A NEW IMPLEMENTATION OF THE ICP ALGORITHM FOR 3D SURFACE REGISTRATION USING A COMPREHENSIVE LOOK UP MATRIX

A NEW IMPLEMENTATION OF THE ICP ALGORITHM FOR 3D SURFACE REGISTRATION USING A COMPREHENSIVE LOOK UP MATRIX A. Almhde, C. Léger, M. Derche 2 and R. Lédée Laboratory of Electroncs, Sgnals and Images (LESI), Polytech Orléans, Unversty of Orléans, France 2 rue de Blos, 45067, Orléans, France 2 Electrcal Engneerng Department, Kng Fahd Unversty, Dhahran, Saud Araba phone: + (33) 238494563, fax: + (33) 23847377, emals: {ahmad.almhde, chrstophe.leger, roger.ledee}@unv-orleans.fr and mderche@kfupm.edu.sa web: http://www.unv-orleans.fr/les ABSTRACT The teratve closest pont algorthm s one of the most effcent algorthms for robust rgd regstraton of 3D data. It conssts n fndng the closest ponts between two sets of data whch are then used to estmate the parameters of the global rgd transformaton to regster the two data sets. All the steps of the algorthm are hghly dependent upon the accuracy wth whch correspondence pars of ponts are found. In ths paper, a new enhanced mplementaton of the ICP algorthm proposes to use a look up matrx for fndng the best correspondence pars. It results n reducng the mnmum mean square error between the two data sets after regstraton, compared to exstng mplementatons. The algorthm was mplemented and tested to evaluate ts convergence propertes and robustness to nose. Performance mprovements are obtaned. The new algorthm has successfully been appled to regster 3D medcal data.. INTRODUCTION The regstraton of 3D data sets s an mportant task n both Computer Vson and Photogrammetry, especally n the medcal feld. A detaled overvew of mage regstraton technques can be found n [], and dfferent methods proposed for medcal mage regstraton have been dscussed n [2, 3]. The teratve closest pont (ICP) algorthm, orgnally proposed by Besl and McKey [4], s one of the most popular methods used for estmatng the rgd transformaton of roughly algned 3D data sets. It s wdely used for the regstraton of free-form surfaces where dense data s assumed, and a good ntal estmate s avalable or can easly be obtaned. Addtonally, all selected scene ponts from the scene surface are assumed to have correspondences n the reference surface. The frst step of the ICP algorthm conssts n choosng correspondng (closest) ponts n the two 3D data sets. Snce the accuracy of the search for correspondence ponts hghly affects the estmaton of the transformaton parameters, the output of the frst step has a major mpact over the followng stages and strongly affects the overall performance of the algorthm. Ths step strongly depends upon both the selecton of the ponts of the two surfaces, and the method used for fndng the correspondence of the selected ponts. The Orgnal ICP algorthm [4], denoted OICP n ths paper, searches for the closest pont n the reference surface for each pont n the scene surface wthout any restrctons. Wdespread nterest n 3D surface regstraton usng the OICP algorthm has motvated the scentfc communty to propose other technques to mprove the dfferent steps of the orgnal algorthm. Many varants have been developed for speedng up the convergence and/or mprovng the performance of the dfferent phases of the algorthm [5]. A good revew of these varants can be found n [6]. There has been sgnfcant nterest regardng the selecton of ponts used for the estmaton of transformaton parameters. In [7], the Trmmed ICP selects only a predefned number of estmated matched pars for the calculaton of the optmal moton. The projecton of the scene ponts onto the reference surface can also be used for the correspondence search [8]. The Pcky ICP (PICP) [9] rejects all ponts prevously estmated to correspond to one reference pont, except the one wth the smallest dstance. Ths approach reduces convergence problems that may arse usng the orgnal ICP algorthm, when a common reference pont s assgned to multple ponts n the scene surface. However, ths