AUTOMATED REGISTRATION OF UNORGANISED POINT CLOUDS FROM TERRESTRIAL LASER SCANNERS

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1 AUTOMATED REGISTRATION OF UNORGANISED POINT CLOUDS FROM TERRESTRIAL LASER SCANNERS Kwang-Ho Bae and Derek D. Lcht Department of Spatal Scences, Curtn Unversty of Technology, GPO Box U1987, Perth, WA, 6845, Australa Commsson V, WG V/2 KEY WORDS: Regstraton, Automaton, Three-dmensonal, Pont Cloud, Laser Scannng, TLS ABSTRACT Terrestral laser scanners provde a three-dmensonal sampled representaton of the surfaces of objects resultng n a very large number of ponts. The spatal resoluton of the data s much hgher than that of conventonal surveyng methods. Snce laser scanners have a lmted feld of vew, n order to obtan a complete representaton of an object t s necessary to collect data from several dfferent locatons that must be transformed nto a common coordnate system. Exstng regstraton methods, such as the Iteratve Closest Pont (ICP) or Chen and Medon s method, work well f good a pror algnment s provded. However, n the case of the regstraton of partally overlappng and unorgansed pont clouds wthout good ntal algnment, these methods are not approprate snce t become very dffcult to fnd correspondence. A method based on geometrc prmtves and neghbourhood search s proposed. The change of geometrc curvature and approxmate normal vector of the surface formed by a pont and ts neghbourhood are used to determne the possble correspondence of pont clouds. Our method s tested wth a smulated pont cloud wth varous levels of nose and two real pont clouds. 1 INTRODUCTION The regstraton of pont clouds s one of the mportant steps n processng data from laser scanners. Excellent revews about the exstng methods can be found n Haralck et al. (1989) and Rusnkewcz and Levoy (2001). To the best of the authors knowledge, a method for the regstraton of partally overlappng pont clouds from terrestral laser scanners wthout good a pror algnment has not developed yet. The most sgnfcant reason s that t s very dffcult to recover the correspondence between pont clouds wthout good a pror algnment. In ths paper a method for the regstraton of two partally overlappng pont clouds measured from dfferent locatons usng geometrc prmtves and neghbourhood search wthout good a pror algnment s proposed. Sharp et al. (2002) proposed a method based on Eucldean nvarant features: curvature, second order moments, and sphercal harmoncs. Our method s based on the change of curvature rather than curvature and s a thresholdng method such as Zhang (1994). The Iteratve Closest Pont (ICP) algorthm s a relable and popular method for pont cloud regstraton (Horn, 1987, Besl and McKay, 1992). Horn developed an adjustment method for recovery of unknown transformaton parameters wth knowledge of the correspondence of pont clouds. Besl and McKay developed the ICP usng Horn s algorthm n conjuncton wth a neghbourhood search algorthm. If approxmate a pror nformaton about the pontto-pont correspondence of two pont clouds s provded, then the ICP can teratvely recover the rgdbody transformaton that algns two pont clouds. It converges monotoncally to a local mnmum, whch may or may not be the global mnmum, and all ponts n a pont cloud are assumed to have correspondence n the other cloud. Suppose that there are two pont clouds, C 1 and C 2, and they are measured from dfferent locatons. A pont cloud C 1 has a set of N C 1 ponts, {p 1 1,, p 1 N }. Bold and normal letters represent a vector and a scalar, respectvely. Let C 1 p 1 p2 j be the dstance between pont p1 of pont cloud C 1 and p 2 j of C2. Let CP (p 1, C2 ) be the correspondng pont n C 2 of a pont p 1. The ICP algorthm can be brefly descrbed as follows. 1. Assume that the pont n C 2 closest to a pont n C 1 s the correspondng pont. 2. Fnd the correspondence of two pont clouds, C = N C 1 =1 {T ter=k(p 1 ), CP (T ter=k(p 1 ), C2 )}, where C s the set of all pars of correspondng ponts, T ter=k s the transformaton of the kth teraton and T ter=0 s an ntal transformaton. C may or may not be one-to-one matchng. 3. Compute the new transformaton T ter=k+1 that mnmzes the sum of square dstances between correspondng pont pars,.e. n ter=k =1 p 1 CP (T ter=k (p 1 ), C 2 ) 2 where n ter=k s the number of sample n kth teraton. Snce the ICP s based only on a local search algorthm to recover correspondence between two pont clouds and t mnmses the sum of square dstances between possble correspondng ponts, t converges slowly sometmes and tends to fall nto local mnma.

