮ 𝑇 𝑇 𝑇 𝑇 𝒙 𝒚 Compute centers 𝑋 𝑌 Centering 𝑋 − 𝒙 𝑌 − 𝒚 𝑋 𝑌𝑆 = 𝐾 = Computer 1st Eigenvector 𝒒 : quaternion 𝑞 Convert 𝑞 to 𝑅 𝒕 = 𝒙 − 𝑅𝒚 1 2 3 4 5
- 10. Algorithmsforregistration Horn’s method • Corresponding point sets are given. • Estimate R and t. ICP (Iterative closest point) • Unknown correspondence. • Fast, standard. • Easily fail due to local minimum. • A lot of variants follow. Softassign • Unknown correspondence. • Robust. • Very slow because of iterations. EM-ICP • Unknown correspondence. • Robust. • Very slow because of iterations. Registration algorithm
- 11. ICP:correspondenceisunknown. 𝑋 𝑌 𝒙1 𝒚1 𝒙2 𝒚2 ⋮⋮ 𝑇 𝑇 𝑇 𝑇 Find closest (nearest) point to 𝒙1 in 𝑌 𝑌∗ 𝒚𝑖 𝒚𝑖 Put the point to 𝑌∗
- 12. ICP:correspondenceisunknown. 𝑋 𝑌 𝒙1 𝒚1 𝒙2 𝒚2 ⋮⋮ 𝑇 𝑇 𝑇 𝑇 Find closest (nearest) point to 𝒙1 in 𝑌 𝑌∗ 𝒚 𝑗 𝒚𝑖 Put the point to 𝑌∗ 𝒚 𝑗 ⋮ Horn’s method with 𝑋 and 𝑌∗ Estimate 𝑅 and 𝒕
- 13. ICP:correspondenceisunknown. 𝑋 𝑅𝑌 + 𝒕 𝒙1 𝒚1 𝒙2 𝒚2 ⋮⋮ 𝑇 𝑇 𝑇 𝑇 Find closest (nearest) point to 𝒙1 in 𝑌 𝑌∗ 𝒚 𝑗 𝒚𝑖 Put the point to 𝑌∗ 𝒚 𝑗 ⋮ Horn’s method with 𝑋 and 𝑌∗ Estimate 𝑅 and 𝒕 Repeat Fast, but easy to fail due to hard correspondence.
- 14. Algorithmsforregistration Horn’s method • Corresponding point sets are given. • Estimate R and t. ICP (Iterative closest point) • Unknown correspondence. • Fast, standard. • Easily fail due to local minimum. • A lot of variants follow. Softassign • Unknown correspondence. • Robust. • Very slow because of iterations. EM-ICP • Unknown correspondence. • Robust. • Very slow because of iterations. Registration algorithm
- 15. Softassign:softcorrespondence. 𝑋 𝑌 𝒙𝑖 𝒚 𝑗 𝑚𝑖𝑗 𝑚𝑖𝑗 = ||𝒙𝑖 − 𝑅𝒚 𝑗 + 𝒕 || 𝑀 Weighted Horn’s method with 𝑋 and 𝑌 Estimate 𝑅 and 𝒕 Repeat Each row and column should be normalized to 1 by Shinkhorn iterations
- 16. Shinkhorniterations 𝑀 Each row and column should be normalized to 1 by Shinkhorn iterations 𝑚𝑖𝑗 sum up to 1 sum up to 1 sum up to 1 ⋮ sum up to 1 Repeat row and column normalization until converge.
- 17. Shinkhorniterations 𝑀 Each row and column should be normalized to 1 by Shinkhorn iterations 𝑚𝑖𝑗 sumupto1 sumupto1 sumupto1 ⋮ sumupto1 Repeat row and column normalization until converge.
- 18. Shinkhorn.GPU(rownormalization) 𝑀 Each row and column should be normalized to 1 by Shinkhorn iterations 𝟏 1 1 1 ⋮ 𝑹 𝑀 Using sgemv of CUBLAS
- 19. Shinkhorn.GPU(rownormalization) 𝑀 Each row and column should be normalized to 1 by Shinkhorn iterations 𝑹 𝑀 UsingCUDA kernel Row-wise division Column normalization is done by the same way.
- 20. WeightedHorn’smethod 𝑋 𝑌𝑆 = 𝑋 𝑌𝑆 = 𝑀 3 3 Normal version Weighted version Using CUBLAS sgemv twice.
