A Probablstc Method for Algnng and Mergng Range Images wth Ansotropc Error Dstrbuton Ryusuke Sagawa Nanaho Osawa Yasush Yag The Insttute of Scentfc and Industral Research, Osaka Unversty 8-1 Mhogaoka, Ibarak, Osaka, 567-0047, JAPAN {sagawa,n-oosawa,yag}@am.sanken.osaka-u.ac.jp Abstract Ths paper descrbes a probablstc method of algnng and mergng range mages. We formulate these ssues as problems of estmatng the mamum lkelhood. By eamnng the error dstrbuton of a range fnder, we model t as a normal dstrbuton along the lne of sght. To algn range mages, our method estmates the parameters based on the Epectaton Mamzaton (EM) approach. By assumng the error model, the algorthm s mplemented as an etenson of the Iteratve Closest Pont (ICP) method. For mergng range mages, our method computes the sgned dstances by fndng the dstances of mamum lkelhood. Snce our proposed method uses multple correspondences for each verte of the range mages, errors after algnng and mergng range mages are less than those of earler methods that use one-to-one correspondences. Fnally, we tested and valdated the effcency of our method by smulaton and on real range mages. 1. Introducton Many researchers have studed modelng shapes of real world objects by scannng them usng three dmensonal dgtzers such as laser range fnders [5, 11, 3] and structured-lght range fnders [22]. For eample, a major subject of research s the modelng of cultural hertage objects [12, 9] because they are seen to be canddates havng the worthest shapes and appearances for modelng. 3D modelng of the shape of an object s accomplshed by followng three steps: 1. Acqurng the range mages (scannng). 2. Algnng the acqured range mages from dfferent vewponts (algnng). 3. Reconstructng a unfed 3D mesh model (mergng). In the frst step, a target object s observed from varous vewponts. In the second step, multple range mages are algned nto a common coordnate system usng regstraton algorthms that establsh pont correspondences and mnmze the total dstance between those ponts; usng, for eample, feature-based [24, 10], or Iteratve Closest Pont (ICP)-based [1, 29, 2, 14, 17, 18] methods. The thrd step s to merge multple pre-algned range mages. Several approaches have been proposed; for eample, mesh-based [25, 23] and volume-based [8, 4, 26, 19] methods. When algnng and mergng multple range mages t s necessary to fnd correspondng ponts between them. There s some varaton among the methods used. WthICPbased methods, the correspondng ponts are defned by the closest Eucldean dstance [1, 29], and the projecton of the source pont onto the destnaton mesh [2, 14, 17, 15]. If an Eucldean dstance s used, t s equvalent to assumng that the error dstrbuton of a range mage s sotropc. However, the error dstrbuton of an actual range fnder does not seem sotropc. If the dstance s computed by projectng the source pont onto the destnaton mesh, the error s dstrbuted on the lne of projecton. It seems plausble because the error dstrbuton of a range fnder s along the lne of sght, especally for a laser range fnder. However, as these methods fnd one-to-one correspondences, t s dffcult to fnd a correct correspondence when the measurement error of a range fnder s large, even f the ntal poston of range mages s a suffcently good guess. The methods for mergng range mages also fnd correspondng ponts by usng the closest Eucldean dstance [25, 8, 26, 19], and projectng a pont onto a mesh [4]. For the same reason as n the case of algnng range mages, t s dffcult to fnd a correct correspondence when the measurement error of a range fnder s large, and the assumpton of the error dstrbuton s far from the actual dstrbuton. Some approaches that model the error dstrbuton of the range measurement have been proposed. Okatan and Deguch[16] proposed a method to algn range mages wth an error model of range mages and modeled t as a normal dstrbuton along the lne of sght. However, they determnstcally computed the one-to-one correspondng ponts. Wllams and Bennamoun [28] proposed a proba- 1
