Model-Based Pose Estimation by Consensus

Transcription

1 Model-Based Pose Estmaton by Consensus Anne Jorstad 1, Phlppe Burlna, I-Jeng Wang, Denns Lucarell, Danel DeMenthon 1 Appled Mathematcs and Scentfc Computaton, Unversty of Maryland College Park Johns Hopkns Appled Physcs Lab Abstract We present a system for determnng a consensus estmate of the pose of an object, as seen from multple cameras n a dstrbuted network. The cameras are ponted towards a 3D object defned by a confguraton of ponts, whch are assumed to be vsble and detected n all camera mages. The cameras are gven a model defnng the 3D confguraton of these object ponts, but do not know whch mage pont corresponds to whch object pont. Each camera estmates the pose of the object, then teratvely exchanges nformaton wth ts neghbors to arrve at a common decson of the pose over the network. We consder eght varatons of the consensus algorthm, and fnd that each converges to a more accurate result than do the ndvdual cameras alone on average. The method exchangng 3D world coordnates penalzed to agree wth the nput model provdes the most accurate results. If bandwdth s lmted, performng consensus over rotatons and translatons requres cameras to exchange only the sx values specfyng the sx degrees of freedom of the object pose, and performng consensus n SE(3) usng the Karcher mean s generally the best choce. We show further that nterleavng pose calculaton wth the consensus teratons mproves the fnal result when the mage nose s large. 1. INTRODUCTION We address the problem of recoverng the pose (poston and orentaton) of an object as vewed by a camera sensor network. It s assumed that the object fts entrely wthn the feld of vew of each camera, and each camera s gven a model of the object. Each camera s able to detect all model ponts n ts mage, but the correspondence between the mage ponts and the model ponts s not known. Each camera determnes ts own estmate of the object pose usng the SoftPost algorthm [], whch determnes the best ft of a known model to a set of feature ponts n an mage, and returns a rotaton matrx and translaton vector from the camera orgn. The cameras then share local pose estmates wth ther neghborng cameras n an teratve fashon, and nformaton s dspersed through the network usng varatons of the consensus algorthm [13]. We consder two classes of consensus approaches: those that pass the 3D world coordnates of the object between the cameras, and those that pass the rotaton matrces and translaton vectors themselves and perform consensus n the space of rotatons. We also look at the effect of nterleavng pose estmaton wth the consensus teratons. Consensus s reached when all cameras agree on a sngle pose, and the results of Monte Carlo smulatons for each consensus varaton are presented. We see that each consensus method converges to a more accurate result than do the ndvdual cameras alone on average, and performng consensus over penalzed world coordnates and usng the Karcher mean n SE(3) offer the best overall results. The dea of nodes n a network teratvely sharng nformaton wth ther neghbors to arrve at a common estmate has been addressed by several message passng algorthms ncludng the consensus algorthm [13]. Snce ts ntroducton, consensus has been appled and extended n several ways. The tradtonal consensus algorthm performs averagng n Eucldean spaces, but we also want to consder the space of rotatons, and an extenson usng a specal defnton of the mean on a Remannan manfolds was used n [17]. An alternate approach [18] was developed for performng consensus on the 3D specal orthogonal group, based on Karcher averagng [10] appled to pose estmaton. Reference [6] looked at the problem of network self-localzaton usng cameras and magnetometers n a dstrbuted fashon. A Kalman flter was cast nto a consensus algorthm n [1] [14] where each sensor estmate s based on ts own and ts neghbors nformaton. A dstrbuted optmzaton method was descrbed n [16] whch can also potentally be used for robust estmaton. Alternate message passng methods such as belef propagaton have also been wdely used for dstrbuted fuson [4] [5]. Computer vson research has made sgnfcant progress n estmatng pose nformaton n a scene observed by a set of cameras [7] [9]. Ths problem has seen the development of many algorthms, most of whch make the assumpton that a central computng entty has access to all cameras nput. Instead, we consder the same problem for a dstrbuted camera sensor network where connectvty between camera nodes s lmted, and there s no centralzed processng unt to collect data from each node and perform computatons. Each camera s responsble for ts own computaton (feature detecton and pose estmaton) and must then share and fuse data wth ts neghbors to collectvely arrve at a common estmate of the scene pose nformaton. Ths type of algorthm s useful for a number of applcatons, ncludng space exploraton (many robots collaboratng to repar a spacecraft), dsaster recovery (robots performng reconnassance and damage assessment after an earthquake), and vdeo survellance /08/$ IEEE 569 ISSNIP 008 Authorzed lcensed use lmted to: Unversty of Maryland College Park. Downloaded on August 0,010 at 17:00:33 UTC from IEEE Xplore. Restrctons apply.

