Distributed Bundle Adjustment

Transcription

1 Dstrbuted Bundle Adustment Karthkeyan Natesan Ramamurthy IBM Research Sharath Pankant IBM Research Chung-Chng Ln IBM Research Raphael Vguer Unversty of Mssour-Columba Aleksandr Aravkn Unversty of Washngton Abstract Most methods for Bundle Adustment (BA) n computer vson are ether centralzed or operate ncrementally. Ths leads to poor scalng and affects the qualty of soluton as the number of mages grows n large scale structure from moton (SfM). Furthermore, they cannot be used n scenaros where mage acquston and processng must be dstrbuted. We address ths problem wth a new dstrbuted BA algorthm. Our dstrbuted formulaton uses alternatng drecton method of multplers (ADMM), and, snce each processor sees only a small porton of the data, we show that robust formulatons mprove performance. We analyze convergence of the proposed algorthm, and llustrate numercal performance, accuracy of the parameter estmates, and scalablty of the dstrbuted mplementaton n the context of synthetc 3D datasets wth known camera poston and orentaton ground truth. The results are comparable to an alternate state-of-the-art centralzed bundle adustment algorthm on synthetc and real 3D reconstructon problems. The runtme of our mplementaton scales lnearly wth the number of observed ponts. 1. Introducton Estmatng accurate poses of cameras and locatons of 3D scene ponts from a collecton of mages obtaned by the cameras s a classc problem n computer vson, referred to as structure from moton (SfM). Optmzng for the camera parameters and scene ponts usng the correspondng ponts n mages, known as Bundle Adustment (BA), s an mportant component of SfM [7, 8, 21]. Many recent approaches for BA can be dvded nto three categores: those that pose BA as non-lnear least squares [10, 13, 21], those that decouple the problem n each camera usng a trangulaton-resecton procedure for estmaton [15, 18], and those that pose and solve BA n a lnear algebrac formulaton [6]. Some mportant consderatons of these methods are reducng the computatonal complexty by explotng the structure of the problem [1, 4, 13], ncorporatng robustness to outler observatons or correspondence msmatches [2, 26], dstrbutng the computatons or makng the algorthm ncremental [9, 11, 23, 24, 5] and makng the algorthm nsenstve to ntal condtons [6]. In ths paper, we develop robust dstrbuted BA over camera and scene ponts. Our approach s deally suted for applcatons where mage acquston and processng must be dstrbuted, such as n a network of unmanned aeral vehcles (UAVs). We assume that each UAV n the network has a camera and a processor; each camera acqures an mage of the 3D scene, and the processors n the dfferent UAVs cooperatvely estmate the 3D pont cloud from the mages. Therefore, we use the terms camera, processor, and UAV n an equvalent sense throughout the paper. We also assume that correspondng ponts from the mages are avalable (possbly estmated usng a dfferent dstrbuted algorthm), and are only concerned about estmatng the 3D scene ponts gven the correspondences. Robust approaches, such as [2, 26], are typcally used to protect world pont and camera parameter estmates from effects of outlers, whch for BA are ncorrect pont correspondences that have gone undetected. In contrast, we use robust formulatons to accelerate consensus n the dstrbuted formulaton. Dependng on how dstrbuton s acheved, every processor performng computaton may see only a small porton of the total data, and attempt to use t to nfer ts local parameters. Small sample means can be extreme, even when the orgnal sample s well-behaved (.e. even when re-proecton errors are truly Gaussan). In the lmtng case, each processor may only base ts computaton on one data pont, and therefore outlers are guaranteed to occur (from the pont of vew of ndvdual processors) as an artfact of dstrbutng the computaton. Hence we hypothe- 2146

