A Scale Stretch Method Based on ICP for 3D Data Registration

Transcription

1 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 6, NO. 3, JULY solutions since they correspond to an objective function with an even saller value, i.e., z(r1; r2) = 010. Nevertheless, this is evidently not the last case. We can, furtherore, have the better but not best or optiu solutions as follows: r1 = ( ) T and r2 = 3; r1 = ( ) T and r2 = 4; r1 = ( ) T and r2 = 5, and so on. V. THE ALGORITHM CORRECTION The proble exposed in the above exaple is resolved, and a set of correct IP forulation to ipleent the transforation is obtained, when (11) (13) are odified as follows: subject to ax z(r1;r2) =r T 1 1 M0 + r2 1 (l T 1 M0 0 g 0 1) (17) r T 1 1 2u + r2 1 l T 1 2u 0 (18) r T 1 1 M0 + r2(l T 1 M0 0 g 0 1) 01 (19) r1; r2 2 + : (20) The reader can verify that the odified forulation can solve the probles with an optial result r1 =( ) T and r2 =1, which lead to an acceptable constraint A Scale Stretch Method Based on ICP for 3D Data Registration Shihui Ying, Jigen Peng, Shaoyi Du, and Hong Qiao Abstract In this paper, we are concerned with the registration of two 3D data sets with large-scale stretches and noises. First, by incorporating a scale factor into the standard iterative closest point (ICP) algorith, we forulate the registration into a constraint optiization proble over a 7D nonlinear space. Then, we apply the singular value decoposition (SVD) approach to iteratively solving such optiization proble. Finally, we establish a new ICP algorith, naed Scale-ICP algorith, for registration of the data sets with isotropic stretches. In order to achieve global convergence for the proposed algorith, we propose a way to select the initial registrations. To deonstrate the perforance and efficiency of the proposed algorith, we give several coparative experients between Scale-ICP algorith and the standard ICP algorith. Note to Practitioners In this paper, we proposed the Scale-ICP algorith, which is used to deal with the registration between two data sets with scale stretches. In practice, there are a large nuber of such probles. For exaple, the registration between the range data sets that have different scanning resolutions deterined by the distances fro the sensor to the object surfaces. The data sets can be not only iage data but also the other easured data, and hence it also can be extended to achining industry besides the coputer vision field. Index Ters Initial registration, iterative closest point (ICP), large-scale stretch, singular value decoposition (SVD), 3D registration. ( ) 1 M (4): VI. CONCLUSION The odified IP forulation is proposed, which is proved to be valid under any case to deterine the feasibility of a transforation and to derive the corresponding solutions. A axially perissive and coputationally efficient policy will be the research focus of our future work. REFERENCES [1] J. Park and S. A. Reveliotis, Liveness-enforcing supervision for resource allocation systes with uncontrollable behavior and forbidden states, IEEE Transa. Robot. Auto., vol. 18, no. 2, pp , Apr I. INTRODUCTION T HE DATA SETS registration is a coon proble in coputer vision and the iterative closest point (ICP) ethod has been shown a best iterative technique to deal with such a registration proble. Especially in recent years, with the iage atching procedure regarded as a registration of two point sets in the position and intensity space [2], [20], [22], the extensive application prospect of ICP ethod has been eerging increasingly in coputer vision. When ICP is applied to the registration between two data sets X and Y in IR n, its goal is to find a transforation F that best aligns X with Y, starting fro an initial guess transforation whose paraeters are known. This is done iteratively as follows: pair each point in X to the closest point in Y, and then copute the next transforation by iniizing a certain distance between the paired points. It has been extensively accepted that the present standard ICP ethod was proposed independently by Besl and McKay [3], Chen and Medioni [4], and Zhang [23], although the rudient of ICP ay /$ IEEE Manuscript received Septeber 01, 2007; revised October 10, 2007 and Deceber 03, First published May 27, 2009; current version published July 01, This paper was recoended for publication by Associate Editor J. Fuh and Editor M. Wang upon evaluation of the reviewers coents. This work was supported in part by NCET and in part by the National Basic Research Progra of China under Grant 2007CB S. Ying and J. Peng are with the Institute of Inforation and Syste Science, Faculty of Science, Xi an Jiaotong University, Xi an , China (e-ail: Yingshihui@gail.co; jgpeng@ail.xjtu.edu.cn). S. Du is with the Institute of Artificial Intelligence and Robotics, Xi an Jiaotong University, Xi an , China (e-ail: dushaoyi@gail.co). H. Qiao is with the Key Laboratory of Coplex Systes and Intelligent Science, Institute of Autoation, Chinese Acadey of Sciences, Beijing , China (e-ail: hong.qiao@ail.ia.ac.cn). Color versions of one or ore of the figures in this paper are available online at Digital Object Identifier /TASE Authorized licensed use liited to: SHANGHAI UNIVERSITY. Downloaded on June 27,2010 at 02:47:11 UTC fro IEEE Xplore. Restrictions apply.