2 . CAMERA AND NETWORK CONFIGURATION The camera network consdered here conssts of eght cameras ponted at a known object, see Fg. 1. The object fts completely wthn the feld of vew of each camera, and each camera s gven a model of the object as collecton of 3D ponts. All object ponts are vsble to all cameras, and no clutter ponts are added. The SoftPost algorthm used to fnd the ntal pose estmaton performs well when these assumptons are relaxed, but addng such extra unknowns complcates the comparson of camera network consensus algorthms. Each camera s able to convert ts pose estmaton to a common reference frame. Fg. 1: Vrtual setup of 8 cameras lookng at object wth 3 model ponts. Three dfferent network topologes were consdered. Dependng on the amount of connectvty between the cameras, the system wll reach consensus at dfferent rates, and the values at convergence wll dffer. Consensus theory tells us that λ, the second smallest egenvalue of the graph Laplacan L, sthe algebrac connectvty of the graph, and a larger λ wll lead to faster convergence [13]. The system wll converge as long as the network s connected. In the frst topology consdered, each camera s connected to two neghbors, formng one sngle graph cycle, and λ =0.5858, see Fg. (a). In the second, three cameras act as hubs, connected to every other camera, whle the other fve cameras are only connected to the hubs, λ =3, Fg. (b). In the thrd, the network s fully connected, λ =8,Fg.(c). (a) Sngle cycle (b) Three hubs (c) Fully connected Fg. : The three network topologes consdered. 3. INITIAL POSE ACQUISITION The SoftPost algorthm [] takes n a model of an object, specfed by a set of 3D object ponts, and a set of potental feature ponts n an mage, and determnes the best mageto-model feature correspondence and pose of the object that fts a subset of the feature ponts. A rotaton matrx R and a translaton vector T from the camera orgn are returned, along wth a correspondence matrx C matchng model ponts to feature ponts n the mage. Ths smultaneous pose and correspondence problem s formulated as a mnmzaton of a global objectve functon: N M E = c jk (d jk α) (1) j=1 k=1 where there are M model ponts, N mage ponts, c jk are elements of the correspondence matrx, d jk s the dstance between the observed mage pont j and the potental projected model pont k, and α s the error allowed n matchng ponts, whch depends on σ, the assumed standard devaton of the mage nose. The tradtonal Post algorthm [3] assumes that pont correspondences between the mage and model are known, and uses a scheme based on updatng scaled orthographc projectons to teratvely solve for the pose. A sngle Post step uses the homogeneous coordnate transformaton equaton to estmate the homogeneous scale from the current rotaton matrx and translaton vector, then uses ths scale to obtan a better estmate for the rotaton and translaton. The full SoftPost algorthm terates between Post, to determne pose, and determnstc annealng, to determne correspondences. Gven an ntal guess for pose, the algorthm determnes the dstances d jk whch are then used to predct the best correspondence c jk wth respect to a slowly ncreasng annealng varable β: c jk = exp [ β(d jk α)]. () Gven the current correspondences, a new predcton for the pose s found usng Post, whch then allows a new set of dstance values to be calculated. 4. CONSENSUS ALGORITHMS Once each camera has obtaned ts own predcton of the pose of the object, these estmates must be combned to generate a system-wde pose consensus. In a dstrbuted network, there s no centralzed processor. Instead, each camera can only communcate wth ts mmedate neghbors, and nformaton s propagated through the nodes. In the tradtonal consensus algorthm over N nodes [13], consensus s reached when all nodes agree on a sngle value, whch wll be equal to the average of all values n the network f the connectons between all nodes are equally weghted. To teratvely update each node n the network, the followng update scheme can be appled: + ɛ x (k+1) = x (k) (x (k) j j N x (k) ), (3) where k s the teraton number, ɛ s the step sze, N s the ndex set of the neghbors of node x, and the algorthm s ntalzed wth nodes x (0). To guarantee convergence, the postve step sze ɛ must be less than 1/Δ, where Δ s the maxmum degree of any node n the network [13]. It can be seen that ths update s a gradent descent method, mnmzng the followng objectve functon: φ(x 1,..., x N )= 1 a j (x x j ), (4),j 570 Authorzed lcensed use lmted to: Unversty of Maryland College Park. Downloaded on August 0,010 at 17:00:33 UTC from IEEE Xplore. Restrctons apply.

