A Method of Generating Multi-Scale Disc-Like Distributions for NDT Registration Algorithm

Transcription

1 International Journal of Mechanical Engineering and Robotics Research Vol. 5, o., January 06 A Method of Generating Multi-Scale Disc-Like Distributions for D Registration Algorith Hyunki Hong and Beohee Lee Departent of Electrical and Coputer Engineering (ASRI), Seoul ational University, Seoul, Republic of Korea Eail: {ckowl, bhlee}@snu.ac.kr By coparison with Iterative Closest Point (ICP), the ost popular registration algorith, the process of searching for correspondence in D is faster since the nuber of distributions is sall [4]. However, the accuracy of D is lower than ICP since the group of distributions is an approxiate representation of structure. In the registering process of D, the new point cloud gets closer iteratively by axiizing the score. heore, new points tend to ove to the ass center of erence distribution where the score is the axiu. he big issue of D is the size of cells. If the side length of regular cell is bigger, the structure of erence point cloud in a cell is ore coplex [], [4]. hus the ass center of the point cloud is possibly far away fro the point cloud. For exaple, in Fig., a point cloud in a cell is a wall in L-shape. he ball in Fig. (a) is an ellipsoid whose radius are Mahalanobis distance of.6. It is obvious that the ass center is hanging in the air, and that distribution is not fitting the point cloud. heore, new points tending to ove to the ass center would lead to the failure of registration or the low accuracy. On the other hand, if the cells are subdivided once ore as shown in Fig. (b), distributions in cells are disc-like, and ean points are alost near the centers of partial point clouds. hus, distributions generated by the sall cells represents the structure better than by the big cells. However, if the nuber of erence points in a cell is too sall, the distribution hardly represent the structure, and it could even be generated. Abstract hick distributions whose ass centers are hanging in the air is one reason lowering the accuracy of oral Distributions ransfor (D). his paper proposes a ethod to solve this proble. he proposed ethod is transforing erence point cloud to disc-like distributions fitting the structure of point cloud. he subdivision of cells is based on octree, and it subdivides point cloud by different sizes of cells. By Principal Coponent Analysis (PCA), it finds fitting plane for the partial erence point cloud in each cell. It copares the Mean Square Error (MSE) to the threshold value to evaluate the flatness of the partial point cloud. If the cloud is not flat, the corresponding cell is subdivided to saller cells, and those cells subdivide the cloud once ore. he proposed ethod recursively subdivides erence point cloud until no ore partial point cloud is possible to be subdivided. he experient evaluates the loss of data and the error related to. In addition, the flat distributions transfored fro erence point cloud is shown. Index ers point cloud, registration, scan atching, noral distributions transfor, D, subdivision, ultiscale, disc-like, flat I. IRODUCIO oral Distributions ransfor (D) is one of the ost faous registration algoriths []-[3]. he algorith registers a point cloud to the erence point cloud. In robotics, it is a ethod of estiating the pose variation of a robot. While the robot continuously scans the environent, it registers each new point cloud to preceding point cloud or the global ap by registration algorith. he transforation vector is obtained by each registration, and it is the estiation of pose variation. D is a registration algorith accelerated by transforing erence point cloud to ultiple distributions. hree processes of D and variants are subdividing the erence point cloud to partial point clouds, transforing partial point clouds into distributions, and registering new point cloud to distributions. At first, D subdivides erence point cloud by regular cells. ext, it coputes eans and covariance atrices of partial point clouds in cells, and it transfors the into noral distributions. Finally, it registers new point clouds by axiizing the su of scores. Figure. (a) A point cloud whose structure is not planar generates a thick distribution. (b) he desired flat disc-like distributions. his paper focuses on the effect of cell size and propose a new ethod to generate ulti-scale disc-like Manuscript received April 0, 05; revised October 4, 05. doi: 0.878/ijerr