- 21. Centering.GPU(weightedversion) 𝑋 𝑹 𝑀 𝟏 1 1 1 ⋮ 𝑋 ∗∗ CUDA kernel CUBLAS sasum 𝑹 𝑀 𝟏 ∗ CUBLAS sasum 𝒙 Weighted center Same as for 𝒚 Weighted sum
- 22. PipelineofSoftassing.GPU 𝑋 𝑌 𝑋 𝑌 𝑀 𝑋 𝑌𝑆 = 𝑀 Compute 𝑀 with CUDA kernel Shinkhorn.GPU Centering.GPU 𝑆 Weighted Horn’s method 𝐾 𝑅 and 𝒕 Solve Eigenvalue problem 𝒙 , 𝒚
- 23. Algorithmsforregistration Horn’s method • Corresponding point sets are given. • Estimate R and t. ICP (Iterative closest point) • Unknown correspondence. • Fast, standard. • Easily fail due to local minimum. • A lot of variants follow. Softassign • Unknown correspondence. • Robust. • Very slow because of iterations. EM-ICP • Unknown correspondence. • Robust. • Very slow because of iterations. Registration algorithm
- 24. EM-ICP:softcorrespondence. 𝑌 𝑋 𝒚𝑖 𝒙𝑗 𝑑𝑖𝑗 𝑑𝑖𝑗 = ||𝒙𝑗 − 𝑅𝒚𝑖 + 𝒕 || 𝐴 Weighted Horn’s method with 𝑋′ and 𝑌 Estimate 𝑅 and 𝒕 Repeat 𝑋′ 𝒙′𝑖 Pseudo correspondence 𝑋′ Each row is normalized once.
- 25. RownormalizationonGPU 𝐴 𝟏 1 1 1 ⋮ 𝑪 Using sgemv of CUBLAS Not normalized yet.
- 27. Computingweights 𝐴 𝟏 1 1 1 ⋮ 𝝀 Using sgemv of CUBLAS Now normalized.
- 28. Pseudocorrespondence 𝑋 𝐴 𝑋′ CUBLAS sgemv Centering: same with Softassing.GPU Now normalized.
- 29. WeightedHorn’smethod 𝑋′ 𝑌𝑆 = 3 Weighted version 0 0𝜆1 𝜆2 ⋱ 𝝀𝑋′ ∗ CUDA kernel 𝑋 𝑋’ 𝑌𝑆 = CUBLAS sgemm 3 Weighted version (2 steps) (not efficient)
- 30. PipelineofEM-ICP.GPU 𝑋 𝑌 𝑋 𝑌 𝐴 Compute 𝐴with CUDA kernel Row normalization on GPU Centering.GPU 𝑆 2 step weighted Horn’s method 𝐾 𝑅 and 𝒕 Solve Eigenvalue problem 𝒙 , 𝒚 𝝀𝑋′ ∗ 𝑋 𝑋′ 𝑌 𝑆 =
- 31. Computingtimeoverdifferentnumberofpoints Successfully aligned 5000 points less than 7 seconds. Slightly fast, but failed. GPU: GeForce8800GT CPU: Intel Core2 Quad + OpenMP (4 cores)
- 32. Summary Implemented 3D registration algorithms on a GPU are: Softassign, EM-ICP, Weighted Horn’s method. EM-ICP.GPU is able to align 5000 points within 7 seconds, 60 times faster than EM-ICP.CPU, more robust than ICP.CPU. Code, binary, and movies are available at: http://home.hiroshima-u.ac.jp/tamaki/study/cuda_softassign_emicp/
- 33. Limitations Number of points Should be less than 8000 for GeForce8800GT with 512MB memory. More memory, more points. Stopping condition requires to store whole matrix 𝑀 or 𝐴, and compare with previous ones: inefficient. Hence, currently, number of iterations is fixed.
- 34. Algorithmsforregistration Horn’s method • Corresponding point sets are given. • Estimate R and t. ICP (Iterative closest point) • Unknown correspondence. • Fast, standard. • Easily fail due to local minimum. • A lot of variants follow. Softassign • Unknown correspondence. • Robust. • Very slow because of iterations. EM-ICP • Unknown correspondence. • Robust. • Very slow because of iterations. Registration algorithm