2 sze that usng robust losses for penalzng re-proecton errors, and quadratc losses for enforcng consensus mproves performance. Fgure 1: Orgnal confguraton of cameras A, B and scene ponts 1,2,3,4,5, dstrbutng both the camera parameter and scene pont estmaton wth the constrants A1 = A2 = A3, B1 = B2 = B3, and 3A = 3B. Our proposed robust BA approach supports a natural dstrbuted parallel mplementaton. We dstrbute the world ponts and camera parameters as llustrated for a smple case of 2 cameras and 5 scene ponts n Fgure 1. The algorthm s developed usng dstrbuted alternatng drecton method of multplers (D-ADMM) [3]. Each processor updates ts copy of a set of parameters, whle the updated estmates and dual varables ensure consensus. Dstrbutng both the world ponts and the camera parameters yelds teratons wth O(l) requred operatons n a seral settng, where l s the total number of 2D observatons. In a fully parallel settng, t s possble to brng the tme complextes down to O(1) per teraton, a vast mprovement compared to tradtonal and sparse versons of BA, whose complextes are O((m+n) 3 ) and O(m 3 +mn) respectvely [13] (wth m and n the number of cameras and 3D scene ponts). We also explot the sparsty of the camera network, snce not all cameras observe all scene ponts. Another optmzaton-based dstrbuted approach for BA was recently proposed [5] 1. Authors of [5] dstrbuted camera parameters, and performed synthetc experments usng an exstng 3D pont cloud reconstructon, perturbng t usng moderate nose, and generatng mage ponts usng known camera models. We go further, dstrbutng both world ponts and camera parameters n a flexble manner, and we mplement the entre BA ppelne for 3D reconstructon: performng feature detecton, matchng correspondng ponts, and applyng the robust dstrbuted D-ADMM BA 1 The ntal verson of our method was proposed at the same tme as [5] technque n real data settngs. The rest of the paper s organzed as follows. We provde background n Secton 2, and present the new formulaton n Secton 3. We show experments and results on synthetc and real data n Secton 4, and conclude wth Secton Background 2.1. The camera magng process We denote themcamera parameter vectors by{y } m =1, the n 3D scene ponts as {x } n =1, and the 2D mage ponts as {z }. Each 2D mage pont z R 2 s obtaned by the transformaton and proecton of a 3D scene pont x R q by the camera y R p. BA s an nverse problem, where camera parameters and 3D world ponts are estmated from the observatons {z }. The forward model s a non-lnear camera transformaton functonf(x,y ). The number of mage ponts s typcally much smaller than mn, snce not all cameras mage all scene ponts. The camera parameter vector (y ) usually ncludes poston, Euler angles, and focal length. In ths dscusson, we assume focal length s known for smplcty, andy R 6 comprses Euler anglesα,β,γ and the translaton vectort R 3. Denote the dagonal focal length matrx as K R 3 3, wth the frst two dagonal elements set to the focal length and the last element set to 1. The rotaton matrx s represented as R = R 3 (γ)r 2 (β)r 1 (α), where R 1,R 2,R 3 are rotatons along the three axes of R 3. The camera transformaton s now gven as z = Rx + t. The fnal 2D mage pont z s obtaned by a perspectve proecton, wth co-ordnates gven by 2.2. Bundle adustment z 1 = z 1 z 3,z 2 = z 2 z 3. (1) Gven the 2D ponts n multple mages that represent the same scene pont, BA s typcally formulated as a nonlnear least squares problem: mn m {x },{y } =1 S() z, f(x,y ) 2 2. (2) The set S() contans f the scene pont s maged by the camera. The number of unknowns n ths obectve s 3n+6m, and hence t s necessary to have at least ths many observatons to obtan a good soluton; n practce the number of observatons s much larger. Problem (2) s solved teratvely, wth descent drecton (δ x,δ y ) found by replacng f n (2) by ts lnearzaton f(x+δ x,y +δ y ) f(x,y)+j(x)δ x +J(y)δ y, where J(x) = x f,j(y) = y f. The Levenberg- Marquardt (LM) algorthm [16] s often used for BA. 2147