blstc method to algn range mages. Though they modeled the error dstrbuton by a covarance matr, they also used determnstc correspondences. Sagawa et al. [20] proposed a method to remove the nose n range mages along the lne of sght, but they dd not smultaneously consder the error dstrbuton of multple range mages. If we assume that the error dstrbuton of a range fnder s along the lne of sght, the nose of each pel can be reduced by smple temporal averagng by obtanng multple range mages from the same vewpont. However, t s not applcable n the case that the sensor or object moves when acqurng those range mages. Our proposed method can be appled even f the sensor s movng, and only one range mage s obtaned for each vew drecton. In ths paper, we propose probablstc methods to algn and merge range mages. We regard algnng range mages as problems of fndng optmal parameters of mamum lkelhood. Our mergng method fnds a surface of mamum lkelhood durng the converson from range mages to a volumetrc representaton. Because we analyze the error dstrbuton of a laser range fnder, we smplfy the model for the measurement error and usng an ansotropc error model we propose algorthms for algnng and mergng range mages. In Secton 2, we formulate the problems as mamum lkelhood estmates. Then, we analyze the error dstrbuton and propose practcal algorthms n Secton 3. We evaluate our method n Secton 4 and fnally summarze ths paper n Secton 5. 2. Probablstc Method for Algnng and Mergng Range Images Ths secton descrbes a method to algn and merge range mages based on a probablstc framework. Frst, we defne a probablstc model of the error dstrbuton of a range mage that s obtaned by a range fnder. Net, we defne the problems wth the model of error dstrbuton as follows: Algnng range mages has the problem that estmates the parameters that have mamum lkelhood. Mergng range mages has the problem that fnds a surface that has a mamum lkelhood. 2.1. Probablstc Model of Error Dstrbuton n Range Measurement A range mage s a set of 3D ponts acqured by a range fnder and whch are connected f ther postons are close to each other [25]. Therefore, a range mage s represented as a mesh model, consstng of vertces and patches. Because of the nose n range measurement, a 3D pont n a range mage has an error. Fgure 1 shows an eample where there are a true surface S and a observed range mage A. s a pont of A. If s a measurement of a pont s of S, the probablty s represented as a posteror probablty P ( s). If a general model of error dstrbuton s assumed, can surface S range mage A s P( s) Fgure 1. A range mage A s observed by scannng atruesurfaces. A pont of A s a measurement of a pont s of S. The posteror probablty s P ( s). range mage B surface S range mage A s P(s B) P( S) Fgure 2. Two range mages A and B are two samples of observng a true surface S. If B s observed, the probablty of S becomes P (S B) = P (s B), s and the probablty that A s observed s P (A S) = P ( S) f S ests. becausedbyallpartsofs. Thus, we denote the posteror probablty as P ( S) and the probablty of a whole range mage A becomes P (A S) = P ( S). (1) 2.2. Algnng Range Images by Mamum Lkelhood Estmaton Fgure 2 shows a stuaton where there are two range mages A, B and a true surface S. Now, we consder the algnment of the range mage A to B by changng the parameters of rotaton R and by translaton t of the range mage A. In a probablstc framework, the parameter s computed by mamzng the probablty P (A B; θ), whereθ s a vector that represents the parameters R and t. P (A B; θ) s computed as follows: P (A B; θ) = P (A S; θ)p (S B)dS, (2) S Ω where Ω represents the scope of the shape of the true surface S. Usng the Epectaton Mamzaton (EM) approach, the parameter θ that mamzes P (A B; θ) s computed by estmatng the followng condtonal epectaton E[log P (S B) A] = P (S A; θ)logp (S B)dS. (3) S Ω 2