3 where a j are elements of the adjacency matrx defnng the network connectvty. The mnmum s attaned when all nodes agree on one consensus value. We study several varatons of the consensus algorthm for pose estmaton. Two classes of consensus schemes are consdered: consensus on 3D world coordnates, and consensus over rotatons. In the next sectons we use the followng notaton: M = number of model ponts; N = number of cameras; R = rotaton matrx from camera to the object; T = translaton vector from camera to the object; X,m = estmated 3D world coordnates of object pont m from camera, n a common reference frame; P m = actual 3D world coordnates of object pont m of the true soluton, n a common reference frame wth orgn P 0 ; Q m = nput 3D coordnates of model pont m, n a reference frame parallel to the common frame wth orgn Q 0. A. Consensus on World Coordnates 1) Unpenalzed Consensus on World Coordnates: The object model can be rotated and translated by the pose computed from SoftPost to provde 3D world coordnates for every model pont of the ntal estmate from each camera. Ths consensus algorthm performs ts averagng over these coordnates drectly, as n (3). The update equaton s: X (k+1),m = X (k),m + ɛ j N ( X (k) j,m X (k),m ), (5) where the algorthm s ntalzed by the SoftPost output ponts X (0),m. Consensus s reached when the dfference between estmates of neghborng connected cameras s suffcently small. ) Penalzed Consensus on World Coordnates Usng the Model: The frst method (5) makes no guarantee that the fnal soluton s consstent wth the object model. To rectfy ths, we add to the mnmzaton equaton (4) a term whch penalzes any dscrepances wth the model, weghted by penalty constant γ [0, 1]. The penalty s measured n relaton to model pont Q 0, the orgn of the model coordnate system. Ths s the pont that the Post algorthm uses to construct the translaton from the camera to the object, and s generally the most accurate pont of the SoftPost estmaton. The objectve functon wth regard to object pont m becomes: φ( X 1,m,..., X N,m )= 1 a j X,m X j,m (6),j + γ 1 ( X,m X,0 ) ( Q m Q 0 ). If γ =0, we have the orgnal consensus as n (5), and as γ gets larger the consensus result conforms more to the model. If γ =1, the consensus algorthm s equvalent to performng consensus over the locatons of the model reference ponts X,0 alone, and then smply addng n the rest of the nput model as defned by the nternal model vectors Q m. The penalty constant γ s therefore a measure of the user s confdence n the model. Convergence wll be reached for all γ [0, 1], as both endponts follow the tradtonal consensus algorthm, and all values n between result n a quadratc programmng problem. In ths study, γ = Applyng the gradent descend method to (6), the penalzed update equaton s: X (k+1),m B. Consensus over Rotatons = X (k),m + ɛ j N ( X (k) j,m X (k),m ) (7) ɛγ[( X (k),m X (k),0 ) ( Q m Q 0 )]. Instead of averagng the 3D world space coordnates, the consensus can take place over the rotatons and translatons themselves. A rotaton and a translaton each have three degrees of freedom, so performng consensus teratons over the rotatons and translatons conssts of passng sx arguments, nstead of the 3D coordnates of M model ponts, and n general, 6 3M. It can be shown [11] that rotatons and translatons can be averaged separately to acheve the correct fnal result. Each element of the translaton vector can be averaged just lke the world coordnates n (5): (k+1) T = T (k) + ɛ j T (k) ). (8) j N ( T (k) Elements of a rotaton matrx cannot be averaged so drectly. Dong so does not produce meanngful values, and generally does not return a vald rotaton matrx. Instead, rotaton calculatons must be completed n an approprate, non- Eucldean space. 