3 The nave LM algorthm requres O((m + n) 3 ) operatons for each teraton, and memory on the order of O(mn(m + n)), snce we must nvert of an O(m + n) O(m + n) matrx at each teraton. However, explotng matrx structure and usng the Schur complement approach proposed n [13], the number of arthmetc operatons can be reduced to O(m 3 + mn), and memory use to O(mn). Further reducton can be acheved by explotng secondary sparse structure [10]. The conugate gradent approaches n [1, 4] can reduce the tme complexty too(m) per teraton, makng t essentally lnear n the number of cameras. Another popular approach to reduce the computatonal complexty nvolves decouplng of the optmzaton by explctly estmatng the scene pont usng back-proecton n the ntersecton step and estmatng the camera parameters n the resecton step [18]. The resecton step decouples nto m ndependent problems, and hence the overall procedure has a cost of O(m) per teraton. A smlar approach, but wth the mnmzaton ofl norm of the re-proecton error was proposed n [15]. It was shown to be more relable and degraded gracefully wth nose compared tol 2 based BA algorthms. Recently Wu proposed an ncremental approach for bundle adustment [23], where a partal BA or a full BA s performed after addng each camera and assocated scene ponts to the set of unknown parameters, agan wth a complexty ofo(m). We use the ADMM framework to develop our approach Alternatng Drecton Method of Multplers ADMM s a smple and powerful procedure well-suted for dstrbuted optmzaton [12], see also [3]. In order to understand D-ADMM, consder the obectve h(x) := n =1 h (x). We ntroduce local varables wth a consensus equalty constrant: mn {x },u =1 n h (x ) subect tox u = 0, {1,...,n}. To solve ths problem, we frst wrte down an augmented Lagrangan [19]: n l φ (x,u,r,ρ) := h (x )+r T (x u)+ ρ 2 φ(x,u), =1 where ρ > 0 s the penalty parameter, r s the Lagrangan multpler for the constrant, and φ(x,u) s the augmentaton term that measures the dstance ndvdual varables x and the consensus varable u. We then fnd a saddle pont usng three steps to update {x }, u, and {r }. Typcally φ(x,u) s chosen to the squared Eucldean dstance n whch case (4) becomes the proxmal Lagrangan [19], but other dstance or dvergence measures can also be used. (3) (4) 3. Algorthmc formulaton 3.1. Dstrbuted estmaton of scene ponts and camera parameters We dstrbute the estmaton among both the scene ponts and the camera parameters as llustrated n Fgure 1. We estmate the camera parameter and the scene pont correspondng to each mage pont ndependently, and then mpose approprate equalty constrants. Eqn. (2) can be wrtten as m mn φ m (z, f(x,y )), (5) {x },{y },{x},{y} =1 S() such thatx = x,, and { : S()}, (6) y = y,, and { S()}. (7) The augmented Lagrangan, wth dual varables r and s, s gven by m =1 S() φ m (z, f(x,y ))+r T (x x )+s T (y y ) +(ρ x /2)φ a (x x )+(ρ y /2)φ a (y y ) Here φ a measures the dstance between the dstrbuted world ponts and ther consensus estmates, and dstrbuted camera parameters and ther consensus estmates. For φ m we compare squared Eucldean and Huber losses, andφ a s always the squared Eucldean loss. The ADMM teraton s gven by (x (k+1) x (k+1),y (k+1) ) := argmn φ m (z, f(x,y )) {x },{y } +r (k)t (x x(k) )+s (k)t (y y (k) ) (8) +(ρ x /2)φ a (x x(k) )+(ρ y /2)φ a (y y (k) ), (9) 1 := : S() 1 := S() y (k+1) r (k+1) s (k+1) S() : S() ( y (k+1) := r (k) +ρ x ( x (k+1) := s (k) +ρ y ( y (k+1) ( x (k+1) ) +(1/ρ x )r (k), ) +(1/ρ y )s (k), x (k+1) y (k+1) (10) (11) ), (12) ). (13) The equaton (9) has to be solved for all S(), {1,...,m}, and t can be trvally dstrbuted across multple processes. Whenφ m s squaredl 2 dstance,(9) can be 2148

4 solved usng the Gauss-Newton method [17], where we repeatedly lnearze f around the current soluton and update (x,y). Whenφ m s the Huber loss, we use lmted memory BFGS (L-BFGS) [17] to update the dstrbuted scene ponts. Upon convergence, we wll obtan the consensus estmates x and y for all scene ponts and cameras Convergence Analyss We show that under certan assumptons the proposed D- ADMM algorthm n Secton 3.1 converges, usng the nonconvex and non-smooth framework developed by [22]. Theorem 1 The D-ADMM algorthm proposed n Secton 3.1 to the statonary pont of the augmented Lagrangan n 8 when: 1. f(.,.) s the perspectve camera proecton model, 2. φ m s any convex, smooth loss functon, and φ a s the squared Eucldean loss. 3. ρ x andρ y are suffcently large. Proof Let d be the stack of {x,y }, and ˆd = [ˆd ],. Smlarly each par of consensus varables are stacked as the vector ĉ = [x T yt ]T, and ĉ = [ĉ ],. ˆd and ĉ are respectvely equvalent toxandy n [22]. We show that the fve assumptons (A1-A5) of [22, Thm. 1] are satsfed. 