2 560 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 6, NO. 3, JULY 2009 be traced back to the early 1980s when B. D. Lucas proposed an iterative technique for iage registration in [14]. There by now has been a large nuber of research works devoted to iproving ICP ethod, towards two ain directions. The first is to accelerate the algorith and iprove the accuracy as far as possible, assuing that the data sets for registration are rigid. In this direction, there have been presented any iproved versions of ICP, such as the hybrid ethod by Rusinkiewicz and Levoy [19], the SIC-range ethod by Hügli and his coauthors [10], the anifold ethod by Krishan and his coauthors [12], the tried ICP algorith by Chetverikov et al. [5], the LM algorith by Fitzgibbon [9], and the picky ICP algorith by Zinßer et al. [24]. The second is to reduce the requireent for the data sets. For exaple, Feldar and his coauthors in [7] and [8] studied the registration proble of free for surfaces by incorporating scale ingredients. For overviews on data sets registration using ICP ethod, we also refer to Pennec and Thirion [15], Sharp et al. [20], Pottann et al. [17], [18], Ezra et al. [6], Liu [13], Zitová et al. [25], and Planitz et al. [16]. However, the standard ICP algorith does not take scale factor into account in the registration proble, while the scale factor always exists in registration. For exaple, the range data set usually has different scanning resolutions deterined by the distances fro the sensor to the object surfaces. This requires that the registration algoriths be able to estiate the scaling paraeter, as well as the general otion paraeters between iages. In this paper, we consider the registration proble of 3D data sets where large-scale stretches and noises exist. We odify the standard ICP algorith by introducing a scale factor s with lower and upper bounds, and then forulate the registration proble into a quadric constraint optiization proble over a 7D nonlinear space. To solve such an optiization proble, we design a fast and accurate iterative procedure based on the singular value decoposition (SVD) technique. For the proposed algorith, we discuss how to select a good initial registration to achieve the global iniu. We also give several coparative experients to verify that the proposed algorith is really fast and accurate for the registration probles of 3D data sets with large-scale stretches. The reainder of this paper is organized as follows. In Section II, we propose a odel for registration of two 3D scale data sets. In Section III, a coonly explicit iterative rule based on the standard SVD and the properties of the quadric function is given with the corresponding algorith following. Moreover, in Section IV, a convergence theore is claied that the Scale-ICP algorith is a locally convergent ethod, and hence a good initial registration is suggested to achieve the global iniu. Several coparative and practical nuerical experients are ipleented in Section V and this paper is concluded and further discussed in Section VI. II. PROBLEM DESCRIPTION AND THE SCALE-ICP MODEL The registration of two 3D data sets X = fx i g and Y = fy j g n j=1 with different scales can be stated below. Proble 1 (Registration of Data Sets With Scale Stretch): Find a rotation R 2 SO(3), a translation T 2 IR 3, and a scale s in a given interval I of IR + such that the resulted set s1rx+t aligns best with Y. When ICP ethod is applied, the above proble can be addressed in two iterative steps. Step 1) For the current rotation R k, translation T k, and scale s k 2 I, search a subset Z k of Y with the sae size as X such that the following objective function is iniized: Step 2) Fixing the obtained