1、35 7 1Vol.35, No.72009 M7ACTA AUTOMATICA SINICA July, 2009c_sM “ 5Lie ZE H1 62 72;3 D4K1YVsc_sM “ 5,| ,K(Iterative closest point,ICP)ZEBV,|c_sM “ 5 Lie 5.YVLie LZE,| / Lie 5B“ =Q?5,K B .ZE .dICPZE y +,7 O ?) ivM “ 5. ,yN1.dZEz Z.K, , B* SM4Z.1oM “ ,_sM, Lie ,=Q?ms |O29; TP39Lie Group Method for Data
2、 Set Registration Problem withAnisotropic Scale DeformationYING Shi-Hui1 PENG Ji-Gen2 ZHENG Kai-Jie2;3 QIAO Hong4Abstract By analyzing the data set registration problem with anisotropic scale deformation, we introduced the con-straints to the model. Combining with the procedure of traditional iterat
3、ive closest point (ICP) method, the registrationproblem was described as a constrained optimization problem. Using parameterized method by Lie group and quadricapproximation to the objective function, the registration problem was translated into a series of quadratic programmingproblems. Then, a nov
4、el scale-registration algorithm was proposed. The numerical simulations showed that such methodnot only was rapid and accurate as the traditional ICP method, but also could deal with the registration problem withlarge scale deformation. By introducing the constraints to the scale parameters, the alg
5、orithm was more robust. A wayfor choosing the initial transformations was proposed to assume the global registration.Key words Data set registration, anisotropic scale deformation, Lie group, quadratic programming “ 5 V9 j $m) 5 B d1 5,7K(Iterative closest point, ICP)ZE1V20 W90 MBeslMckay1aChenMedio
6、ni2Zhang31 4X 5BKidZE.v T1“/ Z : 1)4 E . Fitzgibbon4Levenberg-MarquardtZE E, Jost5YsO q/ # ICPZE. 2)4 E Z. Lee64B 9ICPZE V L,7Sharp74 M+ZE, ICPZEJ ll 2008-04-24 l 2008-12-24Received April 24, 2008; in revised form December 24, 2008SE$?Z9(9739) (2007CB311002)Supported by National Basic Research Progr
7、am of China (973Program) (2007CB311002)1. Zv “ Z200444 2.Yv “d S 710049 3.y =Sv 9 S350007 4.S S1 1001901. Department of Mathematics, School of Science, ShanghaiUniversity, Shanghai 200444 2. Faculty of Science, Xi0an Jiao-tong University, Xi0an 710049 3. School of Mathematics andComputer Science, Fu
8、jian Normal University, Fuzhou 3500074. Institute of Automation, Chinese Academy of Sciences, Bei-jing 100190DOI: 10.3724/SP.J.1004.2009.00867q, Silva88L. Ei 4 ,7Granger9v d 9# E| E E Z. 3) E aS. SICP Ei I nd .7 5M i, ,B8 /, ,vl? 3 M.D10 I n _ % M “W 5. E Zz,V | 5 Z.D11,YV ,v E Z, 7 , D10 D11ZE ?$w:
9、SkRkxl; i = 0SkEsiRkxl; i = 1; ;NsSkRkEriNsxl; i = Ns +1; ;NC0 , vlYK p,yN1 p./ , Hqe.S 2 D+ ,yN, T(9)N(SK)1SL exp(nXi=1siEsi) (SK)1SU (13):Sk = diagfsk1; ;skng, SL = diagfsL1;, sLngSU = diagfsU1 ; ;sUng,5Blog(sLi =ski) asi log(sUi =ski); i = 1; ;Ns(14)yN5(6) /=Q?5minaf(a) = aTHa +2LTa (15)s:t: log(