1. Gven our assumptons, the obectve functon n (6) s coercve,.e., t tends to as ˆd (A1). 2. The feasblty and sub-mnmzaton path condtons are also satsfed snce the constrant matrces are easly seen to be full rank (A2-A3). 3. Each addtve part of the obectveφ m (z, f(x,y )) s restrcted prox-regular f φ m s a smooth convex functon and f s the perspectve camera model. The gradent wll be steep when z 3 n (1) s less than some ǫ > 0 and φ m (z, f(x,y )) s prox-regular for ǫ > 0; hence A4 n [22, Thm. 1] holds. 4. Our obectve wth respect to the consensus varable s dentcally 0, whch s trvally regular (A5). Snce all the assumptons hold, the teratve algorthm n eqns. (9)-(13) converges to a statonary pont of the augmented Lagrangan for suffcently largeρ x and ρ y Tme Complexty Optmzng (9) takes O(l) tme for each round of updates, snce (9) must be solved l tmes, wth each solve requrng constant tme. The tme complexty of the consensus steps for camera parameters and world ponts gven by (10) and (11) are O(m) and O(n) respectvely. For the Lagrangan parameter updates gven by (12) and (13), the tme complexty so(l). Hence the domnant tme complexty of the proposed algorthm s O(l) for each round. Snce the algorthm can be trvally parallelzed, the complexty can be brought down to O(1) for each round, f we dstrbute all the observatons to ndvdual processors Communcaton Overhead Consderng a sparse UAV network, assume that each world pont s maged by d cameras. Each camera needs to mantan a copy of the consensus world ponts x. Therefore to updatex usng (10), each camera needs to obtand 1 ndvdual estmates of x and send ts verson of x to d 1 other cameras. Values r can be updated locally n each camera, gvenx,x and prevous versons ofr usng (12). Hence, for each world pont we have a communcaton overhead of 3(d 1)d floatng ponts per teraton (each world pont s a 3D vector). Hence for n world ponts, the communcaton overhead s 3(d 1)dn floatng ponts per teraton, where d depends on the dstance of the camera from the scene Generalzed Dstrbuted Estmaton The problem (9) requres each processor to estmate p+q > 2 parameters from a sngle 2D observaton. To control the varablty of ndvdual estmates as the algorthm proceeds, we generalze the approach to use more than one observaton and hence more than one scene pont and camera vector durng each update step. Ths generalzed step provdes flexblty to adust the number of 3D scene ponts and cameras based on computatonal capablty of each thread n a CPU or a GPU. We solve (X (k+1),y (k+1) ) := argmn φ m (Z, f(x,y )) {X },{Y } +r (k)t (X X(k) )+s (k)t (Y Y (k) ) +(ρ x /2)φ a (X X(k) )+(ρ y /2)φ a (Y Y (k) ), (14) where [ X (k+1) := Y (k+1) := 4. Experments x (k) 1 x (k) 2 [ y (k) 1 y (k) 2... x (k) π ] T,... y (k) κ ] T (15) We perform several experments wth synthetc data and real data to show the convergence of the re-proecton error and the parameter estmates. We also compare the performance of the proposed approach a the centralzed BA algorthm that we mplemented usng LM. The LM stops 2149

5 when the re-proecton error drops below 10 14, or when the regularzaton parameter becomes greater than We mplement our dstrbuted approach n a sngle mult-core computer and not n a sparse UAV network, but our archtecture s well-suted for a networked UAV applcaton Synthetc Data We smulate a realstc scenaro, wth smooth camera pose transton, and nose parameters consstent wth realworld sensor errors. Usng the smulaton, we evaluate the error n the estmated 3D scene pont cloud and the camera parameters, and nvestgate how estmaton error of camera pose affects the fnal te ponts trangulaton. The camera postons are sampled around an orbt, wth an average radus 1000m and alttude 1500m, wth the camera drected towards a specfc area. To each camera pose, a random translaton and rotaton s added as any real observer cannot move n a perfect crcle