1) Consensus n Axs-Angle Representaton: Any rotaton nduced by a rotaton matrx R can be represented by an angle θ and a normalzed axs of rotaton u [15]: ( ) trace(r) 1 θ = cos 1, (9) u = 1 R 3, R,3 R 1,3 R 3,1 sn(θ) R,1 R 1,, (10) and from Rodrgues rotaton formula: 0 u(3) u() R = u(3) 0 u(1) sn(θ)+[i u u T ]cos(θ)+ u u T. u() u(1) 0 (11) In ths sense rotatons can be summed as follows, usng scaled rotaton vectors θ u : N u sum = θ u, (1) =1 θ ave = 1 N u sum, (13) 571 Authorzed lcensed use lmted to: Unversty of Maryland College Park. Downloaded on August 0,010 at 17:00:33 UTC from IEEE Xplore. Restrctons apply.

4 1 u ave = θ ave N u sum. (14) The consensus update step then becomes: (θ u ) (k+1) =(θ u ) (k) + ɛ j N (θ j u j ) (k) (15) where θ and u at consensus can be determned by normalzng the resultng vector, and these can be converted back to a tradtonal rotaton matrx usng equaton (11). Ths method produces an exact barycentrc average n the space of rotatons, but t cannot take nto account the contnuous equvalence of θ across 0 = π, so boundary condtons skew the result away from the true average [15]. ) Consensus n SE(3) usng the Karcher Mean: In order to calculate the true mean of a set of rotatons n the manfold of rotaton matrces, known as SO(3), we consder the Remannan dstance between rotatons [1]. Ths dstance s the length of the shortest geodesc curve connectng rotatons whch les entrely wthn SO(3). The element of SO(3) whch mnmzes the sum of squared dstances between tself and all members of a set of rotatons s called the Karcher mean [10] of the set. SO(3) s a Le group, and so(3), the space of skew symmetrc matrces, s ts correspondng Le algebra. Ths means that a metrc can be defned over elements n each, and the Karcher mean can be calculated wth the help of these algebrac constructs. The space of rotaton matrces n SO(3) plus 3D vectors n R 3 s referred to jontly as SE(3). Elements n so(3) can be consdered as a matrx, ŵ, or as a vector, w: ŵ = 0 w 3 w w 3 0 w 1, (16) w w 1 0 w = w 1 w. (17) w 3 The log() functon converts elements of SO(3) to so(3). Wth θ defned n (9), log(r) s equal to u n (10): log(r) = u. (18) The exp() functon converts elements of so(3) to SO(3): θ = w, (19) exp(w) =I +ŵsn(θ)+ŵ (1 cos(θ)). (0) It s straghtforward to verfy that exp(w) n (0) s equvalent to the defnton for R n (11), gven w =1.But here the consensus updates wll be performed on the manfold. As n [18], one verson of the consensus update s: R (k+1) = R (k) exp ɛ [ ] (R T ) (k) R (k) j, (1) j N log but the result of these teratons s just an approxmaton to the Karcher mean. To calculate the true mean, we use the output of the above teratons to start each camera s approxmaton wth the same ntal value that s close to the true average. The second set of teratons use the nput R (0) j : R (k+1) = R (k) exp 1 N j N log [ (R T ) (k) R (0) j ]. () These teratons converge to the true Karcher mean n the manfold. Consensus n both cases s reached when the sum of the terms nsde the exp() s suffcently small. Remark: Consensus n Quaternons: Takng a drect lnear average of quaternons s fundamentally an approxmaton [8]. Although the approxmaton s good for small angles, the naccuracy can be sgnfcant for large angles. Our experments demonstrated that consensus n quaternons s sgnfcantly less accurate than all other methods presented here, and quaternons wll not be dscussed further. C. Interleavng Post Iteratons Dependng on the nput and constrants mposed on SoftPost, the pose or correspondences returned mght not be close to the true soluton. Such an ncorrect soluton negatvely nfluences the consensus average, and the pose of these solutons should be further modfed durng the consensus teratons. Ths s acheved by nsertng a Post step between each consensus step. Post s