Z k, search the next rotation R k+1, translation T k+1, and scale s k+1 2 I such that the following objective function is iniized: e k (R; T; s) := ks 1 Rx i + T 0 z k i k 2 : (2) The optiization proble (1) can be easily solved using the progra function dsearchn in Matlab, so we only focus our attention on the proble (2), which is obviously a constraint optiization proble over the 7D nonlinear space SO(3) 2 IR 3 2 I. In the next section, we will develop an approach to such an optiization proble. III. A FAST SCALE-ICP ALGORITHM BASED ON SVD In this section, we give an iterative rule for Proble 1. To this end, we eliinate the translation T at Step 2 of ICP for Proble 1 by calibrating the centers of X and Z k, and so we reduce the optiization proble (2) to an optiization proble with respect to R and s. On the base of this, we eploy the SVD technique to get the explicit expressions of the next rotation R k+1 and the scale factor s k+1. A. Copute the Next Translation T k+1 Denote by x c and z k c, the centers of the test data set X and the set Z k at kth step, naely x c = 1 x i; z k c = 1 z k i : (3) Then, noticing that (R k+1 ; T k+1 ; s k+1 ) is the least square solution of (2), we have T k+1 = z k c 0 s k+1 1 R k+1 x c: (4) B. Copute the Next Rotation R k+1 and Scale s k+1 To copute the next rotation R k+1 and scale s k+1, we let ~x i = x i 0 x c ; ~z k i = z k i 0 z k c : (5) Then, the proble (2) is equivalently siplified into (R k+1 ;s k+1 )=arg in e k (R; s) R2SO(3);s2I := ks 1 R~x i 0 ~z i k k 2 : (6) By the orthogonal property of rotation atrix R, wehave e k (R; s) = s 2 h~x i; ~x ii02shr~xi; ~z k i i + h~z k i ; ~z k i i : (7) That is, e k (R; s) is a quadric function of s 2 I with positive coefficient of quadric ter. Hence, if e k achieves its iniu at (R k+1 ;s k+1 ), then s k+1 necessarily satisfies the following equality: Solving (8), we ek (R; s) =0: (8) " k (Z) := ks k 1 R k x i + T k 0 z i k 2 : (1) s = hr~x i; ~z k i i h~x i; ~x i i (9) Authorized licensed use liited to: SHANGHAI UNIVERSITY. Downloaded on June 27,2010 at 02:47:11 UTC fro IEEE Xplore. Restrictions apply.

3 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 6, NO. 3, JULY which could be the candidate of the next scale s k+1. It should be noted that the s deterined by (9) ay be out of the range of the interval I. Hence, it is quite necessary to discuss the solution s of (9). For this, we assue I=[a; b];a > 0, without loss of generality. 1) If s a(or b), the quadric function e k (R; s) achieves in I its iniu at a (or b). In this case, we set s k+1 = a (or b): (10) Consequently, iniizing e k (R; s) is equivalent to axiizing the following function of R over SO(3): F k (R) = hr~x i ; ~z k i i: (11) 2) If a<s<b, then we set s as the next scale s k+1. Substituting this scale into (6), we get that e k (R; s) =0s k+1 hr~x i ; ~z k i i + h~z k i ; ~z k i i: (12) It follows that iniizing e k (R; s) is also equivalent to axiizing the function of R defined by (11). The above discussion indicates that in either case 1) or case 2), we can copute the next rotation R k+1 by axiizing the sae one-variable objective function F k (R) = hr~x i; ~z i k i other than the bi-variable objective function e k (R; s). Below, we apply the SVD ethod to axiizing such one-variable function F k (R) over the Lie group SO(3). Let H be the atrix defined by H := ~xi(~zk i ) T. According to the singular value decoposition principle [1], [22], there are orthogonal atrices U and V and a diagonal atrix 3 with non-negative eleents such that H = U 3V. By [1], we know that, as the axiu point of the function F k (R), the next rotation R k+1 is given by (13) R k+1 = Accordingly, setting VU T ; det(vu T )= V U T ; det(vu