whle steadly amng always n the same exact drecton. The camera path and the 3D scene ponts for an example scenaro are shown n Fgure 2. In practce, te ponts are usually vsble only wthn a small subset of the avalable vews, and t s generally not practcal to try to match all key ponts wthn each possble par of frames. Instead, ponts are matched wthn adacent frames. In our synthetc data, we create artfcal occlusons or ms-detecton so that each pont s only vsble on a few consecutve frames. We nvestgate convergence of the re-proecton error and parameters for D-ADMM BA, comparng the convergence when φ m s squared l 2 vs. Huber n (5), and φ a always the squared l 2. The number of cameras s 5, the number of scene ponts s 10, and the number of 2D mage ponts (observatons) s 50. We fx the standard devaton for the addtve Gaussan nose durng the ntalzaton of the camera angles and postons to be 0.1. We vary the standard devaton of nose for the scene ponts from 0.2 to 1.7. Introducng robust losses for msft penalty helps the convergence of the re-proecton error sgnfcantly, see Fgure 3, vs.. Ths behavor s observed wth the convergence of the scene ponts, see Fgures 3, vs. (d), and camera parameters. The Huber penalty s used to guard aganst outlers; here, outlers come from processors workng wth lmted nformaton. The performance degrades gracefully wth nose, see Fgure 3, and (d). Rep. errors: φ m n (5) sl 2 MSE: φ m n (5) sl 2 Rep. errors: φ m n (5) s Huber (d) MSE:φ m n (5) s Huber Fgure 3: Choosng φ m loss to be Huber penalty leads to better performance n dstrbuted BA, even when there are no outlers n the orgnal data. Panels and compare reproecton errors, whle and (d) compare MSE of scene ponts. In all fgures, curves correspond to values σ of scene varance, as shown n the legend. Consensus penalty φ a s alwaysl 2. Fgure 2: Camera flght path (blue) and 3D scene ponts (red) for an example synthetc data set Convergence and Runtme We also compare D-ADMM BA wth the centralzed LM BA and present the results n Fgure 4 and. The number of camera parameters and 3D scene ponts are (10,40), (15,100), (25,100), (30,200), (100,200), and (100,250); wth the number of observatons ncreasng as shown on the x-axs of Fgure 4. In most settngs, D-ADMM BA has a better parameter MSE than centralzed LM BA. The runtme of the proposed approach wth respect to the number of observatons and parallel workers s shown n Fgure 4. The parallel workers are confgured n MATLAB, and the runtme s lnear wth respect to the observatons and reduces wth ncreasng workers. Our mplementaton s a smple demonstraton of the capablty of the algorthm a fully parallel mplementaton n a fast language such as C can realze ts full potental. 2150

6 Fgure 4: MSE between the actual and estmated camera parameters, MSE between the actual and estmated scene ponts, runtme of the proposed D-ADMM algorthm wth ncreasng number of processor cores Real Data To demonstrate the performance of D-ADMM BA, we conducted experments on real datasets wth dfferent settngs. All experments are done wth MATLAB on a PC wth a 2.7 GHz CPU and 16 GB RAM. In our SFM ppelne, SIFT feature ponts [14] are used for detecton and matchng. The relatve fundamental matrces are estmated for each par of mages wth suffcent correspondng ponts, whch are used to estmate relatve camera pose and 3D structure. Next, the relatve parameters are used to generate the global ntal values for BA. The datasets were downloaded from the Prnceton Vson Group and the EPFL Computer Vson Lab [20]. Snce there are no ground truth 3D structures avalable for the real datasets, we compare the dense reconstructon results obtaned usng the method of [25]. The frst dataset has fve mages and a sample mage s shown n Fgure 5. After keypont detecton and matchng, centralzed LM BA and D-ADMM BA are gven the same nput. There are a total of 104 world ponts and 252 observatons. The fnal re-proecton error of LM and D-ADMM are 0.93 and 0.67 respectvely. Fgure 5 and (d) shows that the dense reconstructon qualty of LM and the D-ADMM are smlar. Fgure 5 shows the convergence of re-proecton error for the D-ADMM