If S conssts of n ponts s 1, s 2,, s n, (3) becomes P (s j A; θ)log P (s B)ds 1 ds n s 1 s n j = P (s j A; θ) L(s B)ds 1 ds n (4) s 1 s n j where L(s B) s the log lkelhood log P (s S). Because s P (s A; θ) =1for all =1,,n, t s smplfed as E[log P (S B) A] = P (s A; θ)l(s B)ds. (5) s Snce an actual range fnder has a characterstc error dstrbuton, we propose an algorthm to compute and mamze (5) based on the error dstrbuton n Secton 3.2. 2.3. Mergng Range Images by Fndng Surface of Mamum Lkelhood To merge range mages, we compute a sgned dstance feld (SDF) as an ntermedate representaton. Snce t s an mplct representaton, the merged surface becomes a set of ponts that satsfes f() = 0,wheref() s a sgned dstance at. The mplct surface represented by a SDF s converted to a mesh model by the Marchng Cubes algorthm [13]. In ths secton, we propose a method to merge range mages by computng a sgned dstance f() of mamum lkelhood from multple range mages. Fgure 3 shows an eample where there are two range mages A, B, a true surface S and a pont. D () s the dstance from to a pont s of S. The probablty functon for D () d s computed by P (D () d) = P (s A, B; θ)ds, (6) s d where θ s the parameter of algnment. Thus, P (D() = d) s computed by takng average of all neghbor ponts as follows: P (D() =d) = N P (D() d δ) P (D() d+δ) lm δ 0 2δN, (7) where N s the number of neghbor ponts. Snce the magntude of a sgned dstance f() s computed as the dstance from to the nearest neghbor pont of the surface [19], then f s s the nearest neghbor pont, the other ponts, for eample s k n Fgure 3, are farther than s from. Snce the probablty P 1 (D() =d) that only one pont s wthn [d δ, d + δ) s lm δ 0 P (D() d δ) P (D() d + δ), (8) 2δ the probablty densty functon for f() = d s computed by P ( f() = d) =P (D() =d) P 1 (D() =d) (9) By fndng the mamum lkelhood of P ( f() = d), we determne the magntude of the sgned dstance. range mage B surface S s k D () k range mage A P(s A,B) s D () Fgure 3. An eample of mergng two range mages. D () s the dstance from to a pont s of a true surface S. The remanng ssue s to determne the sgn of f(). In [19], the sgn s determned by consderng the angle between p and the normal vector n at p, f the nearest pont s p. Namely, the sgn of f() becomes postve f (p ) n < 0, and negatve otherwse. In our method, f the magntude of f() s d, we determne the sgn by computng a weghted sum of the probablty: W (,d)= P (s p)sgn((p ) n)dpds, p s =d (10) where p s a pont of the range mages and sgn(a) s 1 f a<0and negatve otherwse. Therefore, the sgn of f() becomes sgn(w (,d)). In ths secton, we eplan our method for mergng just two range mages; however, t can easly be etended to cases of three or more range mages. 3. Algorthm wth Ansotropc Error Model Ths secton descrbes a practcal algorthm to algn and merge range mages. Snce t s dffcult to compute the probabltes descrbed n Secton 2 for general probablstc models, we frst analyze the error dstrbuton of an actual laser range fnder and smplfy the model of error dstrbuton. Net, we propose algorthms based on the model. 3.1. Smplfed Model of Ansotropc Error Dstrbuton Here, we used a Canesta DP200 [3] as the laser range fnder to analyze the error dstrbuton. As a prelmnary eperment, we evaluated the error dstrbuton of the sensor by observng a small object. Snce t has a lght source and a camera, t obtans a 2D range mage by observng the reflected lght. The poston and sze of the small object s adjusted so that t s projected wthn a pel of the mage. Fgure 4 shows the results of the measurements. Fgure 4(a) s the hstogram of the number of measurements for each pel along the horzontal as of the mage. Snce most of the measurements are projected onto only one pel, 3
Number of measurements Number of measurements 7000 6000 5000 4000 3000 2000 1000 0 1400 1200 1000 800 600 400 200 0 0 10 20 30 40 50 60 X as (pel) (a) 1245 1252 1259 1266 1273 1280 1287 1294 1301 1308 1315 1321 1328 1335 1342 1349 1355 Dstance (mm) (b) Fgure 4. Hstograms of measurng a small object: (a) along the horzontal as of a range mage, (b) along the lne of sght. t can be assumed that a pel s not affected by other pels, whch s reasonable f an object s n focus. It s equally reasonable to assume that the error ests only along the vew drecton. Fgure 4(b) s the hstogram of the number of measurements for a pel along the lne of sght. Snce the dstrbuton s smlar to a normal dstrbuton, we assume that t s a normal dstrbuton N(d, σ 2 ),whered s the dstance to the true surface and σ s the standard devaton. Snce [21] analyzed for the standard devaton of the error, we use the result. Consequently, snce we assume that the error only ests along the vew drecton, the model of error dstrbuton s smplfed to a 1D normal dstrbuton. Though ths error model s valdated only for the Canesta DP200, t can be appled to other range fnders, especally laser range fnders. 