2 International Journal of Mechanical Engineering and Robotics Research Vol. 5, o., January 06 distributions for D. o decrease the nuber of distributions and represent the structure of environent better, the proposed ethod subdivides erence point cloud by ulti-scale cells. A big plane-like partial point cloud would be represented by one disc-like distribution while the coplex-shaped one would be represented by distributions in ultiple sizes. he ost iportant part of the proposed ethod is the flatness evaluation ethod. It deterines whether the partial point clouds are flat. he plan of the paper is as follows. In Section II the processes of the proposed ethod is described, and the details about octree-based subdivision and flatness evaluation ethod are described. In Section III the results of visualization is deonstrated, and the error and loss of data varied by threshold value is discussed. II. distributions. his paper focus on the second reason and proposes a ethod to iprove the ap representation of distributions. BACKGROUD he process of D could be divided into subdividing, generating, and registering [5]. In the subdividing process, D generates cells and subdivides erence point cloud into partial point clouds by those cells as shown in Fig. (b). D transfors each partial point cloud into a noral distribution whose score function is s(p) exp p C p () where is the ean vector, and C is the covariance atrix. and C are coputed by C n p (), i i n p i,i (3) where p,i is the ith erence point, and n is the nuber of p in the cell. In the generating process, D coputes both and C of each partial point cloud to generate distributions as shown in Fig. (c). In registering process, D coputes the transforation vector ξ * which axiizes the su of scores new S s( (p new,i, ξ)) Figure. (a) A erence point cloud. (b) Octree cells subdividing the erence point cloud. (c) Octree cells and distributions transfored fro erence point cloud. (d) Disc-Like distributions. he color is RGB values apped fro noral vectors. (4) i where p new,i is the ith new point, new is the nuber of new points. (p, ξ) is the transforation function as (p, ξ) R(ξ)p t (ξ) III. (5) he goal of the proposed ethod is generating disclike distributions to represent the structure of point cloud as shown in Fig. (d). It generates disc-like distributions by subdividing point clouds until all of partial point clouds are disc-like. o generate disc-like distributions, it finds the fitting planes for partial point clouds and evaluate their flatness. If a point cloud is flat, it would not be subdivided. On the other hand, if a point cloud is not flat, it would be subdivided into two or ore partial point where R(ξ) is the rotational atrix, and t (ξ ) is the translational vector. o obtain ξ *, ewton ethod and Levenberg-Marquardt algorith are usually used [], [5]. In soe cases, the error between the estiated pose variation and ground truth is not acceptable. One of reasons is that D reaches a local axiu, and another reason is that the global axiu is wrong because of the bad ap representation of noral HE PROPOSED MEHOD 53

3 International Journal of Mechanical Engineering and Robotics Research Vol. 5, o., January 06 clouds by saller cells. It would be recursively repeated until all partial point clouds are flat. he process is as follows:. Copute the side length of the initial cell including all erence points.. Evaluate the flatness of each partial point cloud in a cell. 3. Evaluate flatness of partial point clouds. If a partial point cloud is not flat, then subdivide the corresponding cell. 4. Subdivide the point clouds by subdivided cells. 5. Repeat -4 recursively. erinate recursion if partial point clouds are flat or degenerated. he evaluation at step is based on the flatness evaluation ethod, and the cells and erence point cloud are subdivided by octree-based subdivision ethod at step 3 and 4. he details about two ethods would be described in subsections. A. Flatness Evaluation Method he flatness evaluation of a point cloud is the iportant part of the proposed ethod. It decides whether the point cloud need be subdivided. If the shape of a partial point cloud is not flat, then it need be subdivided. he erence point cloud is recursively subdivided until all of partial point clouds are flat or degenerated. he criteria of flatness evaluation is the Mean Square Error (MSE) of distances between erence points and the corresponding fitting plane. It is obtained as follows:. Copute and C.. Find the fitting plane. 3. Copute MSE and evaluate the flatness. First of all, it coputes ean and covariance of each partial point. ext, it coputes the noral vector n of the fitting plane by principal coponent analysis (PCA) as shown in Fig. 3 [6]. It uses the eigenvector corresponding to the sallest eigenvalue of covariance atrix C as n, and then it obtains a plane which includes as follows. distance d between a point and the plane as shown in Fig. 3 is obtained by MSE is the ean of [7]. MSE i d n ( p p ) n ( p p )( p p ) n (7) d, and it is coputed as follows n ( p p )( p p ) n i (8) ( )( ) n p p p p n i n Cn d As shown in (8), in fact, it reuses C instead of coputing each square of distance. heore, the elapsed tie of flatness evaluation would be reduced. For the proposed ethod, if MSE is bigger than the threshold, it eans that the big nuber of points are further than the threshold distance. In short, the variance of point cloud is too big; theore, it is not disc-like. he ethod has a trade-off between runtie and the accuracy. If is sall, then the exaination becoes strict. As a result, the runtie becoes longer since the erence point cloud is subdivided ore ties. Moreover, the part of point clouds would be degenerated since it transfors partial point clouds to distributions only if the nuber of points is bigger than 4. Figure 3. Flatness evaluation ethod coputes MSE of distance between erence points and the fitting plane to evaluate the flatness of the partial point cloud in a cell. he black points are erence points, and the red point is the ean point. n p n (6) where p is an arbitrary point on the plane. Finally, it coputes the MSE of distances. he square of the Figure 4. he octree structure. Red nodes are subdivided since the distributions are not disc-like. Gray nodes are degenerated since they have points less than 4. Green nodes are not subdivided since the distributions are flat. B. Octree-Based Subdivision Method Octree structure as shown in Fig. 4 is one of subdivision ethods for D [], [8]. It is effective to anage partial point clouds. In addition, it is efficient to search for desired cells and points. For those reasons, the proposed ethod uses octree to generate cells subdividing the erence point cloud. he first step of 54

