分散メモリシステムにおける大規模距離画像の並列同時 ... - 池内研究室

Vol. 46 No. 9 Sep. 2005 †1 †3 †2 †4 †116 PC Parallel Simultaneous Alignment of a Large Number of RangeImages on Distributed Memory SystemTakeshi Oishi, †1 Ryusuke Sagawa, †3 Atsushi Nakazawa, †2Ryo Kurazume †4 and Katsushi Ikeuchi †1This paper describes a method for parallel alignment of multiple range images. Since itis difficult to align a large number of range images simultaneously, we developed a parallelmethod to accelerate and reduce the memory requirement of the process. Although a generalsimultaneous alignment algorithm searches correspondences for all pairs of all range images,by rejecting redundant dependencies, our method makes it possible to accelerate computationtime and reduce the amount of memory used on each node. The correspondence search is performedindependently for each pair of range images. Accordingly, the computations betweenthe pairs are preformed in parallel on multiple processors. All relations between range imagesare described as a pair node hyper-graph. Then, an optimal pair assignment is computedby partitioning the graph properly. The method was tested on a 16 processor PC cluster,where it demonstrated the high extendibility and the performance improvement in time andmemory.1. 3 †1 Institute of Industrial Science, The University of Tokyo†2 Cybermedia Center, Osaka University†3 Institute of Scientific and Industrial Research,Osaka University†4 Graduate School of Information Science and ElectoricalEngineering, Kyushu University 1)∼4) Timeof-flight 1mm 50m 100m 1cm 3 3 1

2 Sep. 2005 Besl ICP (Iterative ClosestPoint) 5) Chen 6) ICP Chen 7) ICP LMedS(Least Median Squares Estimation) 8) 2 Neugebauer 9) Benjemaa 10) z-bufferBergevin 11) Nishino M 12) ICP 2 n O(n 2 ) (kd-tree) 13) kd-tree 14) PC 15) m m × (m − 1) O(m 2 n 2 ) 15) ICP PC 2 3 4 5 6 2. Neugebauer 9) ICP (1) (a) (b) (2) (3) 122.1 1

Vol. 46 No. 9 3CorrespondenceRay directionnxyScene imageModel image() T⃗t = t x t y t z (4)2 9) ∑¯ε =min ‖ A−→ ijk⃗ δ − sijk ‖ 2 (5)δ i≠j,ks ijk = ⃗n ik · (⃗x ik − ⃗y ijk ) (6)⎛⎞Fig. 1Range finder 1 Search correspondences l th 2.2 2 ⃗x ⃗y ⃗n ⃗x ( 1) ε ε = R M⃗n ·{(R S⃗y + ⃗t S) − (R M⃗x + ⃗t M )} (1) R M ⃗t M R S⃗t S (i, j) k ¯ε ∑¯ε =minR,⃗ti≠j,k( Ri⃗n ik ·{(R j⃗y ijk + ⃗t j) − (R i⃗x ik + ⃗t ) 2i)} (2)2.3 2 R⎛⎞1 −c 3 c 2⎜⎟R = ⎝ c 3 1 −c 1 ⎠ (3)−c 2 c 1 1A ijk =C ijk =⎝}{{} 0...06i×1⎛⎝}{{} 0...0(6j×1C ijk}{{}6×1−C ijk} {{ }6×10...0}{{}6(l−i−1)×10...0}{{}6(l−j−1)×1⎠ +⎞⎠ (7)⃗n ik × ⃗y ijk−⃗n ik) T(8)(⃗ δ = ⃗mT0 ···⃗m n−1) T T(9)() T⃗m i = c 1i c 2i c 3i t xi t yi t zi (10) n (5) ⃗ δ R ⃗t ( )∑∑A T ijkA ijk⃗ δ = A T ijks ijk (11)i≠j,ki≠j,k3. Algorithm Pseudo Code of Parallel Alignment// Server processAssignPairs();while(error > threshold){for(i = 0; i < nImage; ++i)for(j = 0; j < nImage; ++j)if(List[i][j]){// Client processSearchCorrespondingPoints(i,j);