3.2. Algnng Range Images Fgure 5 shows a stuaton where there are two range mages A, B to be algned and an assumed true surface S. If we assume the model of error dstrbuton descrbed n Secton 3.1, a verte of the range mage A s a measurement of s of the surface S, whch s on the same lne of sght of the range mage A. The probablty of becomes P ( s )= 1 ep( s 2 2πσ 2σ 2 ). (11) Meanwhle, the measurement of s n the range mage B s y, whch s on the same lne of sght of B. The probablty of y s smlar to (11). Snce and y only depend on s, range mage B surface lne of sght of B y S range mage A P(y s ) s P( s ) lne of sght of A Fgure 5. Correspondence ponts between range mages A, B and an assumed true surface S. s s on the lnes of sght of both and y. (5) becomes E[log P (S B) A] = s P (s ; θ)l(s y )ds. (12) In the mamzaton step of the EM algorthm, the estmated parameter ˆθ becomes by Bayes s rule ˆθ = arg θ ma P (s ; θ)l(s y )ds s = arg θ ma P ( s ; θ)l(y s )ds,(13) s where we drop P (s ),P( ) and P (y ) because they are constant wth respect to the parameters f we assume that there s no constrant on the shape of S. To resolve (13), we compute the weghted sum of several samples of s nstead of computng the ntegral of s as shown n Fgure 6. Snce the error dstrbuton s a normal dstrbuton, t becomes ˆθ =arg θ mn j P (s,j ) y,j s,j 2, (14) where s,j sasampleofs and y,j s the correspondng pont of s,j. The dstrbuton of s,j s gven by user. P (s,j ) s computed by 1 2 erf(t j + t j+1 2 2σ ) erf(t j + t j 1 2 ), (15) 2σ where erf(z) = 2 z π 0 e t2 dt, ands j = + t j v (v s the unt vector along the lne of sght). Therefore, the EM algorthm teratvely mnmzes the dstance of the correspondng ponts. Ths s smlar to the ICP-based methods, especally [14, 15], whch fnd correspondng ponts along the vew drecton. However, snce they do not consder the error dstrbuton, they use only the correspondence between and y 3 n the case of Fgure 6. Thus, our algorthm s the same as [15] for mnmzng an energy functon, but t uses a dfferent energy functon (14). 4
y,4 y,3 y,5 y,2 range mage A y,1 range mage B v s,1 s,2 s,3 s,4 s,5 Probablty 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 P(D()=d_k) P_1(D()=d_k) P b k 0 0.077246 0.081246 0.085246 0.089246 0.093246 Dstance d_k 0.002 0.0018 0.0016 0.0014 0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0 Probablty P_b_k Fgure 6. Computng the weght sums of samples of s,j nstead of computng the ntegral of s. 3.3. Mergng Range Images The net step s to merge range mages. To fnd the mamum of the probablty of P ( f() = d), we create a hstogram nstead of computng the lmt of the probablty epressed n (9). Our algorthm to compute the sgned dstance f() s as follows: 1. Fnd several neghbor ponts of range mages whose dstances from are smaller than d mn +3σ, where d mn s the dstance to the nearest neghbor pont and σ s the standard devaton of measurement. 2. Compute P (D () d k ) for k =1, 2,...n for each found neghbor pont, where the bn of a hstogram s defned between d k and d k+1. 3. Compute the sum and product of the probablty functon for all neghbor ponts: P (D() =d k )= {P (D() d k+1) P (D() d k )} P 1(D() =d k )= P (D() d k+1) P (D() d k) 4. Fnd the bn b k that has the mamum value of P bk = P (D() =d k ) P 1 (D() =d k ). 5. Interpolate the dstance f() of the mamum probablty by fttng a quadratc functon to P bk 1,P bk and P bk+1. 