affects negatvely the performance of the algorthm n nosy stuatons, snce many ponts are dscarded n the estmaton step. Ths paper focuses on the correspondence search of 3D surface ponts. The am s to enhance the performance of the correspondence search step of the OICP algorthm. The use of a new comprehensve look up matrx s nvestgated and evaluated. The proposed CICP (C for comprehensve) algorthm ensures unque matches of correspondence pars. The rest of the paper s organzed as follows. Frst, the orgnal OICP algorthm s summarzed. Next, the new CICP algorthm s descrbed and performance analyses detals are gven. Results of comparsons based on both synthetc and medcal data are then presented to show the performance mprovement of the CICP algorthm. Fnally some concludng remarks are gven. 2. OVERVIEW OF THE OICP ALGORITHM Assume that the gven two surfaces to be regstered can be descrbed as pont sets; the scene data ponts, P, wth N p

ponts, {p, =,, N p }, and the reference data ponts, M, wth N m ponts, {m, =,, N m }. Dependng on the accuracy of the constructed surfaces, N p s not necessarly equal to N m. Furthermore, the pont p of the scene surface does not necessarly represent 3D correspondence to the pont m of the reference surface. The search space, however, s determned by the sze of the scene data set;.e., N p. The OICP algorthm can be summarzed as follows: A. Intalzaton: ) Let the ntal scene surface P 0, be equal to P. 2) Defne the maxmum number of teratons k max. 3) Intalze the translaton vector and the rotaton matrx as follows: 0 0 0 T = 0 and R = 0 0. () 0 0 0 Ths corresponds to zero translaton and no rotaton. B. Iteratons: ) For each pont p ( =,..., N p ) of the scene P, compute the closest pont y M from the model usng the Eucldan dstance. Let y be the pont on M correspondng to the mnmum dstance p. 2) Usng the selected correspondence pars, compute the transformaton, rotaton (R) and translaton (T), that mnmses the mean square error (MSE) of these pars: MSE = N N p p = y R( p ) T. (2) The resultng transformaton from the mnmzaton of the above equaton wll be denoted R k and T k. Ths step also provdes the mnmum dstances whch correspond to the matched pars. 3) Transform P accordng to (R, T) and restart a new teraton f the change n the MSE s above a predefned threshold ζ, and f the maxmum number of teratons k max s not reached. If not, stop the teratons and ext. 3. THE PROPOSED CICP ALGORITHM In prevous varants of the OICP algorthm, the search procedures of correspondng pars of ponts are based on a lneby-lne (vector) search wthn a P-M dstance matrx descrbed n Table, where d,j s the dstance between p and m j. Duplcate matches may hence occur, snce multple m (columns) can be assgned to dfferent p (lnes). The PICP varant ensures unque matches by rejectng all duplcate pars, except the one wth the smallest dstance. Ths can be descrbed as a lne-by-lne followed by a column-by-column search wthn the P-M dstance matrx. Unfortunately, ths may lead to the excluson of good markers from the estmaton procedure. To overcome ths drawback, a more comprehensve search s needed. A novel effectve evaluaton metrc s ntroduced for correspondence search, called comprehensve lookup matrx measure. Ths measure ensures that every selected pont on the scene surface has a unque match n the reference surface. 2 m m 2 m Nm p d, d,2 d,nm p 2 d 2, d 2,2 d 2,Nm : p Np d Np, d Np,2 d Np,Nm Table the P-M dstance matrx. The CICP s dfferent n that t sorts the d,j, dstances n ascendng order wthn the entre P-M dstance matrx. Moreover, the pont m j s not consdered to be a correspondence to p f ether m j or p has been prevously assgned a correspondence. Ths ensures that each pont n the scene surface wll have a dfferent assocaton n the reference surface. The CICP s the only ICP algorthm that makes use of all scene ponts n the search procedure to fnd the best and unque correspondence pars. In other words, the P-M dstance matrx s ntroduced