2 Another algorthm s Chen and Medon s that s a pontto-surface algorthm whereas the ICP s a pont-to-pont algorthm (Chen and Medon, 1992). It mnmses the sum of the square dstances of a pont to ts correspondng surface. Ths algorthm s generally faster than the ICP. However, the pont clouds need to be more closely algned to each other ntally than wth the ICP. The ICP, ts varants, and Chen and Medon s algorthm assume that the closest pont n pont cloud C 2 s a good estmate of the correct correspondng pont n C 1. If two pont clouds are not approxmately algned usng a pror georeferencng nformaton, ths assumpton s not correct. Although ntal algnment can be provded from other means lke surveyng of the laser scanner locatons, t s not always possble. Fndng correspondng ponts and good regstraton of the pont clouds are more dffcult f they only partally overlap. In addton, these adjustment algorthms provdes a closed-form soluton,.e. no teraton, whch s one of reasons for ther popularty, although closed-form methods can not provde statstcal nformaton of ndvdual parameter of rgdbody transformaton as conventonal least square methods do. 2 PROPOSED METHOD Geometrc prmtves, such as surface normal vector, curvature, and the change of curvature and so on, may provde addtonal and useful nformaton to recover the correspondence of two pont clouds. A method to fnd the correspondence of two pont clouds usng geometrc prmtves and a local search algorthm s proposed. Geometrc curvature and the change of curvature s nvarant to three dmensonal rgd moton and surface normal vector can be rotated accordng to the computed transformaton by Horn s or Chen and Medon s algorthm. The angle between normal vectors and the dfference between the changes of curvature of a pont and ts correspondng ponts are our crtera for selecton of correspondng pont par. The angle between approxmate normal vectors of p 1 and p 2 j can be expressed as θ (p 1 ;p 2 j ) = cos 1 (n p 1 n p 2 j ), (1) where n p 1 and n p 2 j are the respectve approxmate normal vectors of the ponts. The dfference n changes of curvature between two ponts can be wrtten β(p 1, p 2 j) = M cc (p 1 ) M cc (p 2 j), (2) where M cc (p 1 ) and M cc(p 2 j ) are the approxmate changes of curvature of p 1 and p2 j. The normal vector of a pont s estmated by covarance analyss of the pont and ts neghbourhood ponts and the change of curvature s estmated as the rato of egenvalues of the covarance matrx. 2.1 Normal vector estmaton The normal vector of a pont s estmated by one of the egenvectors of the covarance matrx of a pont and ts neghbourhood. The covarance of a pont and ts k neghbour ponts, COV (p 1 ), s expressed as COV (p 1 ) = 1 k (p1 p cent neghbour{j=1 k,p 1 })T (p 1 p cent neghbour{j=1 k,p 1 }) (3) where p cent neghbour{j=1 k,p 1 } s the cetrod of p1 and ts k neghbourhood ponts. COV (p 1 ) s a 3 3, real, postve, and sem-defnte matrx, the egenvalues of whch are always greater than or equal to zero. The egenvector of the mnmum egenvalue s the approxmate normal vector of the surface formed by p 1 and ts k neghbourhood ponts (Hoppe et al., 1992). The other egenvectors are the tangental vectors of the surface. If the mnmum egenvalue s close to zero, then the surface consstng of a pont and ts neghbourhood s flat. Ths method s the frst order approxmaton of the normal vectors of the surface. If the level of nose s large or the number of ponts n the neghbourhood, k, s too small, t could provde an ncorrect normal vector. 