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1、A Brief Introduction Of PointCloud Registration MethodPoint Cloud Registration MethodPoint Cloud Registration With Target ControlQuaternion MethodIterative Closest Point(ICP) AlgorithmOutlineTwo point cloud sets of A andQ are conformations of the same points in different coordinate systemsRigid tran
2、sformation:Objective function:F(T) = S + r-eJ2=min仪器科学与工程学疣(a ) Leica standard rotation plane target ( b ) Mensi plane target仪器科学与工程学枕(a ) Leica standard rotation plane target ( b ) Mensi plane target仪器科学与工程学枕(a ) Leica standard rotation plane target ( b ) Mensi plane target(d ) Mensi plane target(c
3、) FARO standard plane targetFigure 1 Different kinds of plane target仪器科学与工程学枕06 spatial similarity transformation parameters 3 angle elements: <i>x 3、3 translation elements: AXs ay. azAdjustment model:仪器科学与工程学枕0仪器科学与工程学枕0OrXYzba2a3» X Ay62b3y+AYc2c3zMBAZa仪器科学与工程学枕0仪器科学与工程学枕0 Y -AY, AZ,x,y
4、,z)z = £ (0® 匕 AX, AK, AZ, x, y. z)Rotation matrix:cos a sina01 cos PR = sin acos a1J Lsin 00 sin B卩 010 IO cos y0cos p 1L0 sinysinycosy(Ad2 c& coscosAT-sinsiivysinrb2 b3 =coswsirMrcl c2 c3 sincosAr+cossiiKysinr-cossi iw_sin0si ntocos*COStDCOSC-sinsinr+cosincxosr-sincoso_sitk»COSC
5、OStt)仪器科学与工程学枕0X=(cos0cos*r_sin0$inQsin K)x_(cos0sin K+sin 0sin qcos 切一 (sin 0cosq)z+AYY=(cos &> sin r)x+(cos <o cos K)y - (sin qjI)z+Z=(sin 0COSK+8S 0sin兀+(-sin 0sin k+cos/sin ecos K)y+(8S0COS妙z+4Z仪器科学与工程学枕The expansion of Taylor's formula:f =扎 + - dAy+d/SZ° 诚 d& dK dX dY 5AZ
6、Error Equation:卫L/AX+卫L/AY+西 dAZ+竺卯+e+色-45AX 3AY 5AZ 30 do) 泳vv 二空KAX + dY 十空NAZ + ddodK-l ySAXDAYSAZ诃deodKyv = cfAX + </AY+-c/it*/25AX0AY3AZ网加弧Derivative value:= L-=2=-Z,)=-尸命0,型=翟=Xsin0 - Zoos© 虫=JT, =Foos050 dododo)二 丁 cos 加05心11血 型 uXcositaQ+Zsin 如K0 =JTsind>-Fsincos dKdfcdKSuppose th
7、at:lx=X-X、a a2 a3r Xl=r-AT_Ar，Yl=b b2 b3yl=Z_AZ-ZZ'z1 cl c2 c3 z 仪器科学与工程学疣XError equation matrix:'dX'如100-z0-厂XdZ=0100Z'X'VT001X、Y90 ' da)Jdie !yGravity-centra lize:dX 如 dAZdo) d/c仪器科学与工程学枕10仪器科学与工程学枕N pairs of corresponding points:100- Z0一乙7 JV/I0100如/刃001£0JAZ “=dX 100
8、- Z0-Y.d(pJ0100da>I.001X0dK kJOrder:仪器科学与工程学枕 Vxl"10 00 1 0叫】0 0 1,B =1 0 00 1 00 0 10-厂dtX'JdhYJY0dhZJ,X =d人,L = 0d<pJd(b0dK» kJ仪器科学与工程学枕1. Point Cloud Registration With Target ControlError equation:V=BX-LAccording to Adjustment of indirect observations, vTpv = min(P is unit wei
9、ght matrix)Solution of the normal eauation:x=(btb)'bYor: X = d& d匕 dLZf g dwf dkT1. Point Cloud Registration With Target ControlNew approximate value:Iteration:AX严必+d% 人丫严乂+吗 AZ =AZo + dAZ 0i =0o+d0】K、= K°+dK0 = (0o+d0)+ dc2)+ ); s =(吗 + "©) + d®)+ ); k = (r0 + dg)+ 加”);A
10、X = (* + dAX J + JAY2 +);AY = (A/o + dAYJ + 时2 +)；AZ = (AZ。+ dAZ J + dZ2 +)；Finally we get the values of parameter:<K 3、K 必 AY. AZ2. Iterative Closest Point(ICP) Algorithm13Basic thought:Firstlyf from a point setf a bar curvef or a surface to find the closest point that corresponds to one pointf
11、and then uses this result to find two corresponding point sets. Finally, we find out the corresponding point set and the corresponding transformation matrix.Basic steps:Stepl:Search for the nearest points Step2:Solve transformation relations Step3:Application transformation Step4:Repeated iteration仪
12、器科学与工程学疣3. Quaternion Method14 What is Quaternion ? Unit quaternions%=% 9i 的Oo2+i2+2+2=L Rotation matrix Rq?七 qq；-q；R= 2©02+%弘）2（务弘-%）2©%”2 ”2 “2% 一 4 +?2 一弘2（$ 込+9（）91）2©%+%）2（%给一編创）”2“2“2丄“2% -q、-込+狗Definite translation vector:?r = q$ 订Complete registration station:Assume that: p=Pi
13、PleR3,i=it2tyn =eJ?3,j = 1,2,-,wMinimum objective function:1 «F=一£|忑-川五）/外The centroid of the set of basic points :Orthogonal covariance matrix of point set: 工j丄£s-以兀-再门=丄£胪门-: n 扫 In f»lAntisymmetric matrix7 s cyclic components: 九=（匚£九QA =血禺 Ai2fConstruct a symmetric matrix:吃丿 &AE+：-吨 pMIThe unit eigenvector corresponding to the maximum eigenvalue of matrix 0（乙）:办=%4蚯ft>=>Translation matrix:4丫 =比-盹,片Conclusions仪器科学与工程学疣18