to comply wth the fact that a rotaton s a bjectve (one to one) functon. Prevous ICP mplementatons are based on vector not matrx analyss of the assgnment problem whch can yeld surjecton correspondences, ncompatble wth correct estmatons of rotaton parameters. In ths case, some elements n M may be mapped by more than one element n P. When the number of ponts n the two sets to be regstered s not the same, the CICP algorthm consders the one wth a smaller number of ponts as a scene data set to ensure bjectvty of the resultng correspondence pars. To reduce computaton tme ntroduced by matrx search procedure, fast assgnment algorthms can be used. The CICP algorthm can be summarzed as follows: ) For each pont p P, (=,..., N p ), the algorthm computes the Eucldan dstance to each pont m j M, (j=,..., N m ). Then, for N p tmes, the algorthm: a. looks for the locaton (,j) that corresponds to the mnmum dstance n the current look up matrx, b. assgns p to m j as a correspondence par, c. removes ths correspondence par from future consderaton by elmnatng the th row and j th column. 2) Usng the estmated correspondence pars, the algorthm computes the transformaton parameters (R, T), and transforms P accordngly. 3) The algorthm computes the MSE between the reference and transformed scene data sets. If the change n the MSE s above a predefned threshold and k max s not reached, the algorthm starts a new teraton. If not, the teratve procedure s stopped. Instead of leavng the decson of rejectng worse pars tll the end of every teraton as n [9], the CICP algorthm makes such a decson at the end of every selecton of par correspondence. Such an approach mproves the accuracy and the convergence of the ICP algorthms, as shown n the followng sectons. 4. PERFORMANCE ANALYSIS In ths paper, the new CICP algorthm wll be compared to the OICP and PICP as benchmarks. The robustness of the CICP algorthm wll be studed under the presence of nose, wth both synthetc and real medcal data.

4. Nose generaton Real data are usually corrupted by nose caused by a wde range of sources, e.g. detector varatons, envronmental varatons, transmsson or quantzaton errors, etc. Here, the performance of the selected ICP algorthms s nvestgated under the effect of Gaussan and mpulsve nose. 4.. Gaussan nose Gaussan nose s added to the orgnal data accordng to the followng method: ) Transform all ponts of the data set from Cartesan to sphercal coordnates: p(x, y, z ) p(θ, φ, ρ ) = {,, N m }, (3) 2) Add nose to each resultng ρ = {,, N m } : SNR _ db / 20 ρ = ρ + ρ - µ ρ 0 rand(.), (4) N = m where µ ρ ρ, rand(.) s a random Gaussan number generator wth mean zero and varance one, and SNR_dB s the requred sgnal to nose rato n db. 3) Transform ponts back to Cartesan coordnates. 4..2 Impulsve nose Impulsve nose s commonly referred to as outlers. In ths case, a set of h% of the data ponts s assumed to be corrupted by mpulsve nose. To generate outlers, replace step 2 of Gaussan nose generaton by: 2a) Randomly select h% of the total data ponts. 2b) Modfy each selected pont: ρ j j= {,..., h N m /00} ρ j = ρ j (+ β ) (5) where β represents the dstrbuton of the outlers relatve to the ponts of the orgnal data set. 4.2 Performance parameters To compare the performance characterstcs of the CICP algorthm to OICP and PICP algorthms, two parameters are taken nto consderaton: the percentage of correct matches and the mean square error (MSE) between the regstered data sets. 4.2. Percentage of correct matches Correct matches analyss can be carred out when exact orentaton of the data sets s known. Under such a hypothess, the percentage of correct assocatons of correspondng ponts from the scene and reference surfaces s counted at each step of the algorthm. Ths measurement s a straghtforward ndcator of the performance of the correspondence par search method. 