4 Sep. 2005Range imagePairProc2Image jImage iProc0Proc1 2 Fig. 2 Rejection of redundant pairsComputeError(i,j);}// Server processComputeTransformationMatrices(all);} 2 2 3.1 3.1.1 (1) 2 (2) 2 θ 1 2Ray directionOverlapping areaS = v′/ v′i,jci,j 3 Fig. 3 Local overlapping areaθ =90 ◦ 3.1.2 v v c 3 i v i ′ j v ci,j ′ i j S i,jS i,j = v ci,j/v ′ i ′ (12) S i,j S j,i 0.030.05 3.1.3 (x min,x max,y min,y max,z min,z max) z = z min 4 i

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6 Sep. 2005Second partition(b)(a)Seed(c)First partition 6Fig. 6Random seeded breath-first search3.2.3 2 KL FMFiduccia and Mattheyses 23) KLFM KLFM KLFM 24) 22) KLFM 2 k 1 1 1 1 G i G j N (i,j),k N (i,j),k G i G i Di,j,k int G j Di,j,kextg i,j,k = Di,j,k int − Di,j,k ext(15) G j g j,i,k D intj,i,k D extj,i,k 7 ::Fig. 7 Range images used for evaluations(left: original model, right: range images)g j,i,k = Dj,i,k int − Dj,i,k ext(16) 2 G i G j 2 1%4. 2 ArhlonMP2400+ 4GB 8 PC PC PC 100Base-TX Cyrax2400 25) 13m 3 ( 7) 50 ( 7) 122029

Vol. 46 No. 9 7112Average of error [m]0.80.60.40.2Threshold distance : 0.1mThreshold distance : 0.5mThreshold distance : 1.0mThreshold distance : 2.0mThreshold distance : 5.0mCalculation time T1 /Tn108642Graph partitioning methodSequential method00 5 10 15 20Number of iterations [times] 8 Fig. 8 Convergence with general method00 2 4 6 8 10 12 14 16Number of processors n 10 Fig. 10 Computational Efficiency17Average of error [m]0.80.60.40.2Threshold distance : 0.1mThreshold distance : 0.5mThreshold distance : 1.0mThreshold distance : 2.0mThreshold distance : 5.0mRequired memory M1 /Mn654321Graph partitioning methodSequential method00 5 10 15 20Number of iterations [times] 9 Fig. 9 Convergence with our method00 2 4 6 8 10 12 14 16Number of processors n 11 Fig. 11 Required memory 12197 83,288 158,376 4.1 50 2450 160 Cyrax 1cm xyz 10cm x y z 0.05 l th 8 94.2 1 10 n T n 1 T 1 7 1 20560ms16

8 Sep. 2005 1 Table 1 Experimental results (Nara)nProc Time[s] Max Mem.[MB] Min Mem.[MB]2 76.0 1287 127516 13.2 292 254 2 Table 2 Experimental results (Bayon Temple)nProc Time[s] Max Mem.[MB] Min Mem.[MB]4 103.9 1608 145616 40.2 559 472 1784ms 11.5 4.3 116 11 1 2 1.2 1.7 7 1 269Mbyte 16 48Mbyte 1 17%5. 1 2 1 Cyrax2400 25) 114 327470 606072 2 Cyrax2500 210 433785 798890 1 2 2 4 16 1 2 1 8 5.75 22.6% 2 4 2.58 34.8% 2 14 16 35% 12 20 1 269 2 812 6. PC CRESTJSA 1) Ikeuchi, K. and Sato, Y.: Modeling from Reality,Kluwer Academic Press (2001).2) Ikeuchi, K.: Modeling from Reality, Proc.Third International Conference on 3D DigitalImaging and Modeling (3DIM) (2001).3) Levoy, M.: The Digital Michelangelo Project,Proc. SIGGRAPH 2000 , pp. 131–144 (2000).4) Sagawa, R., Nishino, K. and Ikeuchi, K.:

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