10、sLi =ski) asi log(sUi =ski); i = 1; ;Ns,“ l T(12), a = (as1; ;asNs;ar1, ,arNr)T 2 RN.ICPZEB,c_sM “ E /:E1 (c_sM “ E).1. “X = fxlgNxl=1Y =fyjgNyj=1.2.S0, R0, T0 0.3.k3.1.l = 1; ;Nx,YVminyj2Y kSkRkxl +Tk yjk2 pzkl ;3.2.9 “k = 1Nx PNxl=1kSkRkxl + Tkzkl k2;3.3YV p=Q?(15) pa;3.4. Sk+1 = Sk exp(PNsi=1 asi
11、Esi), Rk+1 =Rk exp(PNri=1 ariEri ), Tk+1 = zkc Sk+1Rk+1xc;4.9 = 1“k+1=“kTT5.4.1. ,5KS = Sk+1, R = Rk+1T = Tk+1;4.2.5k ( k +1,3.1 E1 l , / . 1. E1/ 9 lB l. . !X = fxigNxi=1Y = fyjgNyj=1Rn “. !Sk, Rk, TkZk =fzigNxi=1sYkMaMa MMK“,5 -/ T“k = 1NxNxXi=1kSkRkxi +Tk zkik2 (16)/ Tek = 1NxNxXi=1kSk+1Rk+1xi +T
12、k+1 zkik2 (17)lek “k.“,i = 1; ;NxkSk+1Rk+1xi+Tk+1zk+1i kkSk+1Rk+1xi+Tk+1zkik“k+1 ek,y70 ek+1 “k+1 ek “k , f“kgk1 h O/.yN, E l f l. V A: K K.yN1 P E1 | ,1| S | l. l ,yN, | B1o5./ c_sM 5B* S (+Y S)4Z. “X = fxigNxi=1Y = fyjgNyj=1. xZ sY:MX = PNxi=1(xi xc)(xi xc)TMY = PNyj=1(yj yc)(yj yc)T. , xcycV U “X
13、Y.:1 2 n1 2 nMXMY+, p1; ;pnq1; ;qnsY +_ . “ bWs+, V4 SM870135 s0i =rii (18) B V |sLi = 0:95s0i, sUi = 1:05s0i. 6, | SR0R0 = q1; ;qnp1; ;pn1 (19)7 S MT0 V | “Mxc yc.3 L Ls s: 1) 2 “ .Bs, sYBeslMckay0s ICPZE1aZhaZE10aICPZE11 E1BF “ L, “ ZET1 . 2) 3 u “W . 3 ICPZEE11 . Matlab6.5,PentiumIV 3.0GHz CPU, 5
14、12M RAM.3.1 2 “ 1 L MPEG7(PartB)20FT L “.sYBeslMckayICPZE1aZhaZE10aICPZE11# E1 , T V1m1 (n/:) U.VV1 V A, - ZE rTA J, Besl 1; ;75,#sYv = 10, 100378F “, |t“T “sYT k “Bun045, dragonStan-dRight 48happyStandRight 48 . P V1,/ RMS:RMS = 11mmXi=1kSRxi +T zik2!12(20), RaTSsYV UKa M,7zibun000, dragonStandRigh
15、t 0happyStandRight 0SRxi +TK.| = 0:001,Bun000Bun045 “ SM9.V2 (n/:) = 0:5; 1:0; 2:0; 10:0100 H SM,V T V3m2 ( = 0:5 H E1 T) (n/:) U.VV3n: E1 E9 r q H,4 .B 1 ,V74 .+Y, F r,yN, E Z dz.B,9 VVm2 A.Bdrag-onStandRight 0dragonStandRight 48, happy-StandRight 0happyStandRight 48 “. T V4V5#m3m4 (n873:) U.4 ZYV
16、_sy0,c_sM “ 5 ,4 ZE. 8, n5YV| ,ICPB|5Lie 5; Lie ZE “Sf =Q/,| 5872135 (a) -(, j(0.0,90.0)(a) Conflgurations beforeregistering, viewpoint(0.0,90.0)(b) -(, j(23.5,6.0)(b) Conflgurations beforeregistering, viewpoint(23.5,6.0)(c) (, j(0.0,90.0)(c) Conflgurations afterregistering, viewpoint(0.0,90.0)(d) (
17、, j(23.5,6.0)(d) Conflgurations afterregistering, viewpoint(23.5,6.0)m2 = 0:5 HBunny “a k “( TFig.2 Conflgurations of model data set, test data set, and registration results of Bunny data setV2 Bunny / SMTable 2 The initial conflgurations of Bunny data set with difierent scales MsS0.5 0.0534 0.4425
18、0.0602 25.7695 0.0437 0.0477 0.0392 0.5046 0.4753 0.4883 0.54611.0 0.0534 0.4425 0.0602 25.7695 0.0057 0.0006 0.0214 1.0092 0.9506 0.9766 1.09232.0 0.0534 0.4425 -0.0602 25.7695 0.0797 0.0972 0.0142 2.0185 1.9013 1.9532 2.184510.0 0.0534 0.4425 -0.0602 25.7695 0.2718 0.8699 0.2993 10.092 9.5064 9.76