the part of the SoftPost algorthm that adjusts the pose, assumng pont correspondences are known. For each camera, the Post teraton realgns the pose to better match the object pont locatons from the orgnal nput mage, to locally rectfy some of the error generated by the consensus process. Algorthm: 1) SoftPost on each camera R (0),T (0) ) Iterate for each camera: a) Consensus: update from neghbors R (ka),t (ka) b) Post: update own estmate from mage R (k b),t (k b) Interleavng Post steps wthn the consensus algorthm helps rectfy faulty pose estmates, as the ncomng nformaton from other cameras can pull a bad estmate out of a local mnmum and encourage t to converge to a more globally accurate soluton. However, as correspondences are not changed, ths addton has only lmted effectveness. In fact, we wll see that addng Post teratons s only a good dea when the amount of nose n the orgnal mages s large. 5. EXPERIMENTS A vrtual system was constructed n whch a known object wth 3 model ponts was vewed by eght arbtrarly placed cameras between three and seven tmes the sze of the object away from the object, see Fg. 1. Each camera s calbrated, facng the object, and the object fts completely wthn each 57 Authorzed lcensed use lmted to: Unversty of Maryland College Park. Downloaded on August 0,010 at 17:00:33 UTC from IEEE Xplore. Restrctons apply.

5 camera s feld of vew. For each camera, mean zero random nose wth standard devaton σ =, 4, 8, 1 was added, and 50 SoftPost trals were run for each case, wth the output then beng fed nto each consensus algorthm. At frst, each camera was allowed to communcate wth only two neghbors, formng a sngle cycle network topology, as n Fg. (a). Each of the descrbed consensus methods was run on ts own, and agan nterleavng Post steps. Trals were repeated wth the dstances from the cameras to the object multpled by three. All errors are computed as the dfference between the coordnates of each estmated object pont at consensus, X m, and the true coordnates n 3D world space, P m. We compare the average error and the maxmum error for each class of trals: E ave = 1 M X m P m, (3) M m=1 E max =max m X m P m. (4) As a baselne, we consder the average error from an ndvdual camera, calculated by takng the average and maxmum of the above errors over all ponts from each camera s SoftPost output, and referred to n the plots as Drect Averagng: E DrectAve = 1 MN M m=1 =1 N X,m P m. (5) The consensus result wll return the average error of each data pont over all cameras, whch s n general smaller than (5): E ConsensusAve = 1 M 1 N X,m P m. (6) M N m=1 =1 The results for the closer cameras are presented n Fg. 3. Smlar results were obtaned for the further cameras, but wth larger overall errors as expected. In Fg. 4, the errors from all methods are plotted together for comparson. We see that all consensus methods result n less error on average than the drect average error from the ndvdual cameras. Performng consensus usng penalzed world coordnates produces the lowest error on average for every class of trals. However, the results are not very dstnct from the other methods when the mage nose level s low or the cameras are close to the object, especally when consderng the overlap of the dstrbutons of errors. Addng Post steps appears to be a bad dea when there s not much mage nose, but a better dea as the nose ncreases. In the trals wth large nose standard devaton, the translaton output from SoftPost s reasonable, but the rotatons are often qute far off, so addng Post teratons benefts some of these cases. If communcatons between cameras have lmted bandwdth, then t s desrable to pass as lttle nformaton as possble between the cameras. Usng penalzed world coordnates, 3M values must be exchanged for a model wth