2.2 Change of geometrc curvature estmaton The change of geometrc curvature at a pont can be estmated from the egenvalues of the covarance matrx. Each egenvalue represents the spatal varatons along the drecton of the egenvector. Let λ and ν be the egenvalues and egenvectors of the covarance matrx, COV (p 1 ), wth the condton of λ 1 λ 2 λ 3. The change of curvature s a parameter of how much the surface formed by a pont and ts neghbourhood devates from the tangental plane formed by ν 2 and ν 3. The rato of the mnmum egenvalue and the sum of the egenvalues approxmates the change of geometrc curvature, M cc (p 1 ) = λ 1 3 =1 λ. (4) Addtonally, the geometrc curvature, M curv (p 1 ), of a pont can be estmated by the normal vectors of the pont and ts neghbourhood M curv (p 1 ) = 1 k k n p 1 n neghbour{j,p 1 } (5) j=1 where n p 1 and n neghbour{j,p 1 } are the normal vectors of p 1 and ts jth neghbourhood, (Lnsen, 2001). The change of curvature, M cc (p 1 ), can be expressed M cc (p 1 ) = 1 k k M curv (p 1 ) M curv (p 1 neghbour{j,p 1}). j=1 Both methods can provde a good approxmaton to the change of curvature. However, the qualty of estmaton depends on how well the neghbourhood ponts are dstrbuted. Snce our algorthm s for the regstraton of unorgansed pont clouds, there s no guarantee that every pont (6)

3 of a cloud has a set of evenly-dstrbuted neghbourhood ponts. Ths problem can be overcome by usng the angle crteron between the neghbour ponts as Lnsen (2001) dd for the trangulaton of pont clouds. Furthermore, we could use a method n whch each pont has dfferent number of neghbourhood pars. The angle crteron has been used n ths paper but usng dfferent number of neghbourhood for each pont was not mplanted. Therefore, a larger number of neghbourhood ponts has been used. 2.3 Algorthm descrpton The amount of data to process n order to fnd correspondence s very large, whch lmts the robustness of regstraton algorthms. The hgher curvature ponts may have more valuable nformaton than the lower curvature ponts snce they could be edges or corners. Therefore, n the early stages of teraton, we only take nto account hgher curvature ponts and then, as teraton proceeds, lower curvature ponts also are ncluded to mprove the regstraton. Our method for the regstraton of three-dmensonal, partally overlappng and unorgansed pont clouds wthout good a pror algnment can be brefly descrbed as follows: 1. Fnd the k neghbourhood ponts of every pont n C 1 and C 2. Estmate the geometrc prmtves of the ponts. 2. Take ntal sample ponts, p 1 {1,,n ter= }, whose change of curvature s greater than Tsample ter= where n ter= s the number of sample n th teraton. 3. Fnd correspondng ponts of p 1 {1,,n. ter=} p2 j s the correspondng pont of p 1 f θ(p 1, p 2 j) T ter= normal β(p 1, p 2 j) T ter= cc where Tnormal ter= ter= and Tcc are the thresholds for the angle between the normal vectors and the dfference n the changes of geometrc curvature between the correspondng ponts, respectvely. 4. Calculate the approxmate transformaton, T ter=, and transform C 1. Rotate the normal vectors of all ponts of C 1 as well. 5. Update the threshold values n order to apply a more strct crteron for determnaton of possble correspondng ponts as follow. T ter=+1 normal T ter=+1 sample T ter=+1 cc = T ter= normal T normal = Tcc ter= T cc = Tsample ter= T sample 6. Calculate the regstraton error, ɛ ter=, whch s defned as the rms dstance of ponts and ther correspondng surfaces n our method. If ɛ ter= s greater than threshold, then go to step 2. Otherwse stop the regstraton. In addton, f ɛ ter= s smaller than T ɛcm, for example, the average dstance of a pont from ts neghbourhood, then Chen and Medon s method s used snce t converges quckly than Horn s algorthm does f the pont clouds are close (Rusnkewcz and Levoy, 2001). Otherwse Horn s method s used. If the ntal algnment s close to the correct one, only a small number of ponts need to be searched. Otherwse a large number of ponts must be searched n order to fnd correct correspondence of sample ponts. The optmal number of ponts beng searched could be evaluated from the statstcal propertes of the dstrbuton of regstraton error metrc (Zhang, 1994). However, the dstance dstrbuton of the correspondng ponts s usually not a unmodal Gaussan but bmodal or multmodal dstrbutons. Furthermore, good ntal algnment s not assumed n the proposed method, t s dffcult to remove outlers n the early stages of teraton. Therefore, a large number of ponts need to be searched n order to determne the correspondence of two pont clouds. 