T )=01 : (13) s k+1 = a; s a b; s b : (14) ; a<s<b hr ~x ;~z i h~x ;~x i We achieve the following algorith for Proble 1. Algorith 1 (Scale-ICP Algorith): Step 1) Given two data sets X = fx ig and Y = fy jg n j=1. Step 2) Initialize rotation R 0, translation T 0, scale factor s 0 and precision >0. Step 3) Iteration (Step k). Step 3.1) Find the point set Z k = fzi k g in Y closest to X = fx ig. Step 3.2) Copute R k+1 by forula (13). Step 3.3) Copute s k+1 by forula (14). Step 3.4) Copute T k+1 by forula (4). Step 4) Terinating rule Copute =10 ek+1 (R k+1 ;T k+1 ;s k+1 ) e k (R k ;T k ;s k ) Step 4.1) If, output the iniu (R 3 ;T 3 ;s 3 ) = (R k+1 ;T k+1 ;s k+1 ). Step 4.2) Else k ( k +1and goto Step 3. IV. INITIAL REGISTRATION AND CONVERGENCE ANALYSIS In this section, we first prove a local convergence theore for the Scale-ICP algorith, which is siilar to that of [3]. Then, we discuss how to select a good initial registration (R 0 ;s 0 ;T 0 ) to achieve the global iniu. Theore 1: The Scale-ICP algorith converges onotonically to a local iniu. Proof: Let X = fx i g and Y = fy j g n j=1 be two data sets in IR 3. Let R k, T k, s k and Z k = fz ig be the rotation atrix, translation, scale, and the closest point set at kth step. Then, the current error is given by " k = and the next error is given by e k = ks k 1 R k x i + T k 0 z k i k 2 (15) ks k+1 1 R k+1 x i + T k+1 0 z k i k 2 : (16) It is easy to check that e k " k, otherwise, we let R k+1 = R k, T k+1 = T k and s k+1 = s k. Siilarly, " k+1 e k because of the inequality ks k+1 1 R k+1 x i 0 zi k+1 kks k+1 1 R k+1 x i 0 zi k k for all i =1;...;. Therefore, we have that 0 111e k+1 " k+1 e k " k 111 8k: (17) This indicates that the square error sequence f" k g k1 is onotone nonincreasing and lower bounded, and so it follows that the algorith converges to a iniu value. Like the standard ICP algoriths, the Scale-ICP algorith is local. In other words, its convergence to the global iniu strongly depends on the initial registration. A coon way to achieve the global iniu is to find the iniu of all the local inia [3], [10]. It is clear that in Algorith 1 the convergence localness is raised fro Step 3.1 finding the closest points, since the other optiization (2) is obviously global by the convex theore [11]. Next, we present a new way of selecting a good initial registration (R 0 ;s 0 ;T 0 ) to achieve the global iniu. It is well known that variability of data sets is able to be described by the covariance atrix whose any properties can be characterized by its eigenvalues and eigenvectors. Given two data sets X = fx i g and Y = fy j g n j=1. Their covariance atrices are evaluated, respectively, by M X := (xi 0 xc)(xi 0 xc)t and M Y := n j=1 (yj 0 y c )(y j 0 y c ) T, where x c and y c stand for the centers of X and Y. Let and be the eigenvalues of M X and M Y, and let p 1;p 2;p 3 and q 1;q 2;q 3 be their noralized eigenvectors correspondingly. Fro the geoetrical eaning, the geoetric distributions of X and Y are siilar if 0 1 i i +1; i =1; 2; 3 (18) 3 holds for soe sall offset 1, where =1=3 i = i. So, to achieve the global iniu, we suggest that the initial scale s 0 be selected close to and the interval I of the scale s be selected as in i = i ; ax i = i. At the sae tie, since eigenvalues represent the distributions of data set along their corresponding i i noralized eigenvectors, we suggest that the initial rotation R 0 be selected close to the vector [q 1 ;q 2 ;q 3 ][p 1 ;p 2 ;p 3 ] 01 and the initial translation T 0 be selected close to y c 0 x c. Here, for the related discussion Authorized licensed use liited to: SHANGHAI UNIVERSITY. Downloaded on June 27,2010 at 02:47:11 UTC fro IEEE Xplore. Restrictions apply.