6. The sgn s determned by computng (10) for the bn b k. To fnd the neghbor ponts, we use a k-d tree [7] to reduce the cost wth a small modfcaton to fnd ponts whose dstances are wthn d mn +3σ. Snce other ponts of the range mages farther than d mn +3σ do not contrbute to the probablty P (D() =d k ), we omt them from the computaton. The wdth of a bn s defned by the user because t s a trade-off between cost and accuracy. Snce our algorthm s based on [19], the poston of s determned by an octree; for detals, please refer to [19]. Fgure 7 shows an eample Fgure 7. An eample of the probabltes P (D() = d k ), P 1(D() =d k ) and P bk accordng to the dstance d k. lne of sght D () = dk v p' φ p range mage Fgure 8. Our method fnds the nearest pont p for each patch. The standard devaton of d k s appromated by σ = σcosφ. of the probabltes P (D() =d k ), P 1 (D() =d k ) and P bk accordng to the dstance d k. The remanng ssue s how to compute P (D () d k ). As shown n Fgure 8, we fnd the nearest pont p for each patch of range mages from. A patch s usually a trangle that connects adjacent vertces. Though the pont s of the true surface ests along the lne of sght, snce the nearest pont p s nterpolated nsde of the patch, the dstrbuton of the nearest pont becomes smaller. In ths paper, we smplfy the computaton of p by appromatng the standard devaton of d k by σ = σcosφ. Therefore, P (D () d k ) s computed by P (D () d k )= 1 t (1 erf( 2 ) (16) 2σ where p =(1+t)(p ). 4. Eperments We evaluate our method usng synthetc data and real range mages. Frst, we compare our algnng and mergng methods wth prevous methods by usng synthetc range mages, to whch noses are added along the lne of sght. Second, we show that our method successfully creates a model of an object from nosy range mages. s 5
Table 1. Robustness of the estmated parameters. Our method Prevous method [15] X-std. 0.051 0.080 Z-std. 0.0025 0.0063 RMS 0.121 0.146 Dr. 5.81 7.29 -as vew drectons Fgure 9. Three eamples of slces of synthetc range mages n the z-plane. Ther shape and vew drectons are ndcated by colors. The orgnal mesh model s a thck black lne. The nterval of grds s 0.02. 4.1. Comparson by usng Synthetc Range Images We created one hundred synthetc range mages by addng nose of normal dstrbuton to a mesh model from ten dfferent vewponts. Fgure 9 shows an orgnal mesh model and three eamples of slces of synthetc range mages n the z-plane. The orgnal model conssts of two planes, whch are perpendcular to the z-plane. In ths case, the standard devaton of nose s 0.05 whle the wdth of a orgnal mesh model s 1.0 and the nterval between the vertces s 0.025. The mamum dfference between vewng angles s 90 degrees. 4.1.1 Algnng Range Images We estmate the robustness of the algnment wth respect to the ntal parameters by comparng our method wth one of the prevous methods [15], whch fnds correspondng ponts along the algn of sght of one of range mages. Snce the orgnal mesh model conssts of two planes, t has ambguty f we algn the models. Thus, we estmate the error of the algnment results by comparng the dfference of translaton along the - and z-aes and the vew drecton from the ground truth. We algn two range mages startng wth varous ntal postons. The offsets from the ground truth are unformly dstrbuted wthn [ 0.25, 0.25] along each as. Table 1 shows the robustness of the estmated parameters. X-std. and Z-std. are the standard devatons of translaton along the - and z-aes. RMS s the root-mean-square dstance from the ground truth. Dr. s the dfference of the vewng drectons from the ground truth n degrees. Snce our method has smaller standard devaton and errors than [15], our method can estmate the parameters robustly wth respect to the varaton of the ntal parameters. 4.1.2 Mergng Range Images To test the proposed mergng algorthm, we merge two range mages that are acqured from dfferent vewponts. The postons of the nput range mages are at the ground truth. Fgure 10 shows two eamples of the results of mergng by the proposed method (red lne) and a prevous -as Fgure 10. The result of mergng 10 range mages. The red lne ndcates the results of the proposed method and the blue lne ndcates the results of the prevous method [19]. method [19] (blue lne), whch uses the average Eucldean dstance to the nearest ponts. To estmate the error of the merged mesh model, we compute the offset from the orgnal model along the z-as of the vertces. The RMS error of the nput range mages s 0.0402. We test ten tmes for each algorthm; the RMS error of the result of our method becomes 0.0268, whle that of the prevous method s 0.0293. Our method obtans a better result than the prevous method and reduced the nose to 66% of the nput range mages. 