2.4 Scale of selected correspondng ponts The scale of selected correspondng ponts s usually assumed to be unty and ths assumpton s reasonable n most cases (Horn, 1987). It can be also used as the error metrc to represent the qualty of regstraton (Croslla and Benat, 2002). The scale can be nterpreted as a parameter for the qualty n the determnaton of the correspondng ponts. For example, f we have ncorrect correspondence nformaton, then the scale s not unty. The scale factor n kth teraton, s ter=k, can be expressed as s ter=k = nter=k nter=k =1 p 1 T ter=k(cp (p 1, C2 )) =1 T ter=k (CP (p 1, (7) C2 )) 2 where T ter=k s the calculated transformaton of the kth teraton, CP (p 1, C2 ) s the poston vector of the correspondng pont of p 1, and n ter=k s the number of samples n the kth teraton. Although the scale of correspondng ponts s unty, t does not guarantee that we have one-toone matched correspondence. If the scale s much greater or smaller than unty, the calculated translaton could be ncorrect. 2.5 Threshold values The lst of threshold values used n the proposed method s shown n Table 1. Tcc ter=0 and T are the most mportant and crtcal thresholds. The other threshold values are not crtcal to the success rate of the proposed method, although they affect the robustness of the regstraton. It s dffcult to state explctly whch values are the optmal values snce they depend on dataset. Currently we are workng on fndng the optmal and generalzed expressons for these thresholds. Our suggestons of Tcc ter=0 and T from the experences wth the proposed method are Tcc ter=0 = < M cc 1 > + < Mcc 2 > (8) 2 T cc = 2 < Mcc 1 > 2 rms + < Mcc 2 > 2 rms (9)

4 threshold k T ter=0 sample T ter=0 cc T ter=0 normal T sample T cc T normal T ɛcm T ɛ descrpton number of neghbourhood ntal samplng threshold for the change of curvature ntal threshold for the dfference n changes of curvature ntal threshold for the angle between normal vectors ncrement for Tsample ter=k ncrement for Tcc ter=k ncrement for Tnormal ter=k threshold for startng Chen and Medon s method threshold for stoppng the regstraton Table 1: The threshold values are used n the propose method. where < M cc > and < M cc > rms are the mean and rms of the change of curvature of C. 3 EXPERIMENTAL RESULTS Three examples were tested wth the proposed method: a smulated pont cloud and two real pont clouds captured wth two dfferent laser scanners. All datasets have partally overlapped. The proposed method was mplemented n C++ and tested on a PC wth Intel Pentum III 450MHz and 516MB RAM. Our program s not yet optmzed so there s room for mprovement n terms of processng speed. For neghbourhood search, we used a kdtree search lbrary developed by Arya et al. (1998) and LAPACK (1999) was used for covarance analyss. 3.1 Smulated data 1996, Maas, 2000, Rusnkewcz and Levoy, 2001). These nclude the change of rotaton angles or translaton, the dstances between correspondng ponts, the dstances between ponts and ther correspondng surfaces, and so on. Whether the regstraton error, ɛ, s reasonable, too optmstc or pessmstc, depends manly on the number of outlers that are used to regster the pont clouds. In addton, the redundancy of correspondence, the spatal densty of data and the percentage of the overlappng regons are mportant factors. Two parameters that represent the error of regstraton were measured: the dstances between correspondng ponts and the dstances of ponts from ther correspondng surfaces. Fgure 2 shows these measures for the smulated dataset. As expected, more teratons are needed n order to mnmse regstraton error, as more nose s added to the pont clouds. The magntude of the dstances between correspondng ponts s about four tmes greater than the dstances between ponts and ther correspondng surfaces. It means that the success rate to fnd the correct pont-to-pont