4 562 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 6, NO. 3, JULY 2009 Fig. 1. RMS errors of registration within 50 iterations by two algoriths. Fig. 2. Portraits of the RMS errors of five registrations. TABLE I REGISTRATION RESULT OF ICP AND SCALE-ICP on the selection of the initial rotation and initial translation, we refer to [3]. We will follow the above principle to select the initial registration (R 0 ;s 0 ;T 0 ) in siulations of the next section, where the algorith convergence is really global. V. NUMERICAL EXPERIMENT This section is divided into two parts: 1) the coparison between the standard ICP algorith and Scale-ICP algorith for the registration of 3D range data sets without scale stretch and 2) the registration between two range data sets with scale stretch, by Scale-ICP algorith. All progras are written in Matlab6.5 and run by PC with PentiuIV3.0GHz CPU and 512M RAM. 1) Part I: Coparison between the standard ICP algorith and Scale-ICP algorith. In this part, Stanford 3D Scanning Repository 1 is eployed to copare the standard ICP algorith with the Scale-ICP algorith. As an explanation experient, we only display and explain the result of registration between bun045 (40097 points) and bun000 (40256 points) of the Stanford Bunny data set [9]. With the precision =0:001, the initial rotation axis ( 00:0534 0: :0602 ) and rotation angle , and the initial translation ( 00:0557 0: :0214 ), the standard ICP algorith arrives the optia at the 28th step consuing s, while the Scale-ICP algorith, with the additional initial scale s 0 =1:0092 and the scale interval I=[0:9506; 1:0923], arrives the optia at the 30 step consuing s. The coparison results are displayed in Table I and portrayed in Fig. 1. With the resulted RMS errors displayed in Table I, it is coputed that the registration accuracy is iproved by 3:844%(= RMS ICP 0 RMS ScaleICP =RMS ICP 2 100%) by the Scale-ICP ethod. Fig. 1 exhibits the accuracy iproveent. The nuerical results deonstrate that, for registration of the data sets with noises, 1 both algoriths are efficient in the sae way, but the Scale-ICP algorith is able to iprove the registration accuracy. 2) Part II: Registration of two range data sets with scale stretch. To deonstrate the advantage of Scale-ICP algorith for registrations between two data sets with different scales, we consider the group of registrations between the data set bun045 and each data set generated by ultiplying bun000 by soe scales, respectively. For the sake of the error coparability, we adopt the following odified RMS error RMS = 1 1 ks 3 R 3 x i + T 3 0 z 3 i k 2 1=2 (19) where R 3, T 3, and s 3 are the iteration terinal rotation, translation and scale, and z 3 i is the closest point to the point s 3 R 3 x i + T 3. With the precision = 0:001 and the respective initial registrations with = 0:5; 1:0; 2:0; 10:0, and (see Table II), we obtain the corresponding nuerical results displayed in Table III and portrayed in Figs Fro Table III, it is seen that: 1) the five terinal RMS errors are all identical, that is, all five iteration sequences approxiate to the sae value in the given iteration length; 2) the five terinal rotations are all identical, which verifies the fact that the scale stretches introduced by us do not change the rotation stages of data sets; and 3) the five terinal scales s 3 are quite close to the actual ones, which deonstrates the perforance of Scale-ICP algorith robust to the scale bias. Moreover, that the displayed five translations appear to be different is because the translations occur after the scale stretches. In fact, if every terinal T 3 is divided by its corresponding, then we can find that the resulted translations are all the sae in the given precision. Five RMS error sequences are portrayed in Fig. 2. The initial configurations of the data set bun045 (red) and the data set generated by ultiplying bun000 by =0:5 (blue) are portrayed in Figs. 3 and 4 exhibits the configurations of two data sets after registration. We further consider the group of registrations between the data set dragonstandright_48 (22092 points) and each data set generated by ultiplying dragonstandright_0 (41841 points), respectively, by the scales = 0:5; 1:0; 2:0; 10:0 and 100:0. The corresponding nuerical results for = 0:5 are portrayed in Figs. 5 and 6. VI. CONCLUSION AND FURTHER DISCUSSION In this paper, we addressed ourselves to the registration of two 3D data sets with noises and large-scale stretches. By incorporating a Authorized licensed use liited to: SHANGHAI UNIVERSITY. Downloaded on June 27,2010 at 02:47:11 UTC fro IEEE Xplore. Restrictions apply.