19、68 10.923100.0 0.0534 0.4425 -0.0602 25.7695 2.4337 9.5625 3.5061 100.92 95.064 97.668 109.23V3 Bunny / “ TTable 3 Registration results of Bunny data set with difierent scales E E1RMS HW(s)RMS HW(s)0.5 0.48999 0.00194 61 82.564 0.48956 0.48995 0.49046 0.00186 65 88.4531.0 0.97998 0.00194 30 40.053 0
20、.97912 0.97990 0.98092 0.00186 31 45.2492.0 1.95996 0.00194 39 53.865 1.95824 1.95980 1.96184 0.00186 35 45.89610.0 9.79982 0.00194 43 61.187 9.79122 9.79902 9.80922 0.00186 42 59.974100.0 97.9982 0.00194 43 63.756 97.9122 97.9902 98.0922 0.00186 45 66.165V4 Dragon / “ TTable 4 Registration results
21、of Dragon data set with difierent scales E E1RMS HW(s)RMS HW(s)0.5 0.49012 0.00581 49 42.405 0.46882 0.49995 0.50136 0.00532 55 50.5431.0 0.98024 0.00581 52 48.267 0.93764 0.99990 1.00272 0.00532 61 55.8412.0 1.96048 0.00581 41 38.418 1.87528 1.99980 2.00544 0.00532 48 45.96210.0 9.80242 0.00581 55
22、52.561 9.37641 9.99903 10.0272 0.00532 57 55.356100.0 98.0242 0.00581 55 55.026 93.7641 99.9903 100.272 0.00532 63 58.1447 H:c_sM “ 5Lie ZE873(a) -(, j(0.0,90.0)(a) Conflgurations beforeregistering, viewpoint(0.0,90.0)(b) -(, j(10.0,30.0)(b) Conflgurations beforeregistering, viewpoint(10.0,30.0)(c)
23、(, j(0.0,90.0)(c) Conflgurations afterregistering, viewpoint(0.0,90.0)(d) (, j(10.0,30.0)(d) Conflgurations afterregistering, viewpoint(10.0,30.0)m3 Dragon “a k “( TFig.3 Conflgurations of model data set, test data set, and registration results of Dragon data set(a) -(, j(0.0,90.0)(a) Conflgurations
24、 beforeregistering, viewpoint(0.0,90.0)(b) -(, j(45.0,10.0)(b) Conflgurations beforeregistering, viewpoint(45.0,10.0)(c) (, j(0.0,90.0)(c) Conflgurations afterregistering, viewpoint(0.0,90.0)(d) (, j(45.0,10.0)(d) Conflgurations afterregistering, viewpoint(45.0,10.0)m4 HappyBuddha “a k “( TFig.4 Con
25、flgurations of model data set, test data set, and registration results of HappyBuddha data setV5 HappyBuddha / “ TTable 5 Registration results of HappyBuddha data set with difierent scales E E1RMS HW(s)RMS HW(s)0.5 0.49009 0.00332 55 89.847 0.48972 0.49011 0.49042 0.00314 62 92.7581.0 0.98018 0.0033
26、2 59 96.217 0.97944 0.98022 0.98084 0.00314 69 100.4192.0 1.96036 0.00332 52 85.624 1.95888 1.96044 1.96168 0.00314 59 88.17410.0 9.80183 0.00332 62 98.825 9.79442 9.80224 9.80841 0.00314 66 93.754100.0 98.0183 0.00332 51 81.467 97.9442 98.0224 98.0841 0.00314 58 86.675B“ =Q?5.V7 E. S L ,ZE ICPZE y