M model ponts. However, usng ether of the methods that perform consensus over rotatons, only 6 values need to be exchanged (a 3D translaton vector plus 3 angles of rotaton). Consensus (a) std of mage nose = (b) std of mage nose = 4 (c) std of mage nose = 8 (d) std of mage nose = 1 Fg. 3: The average errors (left plot) and maxmum errors (rght plot) n pxels are presented for each nose standard devaton, wth an error bar denotng the 1st quartle, medan, 3rd quartle, wth cameras placed 3 to 7 tmes the sze of the object away from the object. Fg. 4: Average and maxmum errors from each consensus method plotted together, wth cameras placed 3 to 7 tmes the sze of the object away. 573 Authorzed lcensed use lmted to: Unversty of Maryland College Park. Downloaded on August 0,010 at 17:00:33 UTC from IEEE Xplore. Restrctons apply.

6 (a) std of mage nose = 4 (b) std of mage nose = 8 Fg. 5: For each network topology, each consensus method s plotted aganst the average number of teratons (left plot), and the average errors (rght plot), for two mage nose standard devatons. The consensus methods are ordered along the x-axs as defned n Fg. 3. n SE(3) usng the Karcher mean s seen to perform slghtly better on average than the Axs-Angle method. When comparng the three topologes from Fg., all show smlar errors, but the fully connected network wth the largest λ converges n the fewest number of teratons, whle the sngle cycle network wth sgnfcantly fewer edges converges the slowest. The error between the true soluton and the fnal result from the sngle cycle s on average slghtly less than the others for the nonstandard consensus methods. See Fg. 5 for results. The bottleneck of the proposed algorthm s the SoftPost step, whch s really preprocessng for the consensus methods. Each consensus method takes very lttle tme to converge, see Table 1. Addng Post steps takes approxmately 30% longer to converge. Performng consensus over rotatons s faster than over world coordnates, because only sx data ponts need to be exchanged at each step. TABLE 1: AVERAGE TIMES TO CONVERGENCE FOR EACH METHOD WITH NO POSIT STEPS, AS RUN IN MATLAB ON A DESKTOP PC. World Coordnates Penalzed WCs Axs-Angle SE(3) Tme (seconds) CONCLUSIONS We have presented a set of algorthms for determnng the pose of a known object as seen from several cameras n a decentralzed network. All consensus methods presented converge to a more accurate result than do the ndvdual cameras alone on average. Performng consensus over world coordnates penalzed by an object model was found to result n the lowest overall error, but ths method requres exchangng three values for every model pont at every teraton, whch can be slow and requre sgnfcant bandwdth. If the system s bandwdth-lmted, t s nstead desrable to perform consensus over the rotaton and translaton of the object n a common reference frame, as ths requres exchangng only sx values n total. We see that performng consensus n SE(3) usng the Karcher mean produces good results n ths lmted bandwdth settng. The average error of ths method s comparable to that of the penalzed world coordnates method for low and mdrange mage qualtes, and ths method performs the fastest out of all tested. As the nose level of the nput mages ncreases, t s benefcal to add Post steps whch reference the orgnal nput mages between consensus steps, to help rectfy some of the error returned from the orgnal SoftPost pose estmate. The number of teratons requred for convergence can be decreased by ncreasng the connectvty of the camera network. ACKNOWLEDGMENTS We wsh to thank R. Tron, R. Vdal and A. Terzs for kndly makng ther code avalable as descrbed n [18]. REFERENCES [1] R. Carl, A. Chuso, L. Schenato, S. Zamper. 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