5 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 6, NO. 3, JULY TABLE II INITIAL REGISTRATION FOR DATA SETS WITH DIFFERENT SCALES TABLE III RESULTS OF REGISTRATION FOR DATA SETS WITH DIFFERENT SCALES Fig. 3. Original configuration of two data sets when =0:5. Fig. 4. Configurations of two data sets after registration. scale factor to the standard ICP algorith, we developed a new ICP algorith, naed Scale-ICP algorith, whose advantages ainly exist in: 1) for the registration of data sets with the sae scale, the presented coparative experients deonstrated that, without loss of coputational efficiency, it is ore accurate than the standard ICP algoriths; 2) for the registration of data sets where exist large scales, it can ipleent registration accurately, while the standard ICP algorith cannot do that; and 3) its iterations are of closed for. In the authors opinion, the proposed Scale-ICP algorith is distinguished fro the standard ICP algorith in that it has one ore degree of freedo than the standard ICP, which ay be why the Scale-ICP algorith is ore accurate. Authorized licensed use liited to: SHANGHAI UNIVERSITY. Downloaded on June 27,2010 at 02:47:11 UTC fro IEEE Xplore. Restrictions apply.

6 564 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 6, NO. 3, JULY 2009 Fig. 5. Original configuration of two data sets when =0:5. Fig. 6. Configuration of two data sets after registration. However, as sae as all kinds of ICP algoriths, the algorith we proposed is also local. Towards achieving a global optiu, we proposed a way of selection of initial registration. A point worth noticing is that the stretches existing in the data sets we considered in this paper are isotropic, which akes us able to use 1D scale factors to odel the stretches. We have done our best to extend the Scale-ICP ethod to the anisotropic case, but unsuccessful. Indeed, in the anisotropic case, the noncoutativity between the scale atrix S =diag(s1;s2;s3) and the rotation atrix R akes us unable to apply the SVD technique in solving the resulted optiization in e k (R; S) = ~x T i R T Diag(s1;s 2 2;s 2 2 3)R~xi 02 ~z i T Diag(s1;s2;s3)R~xi + ~z i T ~zi: It sees to be not easy to solve such an optiization proble using the coonly used ethods. However, it is really a very significant research issue. REFERENCES [1] K. S. Arun, T. S. Huang, and S. D. Blostein, Least square fitting of two 3-D point sets, IEEE Trans. Pattern Anal. Mach. Intell., vol. 9, no. 5, pp , [2] M. Á G. Ballester, X. Pennec, M. G. Linguraru, and N. Ayache, Generalized iage odels and their application as statistical odels of iages, Med. Iage Anal., vol. 8, pp , [3] P. J. Besl and N. D. McKay, A ethod for registration of 3-D shapes, IEEE Trans. Pattern Anal. Mach. Intell., vol. 14, no. 2, pp , [4] Y. Chen and G. Medioni, Object odeling by registration of ultiple range iage, in Proc. IEEE Conf. Robot. Auto., 1991, vol. 3, pp [5] D. Chetverikov, D. Stepanov, and P. Krsek, Robust Euclidean alignent of 3D point sets: The tried iterative closest point algorith, Iage and Vision Coput., vol. 23, pp , [6] E. Ezra, M. Sharir, and A. Efrat, On the ICP algorith, in Proc. the 22rd Ann. Syp. Coputational Geoetry, 2006, pp [7] J. Feldar and N. Ayache, Rigid, affine and locally affine registration of free-for surfaces, Int. J. Coput. Vision, vol. 18, pp , [8] J. Feldar, G. Malandain, J. Declerck, and N. Ayache, Extension of the ICP algorith to non-rigid intensity-based registration of 3D volues, in Proc. Workshop Math. Methods Bioed. Iage Anal., 1996, pp [9] A. W. Fitzgibbon, Robust registration of 2D and 3D point sets, Iage and Vision Coput., vol. 21, pp , [10] H. Hügli and C. Schütz, Geoetric atching of 3D objects: Assessing the range of successful initial configurations, in Proc. IEEE Int. Conf. Recent Advances in 3D Digital Iaging and Modeling, 1997, pp Authorized licensed use liited to: SHANGHAI UNIVERSITY. Downloaded on June 27,2010 at 02:47:11 UTC fro IEEE Xplore. Restrictions apply.