algorthm. Fgure 6 shows the convergence of re-proecton error for dfferent values of ρ = ρ x = ρ y. Settng ρ to a hgh value accelerates convergence. We also estmate camera parameters and scene ponts, applyng the approach of Secton to the same data set. Fgure 6 shows that as the number of scene ponts per teraton ncrease, the runtme decreases, wth 32 scene ponts per teraton gvng the fastest convergence, see fgure 6. Fgure 6 (d) compares re-proecton errors wth dfferent number of cameras n each teraton. Intal values are the same as n the castle-p30 experment (Fgure 11), and the number of scene ponts n each teraton s 64. Reproecton errors decrease faster as the number of cameras n each teraton ncreases. We perform dstrbuted BA on the Herz-Jesu dataset data set provded n [20] usng the approach n Secton Ths data set has seven mages, 1140 world ponts, and 2993 observatons. In ths experment, the LM BA algorthm usng the same settng as n prevous experments does not converge and has the fnal re-proecton error about Therefore, the dense reconstructon result s not presented. D-ADMM BA wth eght scene ponts n each update step has a fnal re-proecton error of Fgure 7 shows the dense 3D pont cloud estmated wth D-ADMM BA. Addtonal results on other datasets (fountan-p11, entry- P10, Herz-Jesu-P25, and castle-p30) are presented n Table 1, Fgure 8, 9, 10 and 11. σ s mean re-proecton error. Fgure 8, 9, 10 and 11 present dfferent perspectves of the dense reconstructon results to show the robustness of 3D parameter estmatons. Table 1: The dataset nformaton and experment results. Dataset Images Scene pts Obs σ fountan-p entry-p Herz-Jesu-P castle-p Settngs are fxed across experments, and the maxmum teraton counter s set to The experments on fountan-p11 and Herz-Jesu-P25 dataset (Fgure 8 and 10) have better dense reconstructon results snce there are more mages coverng the same regons. The real data experments show D-ADMM BA acheves smlar obectve values (mean re-proecton error < 1) as the number of observatons ncreases; t s not necessary to ncrease the number 2151

7 (d) Fgure 5: Orgnal 2D mage, re-proecton error for D-ADMM BA, dense 3D pont cloud estmated wth LM BA (mean re-proecton error = 0.93), (d) dense 3D pont cloud estmated usng D-ADMM BA (mean re-proecton error = 0.67). Fgure 7: Orgnal 2D mage, dense 3D pont cloud estmated wth D-ADMM BA (mean re-pro. error =0.76). Fgure 6: The re-proecton error for dfferent values of ρ, usng generalzed dstrbuton approach, runtme of D-ADMM BA re-proecton errors wth ncreasng number of scene ponts, (d) re-proecton errors for multple cameras per estmaton vector. (d) ntal data, robust losses are helpful because estmates of processors workng wth lmted nformaton can stray far from the aggregate estmates, see Fgure 3. Formulaton desgn for dstrbuted optmzaton may yeld further mprovements; ths s an nterestng drecton for future work. Results obtaned wth D-ADMM BA are comparable to those obtaned wth state-of-the-art centralzed LM BA, and D-ADMM BA scales lnearly n runtme wth respect to the number of observatons. Our approach s well-suted for use n a networked UAV system, where dstrbuted computaton s an essental requrement. of teratons as the sze of the data ncreases. D-ADMM BA scales lnearly wth the number of observatons and can be parallelzed on GPU clusters. 5. Conclusons We presented a new dstrbuton algorthm for bundle adustment, D-ADMM BA, whch compares well to centralzed approaches n terms of performance and scales well for SfM. Expermental results demonstrated the mportance of robust formulatons for mproved convergence n the dstrbuted settng. Even when there are no outlers n the 2152

8 Fgure 8: Reconstructed Fountan-P11 vews (11 mages, 1346 world ponts, 3859 obs., mean re-pro. error = 0.5). Fgure 9: Reconstructed Entry-P10 vews (10 mages, 1382 world ponts, 3687 obs., mean re-pro. error = 0.7). Fgure 10: Reconstructed Herz-Jesu-P25 vews (25 mages, 2161 world ponts, 5571 obs., mean re-pro. error = 0.87). Fgure 11: Castle-P30 (30 mages, 2383 world ponts, 6453 obs., mean re-pro. error = 0.84). 2153

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