4.2.2 Mean Square Error The convergence property of the algorthm can be estmated by computng the mean square error (MSE) between the reference and the regstered data sets, at each teraton of the algorthm. When orentaton s known for both regstered surfaces, MSE s gven by computng the mean square dstances between correspondence pars of ponts. Otherwse, 3D to 2D mesh nterpolaton can be ntegrated, and MSE can be calculated wth the correspondng non-zero elements of the resultng 2D meshes 5. SIMULATED SCENE SURFACE RESULTS The CICP algorthm was tested under a nose-free stuaton as well as wth Gaussan nose (wth SNR of 5 db, 0 db, 5 db and 20 db) and outlers (wth percentage of outlers of 5 %, 0%, 5 % or 20 % of the data set) condtons. The convergence propertes and the accuracy of the proposed CICP algorthm have been evaluated usng two types of pont sets: synthetc and real medcal data sets. 5. Regstraton of synthetc data In the frst experment, the Rabbt synthetc data set was used as a reference, as shown n Fgure a. Fgure b shows an example of nose-free scene data, rotated -29, -4 and 8 around the x-axs, y-axs and z-axs, respectvely. The number of reference and scene data ponts s the same,.e., 000 ponts. The OICP, PICP and CICP algorthms were tested to regster the reference surface (Fgure a) wth the scene surface (Fgure b), corrupted wth Gaussan and outler nose. Fgure Bunny data. a) Reference surface. b) Scene surface (rotaton, nose free). 5.. Correct matches evaluaton Fgure 2 and Fgure 3 show the percentage of correct matches at each teraton of the OICP, PICP and CICP algorthms. It can be seen that the CICP gves the hghest number of correct matches, n fewer teratons, compared to both OICP and PICP algorthms, for most nose stuaton tests. 5..2 MSE comparson Fgure 4 and Fgure 5 show the evoluton of the mean square error computed at each step of the OICP, PICP and CICP algorthms. Under 5dB Gaussan nose data corrupton, the CICP algorthm reaches mnmum MSE faster (4 teratons) than the OICP and PICP algorthms (respectvely 7 and 37 teratons). Smlar results are obtaned for other amounts of Gaussan nose, as well as for nose-free scene data. In the case of outlers, the CICP algorthm reaches mnmum MSE faster than for the OICP and PICP algorthms. Moreover, the change n outler rato slghtly affects the number of teratons of the CICP, compared to OICP and PICP algorthms. To evaluate the repeatablty of the results presented above wth partcular scene data rotaton, the same experments were carred out on more than twenty randomly selected rotatons, wthn 30 around each x, y and z axs. In all cases, the CICP algorthm showed mprovements compared to the OCIP and PICP algorthms. Specal thanks go to Mr A. Ajmal, at The Unversty of Western Australa, for provdng us wth the data.

5.2 Regstraton of medcal data The second experment consders a set of 922 ponts of real data of human Lung 2 as a reference surface (Fgure 6a). The scene data set s smulated by rotatng the reference scene data of -29, -4 and 8 around the x-axs, y-axs and z-axs, respectvely (Fgure 6b). In the case presented, the number of ponts n the reference and scene data sets s equal. Fgure 2 Accuracy performance n the case of Gaussan nose. Fgure 6: Lung data. a) Reference lung data. b) Scene lung data (rotaton, nose-free). The MSE between the reference and regstered surfaces s measured at each teraton of the OICP, PICP and CICP algorthms, consderng scene surface degraded wth Gaussan nose (Fgures 7) and outlers (Fgure 8). Fgure 7 confrms the performance enhancement of the CICP algorthm, whch reaches the mnmum MSE faster than the OICP and PICP algorthms. Fgure 8 shows smlar results wth outlers. Fgure 3 Accuracy performance n the case of outlers. Fgure 7: Convergence comparson n the case of Gaussan nose. Fgure 4 Convergence comparson n the case of Gaussan nose. Fgure 8: Convergence comparson n the case of outlers. Fgure 5 Convergence comparson n the presence of outlers. 2 Specal thanks go to Dr Fabenne Théran, Chef du servce de Médecne Nucléare, Centre Hosptaler Régonal d Orléans, France.