7 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 6, NO. 3, JULY [11] K. Kanatani, Analysis of 3-D rotation fitting, IEEE Trans. Pattern Anal. Mach. Intell., vol. 16, no. 5, pp , [12] S. Krishnan, P. Y. Lee, J. B. Moore, and S. Venkatasubraanian, Global registration of ultiple 3D point sets via optiization-on-a-anifold, in Proc. Eurographics Syp. Geoetry Processing, 2005, pp [13] Y. H. Liu, Iproving ICP with easy ipleentation for free-for surface atching, Pattern Recogn., vol. 37, pp , [14] B. D. Lucas and T. Kanade, An iterative iage registration technique with an application to stereo vision, in Proc. 7th Int. Joint Conf. Arti. Intell., 1981, pp [15] X. Pennec and J. P. Thirion, A fraework for uncertainly and validation of 3D registration ethods based on points and fraes, Int. J. Coput. Vision, vol. 25, no. 3, pp , [16] B. M. Planitz, A. J. Maeder, and J. A. Willias, The correspondence fraework for 3D surface atching algorith, Coputer Vision and Iage Understanding, vol. 97, pp , [17] H. Pottann and M. Hofer, Geoetry of the squared distance function to curves and surface, in Visualization and Matheatics III, H. C. Hege and K. Polthier, Eds. New York: Springer, 2003, pp [18] H. Pottann, Q. X. Huang, Y. L. Yang, and S. M. Hu, Geoetry and convergence analysis of algoriths for registration of 3D shapes, Int. J. Coput. Vision, vol. 67, no. 3, pp , [19] S. Rusinkiewicz and M. Levoy, Efficient variants of ICP algorith, in Proc. 3rd Int. Conf. 3D Digital Iaging and Modeling, 2001, pp [20] G. C. Sharp, S. W. Lee, and D. K. Wehe, ICP registration using invariant features, IEEE Trans. Pattern Anal. Mach. Intell., vol. 24, no. 1, pp , [21] S. Ueyaa, Least-squares estiation of transforation paraeters between two point patterns, IEEE Trans. Pattern Anal. Mach. Intell., vol. 13, no. 4, pp , [22] P. Yan and K. W. Bowyer, A fast algorith for ICP-based 3D shape bioetrics, in Proc. 4th IEEE Workshop on Auto. Identif. Adv. Technol., 2005, pp [23] Z. Zhang, Iterative point atching of free-fro curves and surfaces, Int. J. Coput. Vision, vol. 13, no. 2, pp , [24] T. Zinßer, J. Schidt, and H. Nieann, A refined ICP algorith for robust 3-D correspondence estiation, in Proc Int. Conf. Iage Process., 2003, pp [25] B. Zitová and J. Flusser, Iage registration ethods: A survey, Iage and Vision Coput., vol. 21, pp , Visual-Based Ipedance Control of Out-of-Plane Cell Injection Systes Haibo Huang, Dong Sun, Jaes K. Mills, Wen J. Li, and Shuk Han Cheng Abstract In this paper, a vision-based ipedance control algorith is proposed to regulate the cell injection force, based on dynaic odeling conducted on a laboratory test-bed cell injection syste. The injection force is initially calibrated to derive the relationship between the force and the cell deforation utilizing a cell ebrane point-load odel. To increase the success rate of injection, the injector is positioned out of the focal plane of the caera, used to obtain visual