27、+,7 O ?)ivM “ 5.N,1.dICPZE 1M ,yN z.d 58s _ 5,7_ 5B V 4 5 Vr,yN,9 B pd 5B5.T, 1 dMZE, d 5 p.References1 Besl P J, McKay N D. A method for registration of 3-Dshapes. IEEE Transactions on Pattern Analysis and Ma-chine Intelligence, 1992, 14(2): 239256874135 2 Chen Y, Medioni G. Object modeling by regi
28、stration ofmultiple range image. In: Proceedings of the IEEE Confer-enceon Robotics and Automation. Sacramento, USA: IEEE,1991. 272427293 Zhang Z Y. Iterative point matching for registration of free-from curves and surfaces. International Journal of ComputerVision, 1994, 13(2): 1191524 Fitzgibbon A
29、W. Robust registration of 2D and 3Dpoint sets. Image and Vision Computing, 2003, 21(13-14):114511535 Jost T, Hugli H. A multi-resolution ICP with heuristic clos-est point search for fast and robust 3D registration of rangeimages. In: Proceedings of the 4th International Conferenceon 3D Digital Imagi
30、ng and Modeling. Washington D. C.,USA: IEEE, 2003. 4274336 Lee B U, Kim C M, Park R H. An orientation reliability ma-trix for the iterative closest point algorithm. IEEE Trans-actions on Pattern Analysis and Machine Intelligence, 2000,22(10): 120512087 Sharp G C, Lee S W, Wehe D K. ICP registration
31、usinginvariant features. IEEE Transactions on Pattern Analysisand Machine Intelligence, 2002, 24(1): 901028 Silva L, Bellon O R P, Boyer K L. Precision range imageregistration using a robust surface interpenetration mea-sure and enhanced genetic algorithms. IEEE Transactionson Pattern Analysis and M
32、achine Intelligence, 2005, 27(5):7627769 Granger S, Pennec X. Multi-scale EM-ICP: a fast and ro-bust approach for surface registration. In: Proceedings ofthe 7th European Conference on Computer Vision. Copen-hagen, Denmark: Springer, 2002. 697310 Zha H B, Ikuta M, Hasegawa T. Registration of range i
33、m-ages with difierent scanning resolutions. In: Proceedings ofthe IEEE International Conference on Systems, Man, andCybernetics. Nashville, USA: IEEE, 2000. 1495150011 Ying S H, Peng J G, Du S Y, Qiao H. A scale stretch methodbased on ICP for 3D data registration. IEEE Transactionson Automation Scie
34、nce and Engineering, to be published12 Barber C B, Dobkin D P, Huhdanpaa H. The quickhull algo-rithm for convex hulls. ACM Transactions on MathematicalSoftware, 1996, 22(4): 46948313 Helgason S. Difierential Geometry, Lie Groups, and Sym-metric Spaces. New York: Academic Press, 2001 H Zv “ =.2008 MY
35、v pV.1Z_ T MY9 j $. E-mail: (YING Shi-Hui Lecturer at theSchool of Science of Shanghai Univer-sity. He received his Ph.D. degree inscience from Xi0an Jiaotong Universityin 2008. His research interest covers pattern recognitionand computer vision.)6Yv “d S q.1Z_dLWfs, “ .YT.E-mail: (PENG Ji-Gen Professor at the In-stitute of Information and S