4 International Journal of Mechanical Engineering and Robotics Research Vol. 5, o., January 06 octree is setting the range. hat is, it need copute the side length of initial cell including all of erence points. he top square in Fig. 4 stands for this initial cell. o balance the tree, it is a strategy setting the center at the average point of erence points. Each cell is split to 8 cells by xy, yz, and xz planes through the center of cell. And then, erence points in that cell are subdivided by those subdivided cells. In Fig. 4, squares in red ean that the partial point clouds should be subdivided since their structures are not flat. he green squares are stand for flat point clouds, and the gray ones are degenerated point clouds, whose nuber of points are less than 4. IV. EXPERIMES he proposed ethod is ipleented based on Point Cloud Library (PCL) [9]. he benchark data set is the scenario provided by Karlsruhe Institute of echnology, and it is scanned outdoor by Velodyne range sensor [0]. 00 scans are randoly chosen for the experient. In this experient, the average percentage of loss of points and the average errors of fitting planes related to threshold are evaluated. flatness evaluation becoes strict as the threshold value decreases. heore, log is better saller than -7. In the interval fro - to -8, the variation of loss of data becoes large. Cases whose loss of data increases steeply as log decreases are scanned at coplex places. By the experient, the partial point clouds scanned at ore coplex environents are uch easier to be degenerated by the strict threshold value, and it leads to the big loss of 6 data. Fig. is an exaple when is set to 0. he radius of ellipsoids standing for distributions are Mahalanobis distance of.8. he different sizes of distributions are all disc-like. he RGB color of distributions are related to the noral vectors of fitting plane. V. COCLUSIO his paper proposes a ethod to generate ulti-scale disc-like distributions for D. It uses octree algorith to easily subdivide erence point cloud. In addition, it uses PCA to find the noral vector for a fitting plane. MSE is used to evaluate the flatness of point clouds. If a partial point cloud is not flat, then it subdivides the point cloud recursively. As the results shown in the experient, the loss of data steeply increases as the threshold value is decreased. In addition, the error increases as the threshold value is increased. ACKOWLEDGME his work was supported in part by the ational Research Foundation of Korea (RF) grant funded by the Korea governent (MSIP) (o.03raaa ), in part by ASRI, and in part by the Brain Korea Plus Project. (a) (b) Figure 5. (a) log - loss of data. (b) log - error. he results are as shown in Fig. 5. he error decreased sharply as log becoes sall. When log is saller than -6, the error is acceptably sall. However, as log decrease, the loss of data increases. he reason is that ore partial point clouds are degenerated since the REFERECES [] P. Biber and W. Straßer, he noral distributions the noral distributions, in Proc. IEEE/RSJ Int. Conf. on Intell. Robot. and Syst., Las Vegas, 003, pp [] M. Magnusson, A. Lilienthal, and. Duckett, Scan registration for autonoous ining vehicles using 3d-ndt, J. Field Robot., vol. 4, no. 0, pp , 007. [3] E. akeuchi and. subouchi, A 3-d scan atching using iproved 3-D noral distributions transfor for obile robotic apping, in Proc. IEEE/RSJ Int. Conf. on Intell. Robot. and Syst., Beijing, 006, pp [4] M. Magnusson, A. uchter, C. Lorken, A. J. Lilienthal, and J. Hertzberg, Evaluation of 3d registration reliability and speed-a coparison of icp and ndt, in Proc. IEEE/RSJ IEEE International Conference on Robotics and Autoation, 009, pp [5] Int. Conf. on Intell. Robot. and Syst., Kobe, 009, pp [6] C. Ulaş and. Hakan, 3D ulti-layered noral distribution transfor for fast and long range scan atching, J. Intell. & Robot. Sys., vol. 7, no., pp , 03. [7] R. O. Duda and E. H. Peter, Pattern Classification and Scene Analysis, st ed. ew York: Wiley, 973. [8] J. Poppinga,. Vaskevicius, A. Birk, and K. Pathak, Fast plane detection and polygonalization in noisy 3D range iages, in Proc. IEEE/RSJ Int. Conf. on Intell. Robot. and Syst., ice, 008, pp [9] A. Hornung, K. M. Wur, M. Bennewitz, C. Stachniss, and W. Burgard, OctoMap: An efficient probabilistic 3D apping fraework based on octrees, Autonoous Robots, vol. 34, no. 3, pp ,

5 International Journal of Mechanical Engineering and Robotics Research Vol. 5, o., January 06 [0] A. Aldoa, Z. C. Marton, F. obari, W. Wohlkinger, C. Pothast, B. Zeisl, et al., Point cloud library, IEEE Robotics & Autoation Magazine, 0. [] F. Moosann and C. Stiller, Velodyne SLAM, in Proc. IEEE Intelligent Vehicles Syposiu, 0, pp Hyun K. Hong received his B.S. in Electrical and Coputer Engineering fro Seoul ational University in 0. He is currently a M.S candidate in the Departent of Electrical and Coputer Engineering at Seoul ational University. His current research interests include graph SLAM and ulti-agent control. Boe H. Lee received the B.S. degree and M.S. degree in electronics engineering fro Seoul ational University, Seoul, the Republic of Korea in 978 and 980, respectively, and the Ph.D. degree in coputer, inforation, and control engineering fro the University of Michigan, Ann Arbor, MI in 985. He was an Assistant Professor with the School of Electrical Engineering at Purdue University, West Lafayette, I fro 985 to 987. He joined Seoul ational University in 987. He is currently a Professor with the Departent of Electrical and Coputer Engineering. His research interests include ulti-agent syste coordination, control, and application. Prof. Lee has been a Fellow of the Robotics and Autoation Society since