feedback for the injection process. In this out-of-plane injection process, the total cell ebrane deforation is estiated, based on the coordinate frae deforation of the cell, as easured with a icroscope, and the known angle between the injector and the plane. Further, a relationship between the injection force and the injector displaceent of the cell ebrane, as observed with the caera, is derived. Based on this visual force estiation schee, an ipedance control algorith is developed. Experiental results of the proposed injection ethod are given which validate the approach. Note to Practitioners In biological cell injection, the control of injection forces is iportant since excessive anipulation force ay destroy the ebrane or tissue of the biological cell, and lead to failure of the bioanipulation task. Although the injection force is a very iportant factor that affects the survivability of the injected cells, research on inially invasive cell injection, based on understanding of the echanical properties of the cellular ebrane, are rare. To resolve this proble, a ethodology is presented here to regulate cell injection forces using geoetry of cell deforations and icro vision feedback. With the visually easured injection force applied to an injected cell, an ipedance control strategy is then developed to regulate the cell injection force through the desired ipedance. One advantage of using the ipedance control in icroanipulation is that tiny forces exerted on the cell ebrane can be reflected in the control syste by adjusting the controller ipedance coefficients. Index Ters Bioanipulation, cell injection, ipedance control, icrorobotic syste. I. INTRODUCTION Since its invention during the first half of the last century, biological cell injection has been widely applied in gene injection [1], in-vitro fertilization (IVF) [2], intracytoplasic sper injection (ISCI) [3], and drug developent [4]. Until very recently, however, existing research /$ IEEE Manuscript received August 13, 2007; revised Noveber 22, First published May 12, 2009; current version published July 01, This paper was recoended for publication by Associate Editor M. Zhang and Editor D. Meldru upon evaluation of the reviewers coents. This work was supported by a grant fro the Research Grants Council of the Hong Kong Special Adinistrative Region, China (Reference No. CityU ) and a grant fro the City University of Hong Kong (Project No ). H. B. Huang was with the Departent of Manufacturing Engineering and Engineering Manageent, City University of Hong Kong, Kowloon, Hong Kong. He is now with the Departent of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON M5S 3G8, Canada (e-ail: hbhuang@ie. utoronto.ca). D. Sun is with the Departent of Manufacturing Engineering and Engineering Manageent, City University of Hong Kong, Kowloon, Hong Kong (e-ail: edsun@cityu.edu.hk). J. K. Mills is with the Departent of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON M5S 3G8, Canada (e-ail: ills@ie. utoronto.ca). W. J. Li is with the Centre for Micro and Nano Systes, Chinese University of Hong Kong, Shatin, NT, Hong Kong (e-ail: wen@acae.cuhk.edu.hk). S. H. Cheng is with the Departent of Biology and Cheistry, City University of Hong Kong, Kowloon, Hong Kong (e-ail: bhcheng@cityu.edu.hk). Digital Object Identifier /TASE Authorized licensed use liited to: SHANGHAI UNIVERSITY. Downloaded on June 27,2010 at 02:47:11 UTC fro IEEE Xplore. Restrictions apply.