correspondence s much smaller than that to fnd the correct pont-to-surface correspondence. Ths s not surprse f we consder that the test pont clouds are parts of a cube,.e. most of overlappng regons of the pont clouds possess low curvature area. Therefore, we use the dstance between pont and ts correspondng surface as the error metrc of our method. Although we use ths as error metrc, the dstance between correspondng ponts wll stll provde good nformaton to ncrease the effcency of our algorthm snce we may remove outlers based on that nformaton. The scales of selected correspondng ponts n each teraton of the regstraton of smulated pont clouds wth varous standard devatons of zero-mean Gaussan nose are shown n Fgure 3. In early stage of regstraton, scales are much greater than unty snce we do not have good a pror algnment. After about fve teratons, all scales of the dfferent levels of nose become approxmately unty, whch s a good ndcaton of success n fndng correspondences. However, there are some dfferences between the scales n the presence of nose as shown n Fgure 3(b). 3.2 Real pont clouds Fgure 1: Before the regstraton of the pont clouds of the parts of a cube The smulated pont clouds are parts of a cube, havng dmensons of 1m 1m 1m, and partally overlapped. The number of ponts n the pont clouds are 2640 and One pont cloud was translated wth (x,y,z)=(0.2m, 0.1m, 0.5m) and rotated 30 around z-axs from regstered state as shown n Fgure 1. Zero-mean Gaussan nose wth varous standard devatons was added ndependently to each pont of the pont clouds. In the case of zero standard devaton,.e. no nose, all ponts n the overlappng regon have exact correspondng ponts. Many dfferent error metrcs have been defned to measure how well two pont clouds are regstered (Smon, The second example s the regstraton of two real pont clouds from a Buddha statue (Ayuthaya, Thaland), scanned wth Regl LMS-Z210 that has angular samplng nterval s (Regl, 2004). Fgure 4 shows the pont clouds as before and after regstraton usng our method. The thrd example s a scene contanng a buldng and trees measured by Mens GS200 whose angular samplng nterval s (Mens, 2004). In ths example, three pont clouds are regstered as shown n Fgure 5. The results of regstraton are lsted n Table 2. In case of the smulated data wthout nose, the regstraton error after seven teratons s 0.04mm. In the cases of smulated pont clouds wth zero-mean Gaussan nose, regstraton errors are smlar wth the standard devatons of Gaussan nose. The executon tme of σ = 0.06 s faster than the other cases. All regstraton errors of both smulated and

5 sgma 0.00 sgma 0.01 sgma 0.03 sgma dstance (m) (a) (b) teraton (a) rms dstances between correspondng ponts dstance (m) teraton sgma 0.00 sgma 0.01 sgma 0.03 sgma 0.06 (b) rms dstances of ponts from ther correspondng surfaces Fgure 2: Two knds of regstraton errors of smulated pont clouds wth dfferent levels of nose. σ s the standard devaton of zero-mean Gaussan nose. scale scale (a) teraton sgma 0.00 sgma 0.01 sgma 0.03 sgma teraton (b) Fgure 3: The scale of selected correspondng ponts n each teraton of the regstraton of smulated pont clouds wth zero-mean Gaussan error. (b) s the magnfed fgure of (a). (c) Fgure 4: A Buddha statue scanned by Regl LMS-Z210. (a) and (b) are before the regstraton. (c) and (d) are after the regstraton. real pont clouds are much smaller than the pont spacngs of pont clouds defned as the average dstance from a pont from ts neghbourhood. The regstraton errors of the two real pont clouds are the order of centmetre. In the cases of the buldng and trees captured by the Mens GS200, regstraton s successful as ndcated by the regstraton error, ɛ, despte the dfference of the pont spacngs of two pont clouds beng about the order of 10cm and the presence of many trees, whch hnders the regstraton of the pont clouds. n 1 k t ɛ d 1 n 2 (sec) (m) d 2 Cube σ = Cube σ = Cube σ = Cube σ = Ayuthaya buldng (2+1) buldng (2+3) Table 2: Results of experments wth smulated and real pont clouds. n s the total number of ponts of pont cloud C. k and are the numbers of the neghbourhood of a pont and total teratons, respectvely. t and ɛ are the executon tme and the regstraton error. d s the pont spacng whch s defned as the average dstance of a pont from ts neghbourhood. 4 CONCLUSION A method for the regstraton of two partally overlappng pont clouds from dfferent locatons wthout good a pror algnment was proposed and tested wth a smulated pont cloud wth dfferent levels of Gaussan nose and two (d)