4.2. Algnng and Mergng Real Range Images Net, we algn and merge real range mages captured by the Canesta DP200, whch obtans range mages at 30 frames/sec, but s less accurate than other slower laser range fnders. The standard devaton s 8mm along the lne of sght f an object s at about 1m from the sensor. We sequentally captured 72 range mages of an object shown n Fgure 11 from varous drectons by rotatng t on a turntable. Fgure 12 shows one of the range mages. From the top vew, we can see that the range mage contans large nose compared to the object sze, whch s about 420mm wde. We sequentally algned the range mages by parwse regstraton. The mages are algned automatcally by settng the ntal parameter of rotaton and translaton to be the same as the algned result of the prevous frame. Fgure 13 shows the estmated sensor poston and vewng drecton. Because the result of our method (red lne) becomes an accurate crcle, our method successfully algned the range mages whle the result of the prevous method [15] (blue lne) collapsed durng regstraton. 6
0-200 -400 Fgure 11. A stuffed bear on a bo. z-as (mm) -600-800 -1000-1200 -1400-1600 -1800-1000 -500 0 500 1000 -as (mm) (a) Front vew (b) Top vew Fgure 13. The estmated sensor poston and vewng drecton by our proposed method (red) and the prevous method [15] (blue). Fgure 12. A range mage of the stuffed bear contans large nose compared to the object sze. Fgure 15(a) shows the results of algnment, and Fgure 14(a) s a horzontal slce of the algned range mages. Though the shape of the object can be recognzed from them, t s represented as a cloud of ponts or meshes. By applyng our mergng method to the range mages, we construct a SDF. In Fgure 14(b) the red lnes are a slce of the mesh model etracted from the SDF. By comparng the merged model by the prevous method [19] (blue lnes), our method reduces the nose n the merged model. Fgure 15(b) s the result of mergng range mages. Snce our method does not assume any pror model of the object, the mesh model s not smooth. After smoothng the SDF by a mean flter, a smooth mesh model s etracted as shown n Fgure 15(c). 5. Concluson Ths paper proposed a probablstc method of algnng and mergng range mages. We formulated the ssues nvolved as mamum lkelhood estmates. By eamnng the error dstrbuton of a range fnder, we modeled t as a normal dstrbuton along the lne of sght. For algnng range mages our method estmates the parameters based on the EM approach. By assumng an error model, the algorthm s mplemented as an etenson of an ICP-based method. For mergng range mages, our method computes the sgned dstances by fndng the dstances of mamum lkelhood, and successfully reduced the sensor nose. Snce our proposed method uses multple correspondences for each verte of range mages, the errors after algnng and mergng range mages are less than those of earler methods that use one-to-one correspondences. In ths paper, we do not assume any pror model of the shape of a true surface. Thus, the resultng surface of mergng may not be smooth, as was (a) (b) Fgure 14. Horzontal slces of models: (a) algned range mages, (b) merged models by the prevous method [19] (blue) and the proposed method(red). shown n the results. Earler models have also been ntroduced durng the reconstructon of 3D models [27, 6]. In future work we ntend to ncorporate a pror model of the surface nto our framework to generate a smooth surface. References [1] P. Besl and N. McKay. A method for regstraton of 3-d shapes. IEEE Trans. Patt. Anal. Machne Intell., 14(2):239 256, Feb 1992. [2] G. Blas and M. Levne. Regsterng multvew range data to create 3d computer objects. IEEE Transactons on Pattern Analyss and Machne Intellgence, 17(8):820 824, 1995. [3] Canesta, Inc. CanestaVson EP Development Kt. http://www.canesta.com/devkt.htm. [4] B. Curless and M. Levoy. A volumetrc method for buldng comple models from range mages. In Proc. SIG- GRAPH 96, pages 303 312. ACM, 1996. [5] Cyra Technologes, Inc. Cyra 2500. http://www.cyra.com. [6] H. Dnh, G. Slabaugh, and G. Turk. Reconstructng surfaces usng ansotropc bass functons. In Proc. Internatonal Conference on Computer Vson, pages 606 613, Vancouver, Canada, July 2001. 7
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