6. MEDICAL APPLICATION In ths experment, we consder the regstraton of the real lung data of Fgure 6a, to a lung atlas presented n Fgure 9a. The atlas data set conssts of 50 ponts, whereas a set of 922 ponts were unformly selected from the scene lung data. The moton parameters that best algn the lung data set wth the atlas are estmated usng the three OICP, PICP and CICP algorthms. The correspondng regstered data are dsplayed n Fgure 9. For the experment carred out, the MSE threshold ζ between the reference surface and the regstered scene surface s set to a low value (0 3 ), and the maxmum number of teratons (k max ) s not lmted. Ths ensures that the estmaton of the moton parameters s correct. The algorthm stops when the estmated moton parameters are constant wthn several teratons, and MSE between the reference and regstered surfaces s computed. Wth lung atlas and lung data, the CICP, PICP and OICP algorthms acheve convergence n 3, 35 and 84 teratons, respectvely. In Fgure 9, when stablty of moton parameters s reached, the CICP algorthm (Fgure 9d) seems to gve a better regstraton of the two data surfaces (Fgure 9a), compared to the OICP (Fgure 9b) and PICP (Fgure 9c) algorthms. Ths qualtatve result was confrmed by an expert n the feld of medcal magng. Further experments wll be conducted to quantfy precsely these prelmnary results. (c) Fgure 9: a) Intal vews: lung atlas (reference surface, black) and lung data (scene surface, red). Regstered data usng OICP, (c) PICP, (d) CICP algorthms. (d) 7. CONCLUSION In ths work, a novel enhanced mplementaton of the ICP algorthm s presented. The use of the complete look-up dstance matrx durng the pont assocaton procedure guarantees that unque matches are obtaned for all ponts from the scene data. The substtuton of a vector by a matrx based search of correspondence pars ensures correct rgd regstraton, n agreement wth the bjectve property of the rotaton. Computng tme expanson due to the swtch from vector to matrx search can be lmted by usng many valuable technques used to solve assgnment problems. Compared to other ICP mplementatons, the proposed CICP algorthm provdes: a faster convergence, n terms of number of teratons, a more precse estmaton of par of ponts correspondence, and a better reslence to addtve Gaussan nose and outlers. The accuracy of the proposed CICP has been nvestgated and very promsng results have been shown for 3D real medcal data regstraton. REFERENCES [] B. Ztová and J. Flusser, "Image regstraton methods: a survey," Image and Vson Computng, vol. 2, pp. 977-000, 2003. [2] J. B. Mantz and M. A. Vergever, "A survey of medcal mage regstraton," Medcal Image Analyss, vol. 2, pp. -36, 998. [3] M. A. Audette, F. P. Ferre, and T. M. Peters, "An algorthmc overvew of surface regstraton technques for medcal magng," Medcal Image Analyss, vol. 4, pp. 20-7, 2000. [4] P. J. Besl and H. D. McKay, "A method for regstraton of 3-D shapes," IEEE Transactons on Pattern Analyss and Machne Intellgence, vol. 4, pp. 239-256, 992. [5] S. Rusnkewcz and M. Levoy, "Effcent varants of the ICP algorthm," Proceedngs of the Thrd Internatonal Conference on 3-D Dgtal Imagng and Modelng, Quebec Cty, Quebec, Canada, 200. [6] G. C. Sharp, S. W. Lee, and D. K. Wehe, "ICP regstraton usng nvarant features," IEEE Transactons on Pattern Analyss and Machne Intellgence, vol. 24, pp. 90-02, 2002. [7] D. Chetverkov, D. Svrko, D. Stepanov, and P. Krsek, "The Trmmed Iteratve Closest Pont algorthm," presented at 6th Internatonal Conference on Pattern Recognton (ICPR'02) 2002. [8] B. Kwang-Ho and L. D. D., "Automated Regstraton of Unorgansed Pont Clouds From Terrestral Laser Scanners," presented at XX th ISPRS Congress, Istanbul, Turkey, 2004. [9] T. Znsser, J. Schmdt, and H. Nemann, "A refned ICP algorthm for robust 3-D correspondence estmaton," Proceedngs of IEEE Internatonal Conference on Image Processng Barcelona, Span, 2003.