6 Arya, S., Mount, D., Netanyahu, N. S., Slverman, R. and Wu, A. Y., An optmal algorthm for approxmate nearest neghbour searchng. Journal of the ACM (Assocaton for Computng Machnery) 45, pp Besl, P. J. and McKay, N. D., A method for regstraton of 3-D shapes. IEEE Transactons on Pattern Recognton and Machne Intellgence 14(2), pp (a) Chen, Y. and Medon, G., Object modellng by regstraton of multple range. Image and Vson Computng 10(3), pp Croslla, F. and Benat, A., Use of generalsed Procrustes analyss for the photogrammetrc block adjustment by ndependent models. The ISPRS Journal of Photogrammetry and Remote Sensng 56, pp (b) Fgure 5: A buldng and trees scanned by Mens GS200. (a) and (b) are before and after the regstraton, respectvely. real pont clouds from two dfferent scanners. The dstance from a pont and the correspondng surface was used as the error metrc of regstraton. The regstraton errors for real pont cloud regstraton were the order of centmetre and that of a smulated dataset were smlar wth the standard devatons of zero-mean Gaussan nose. Several ways are possble to mprove our method. In terms of executon tmes, we can modfy our method to use dfferent neghbour ponts for each pont dependng on the dstrbuton or the area of the regon covered by the pont and the neghbourhood. Regardng threshold values, the propertes of the threshold values used n the proposed method can be nvestgated n order to provde crtera for the selecton of the optmal threshold values. Furthermore, corner ponts of pont clouds can be detected usng geometrc prmtves that have used n the proposed method and they can be used as ntal samples for the regstraton. In addton, the scale of correspondng ponts may be a good ndcaton of the qualty of samplng for regstraton. 5 ACKNOWLEDGEMENT Ths research was supported by an Australan Research Councl (ARC) Dscovery grant DP The authors thanks to Mens for provson of GS200 dataset. REFERENCES Anderson, E., Ba, Z., Bschof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarlng, S., McKenney, A. and Sorensen, D., LAPACK Users Gude. Thrd edn, Socety for Industral and Appled Mathematcs, Phladelpha, PA. Haralck, R. M., Joo, H., Lee, C. H., Zhang, X., Vadya, V. G. and Km, M. B., Pose estmaton from correspondng pont data. IEEE Transactons on Systems, Man, and Cybernetcs 19(6), pp Hoppe, H., DeRose, T., Duchamp, T., McDonald, J. and Stuetzle, W., Surface reconstructon from unorganzed ponts. Computer Graphcs 26, pp Horn, B. K. P., Closed-form soluton of absolute orentaton usng unt quaternons. Journal of the Optcal Socety of Amerca 4(4), pp Lnsen, L., Local versus global trangulaton. Proceedngs of EUROGRAPHICS Maas, H. G., Least-sqaures matchng wth arborne laser scannng data n a TIN structure. The ISPRS Internatonal Archves of the Photogrammetry, Remote Sensng and Spatal Informaton Scences 33(3A), pp Mens, Aprl accessed on 20th Regl, accessed 20th Aprl Rusnkewcz, S. and Levoy, M., Effcent varant of the ICP algorthm. Proceedngs of 3-D Dgtal Imagng and Modellng (3DIM). Sharp, G. C., Lee, S. W. and Wehe, D. K., ICP regstraton usng nvarant features. IEEE Transactons on Pattern Recognton and Machne Intellgence 24(1), pp Smon, D., Fast and Accurate Shape-Based Regstraton. PhD thess, Robotcs Insttute, Carnege Mellon Unversty. Zhang, Z., Iteratve pont matchng for regstraton of free-form curves and surfaces. Internatonal Journal of Computer Vson 13(2), pp