Efficient, General Point Cloud Registration with Kernel Feature Maps

Transcription

1 Effcent, General Pont Cloud Regstraton wth Kernel Feature Maps Hanchen Xong, Sandor Szedmak, Justus Pater Insttute of Computer Scence Unversty of Innsbruck 30 May 2013 Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

2 Outlne 1 Background 2 Rgd Transformaton n Hlbert Space 3 Rgd Transformaton n R 3 4 Experment Results 5 Concluson Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

3 Background 1. Background Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

4 Problem statement Background 3D pont cloud regstraton Gven two pont clouds X 1 = {x (1) } l 1 =1, X 2 = {x (2) j } l 2 j=1, fnd the correct correspondences between x (1) and x (2) j, based on whch two pont clouds can be algned. Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

5 Background Related Work Regstraton Iteraton Closest Pont (ICP); - match nearest neghbours as correspondences mnmze the dstances between correspondences Gaussan Mxture; - ft pont clouds to dstrbutons + correlaton, L2 dstance or kernel methods SoftAssgn / EM-ICP - one-to-many correspondences - optmze w.r.t. correspondence matrx optmze w.r.t. transformaton. Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

6 Related Work, cont. Background {R, b } = arg mn R,b l 1 l 2 =1 j=1 where R, b denote rotaton and translaton n R 3. ( ) Rx (1) + b x (2) 2w,j j (1) - ICP: w,j {0, 1}, determned by shortest-dstance crteron; - Guassan Mxtures: and w,j = 1 l 1 l 2 for all, j (unformly); - SoftAssgn/EM-ICP: w,j s nterpreted as the probablty of the correspondence; Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

7 Rgd Transformaton n Hlbert Space 2. Transformaton n Hlbert Space Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

8 Rgd Transformaton n Hlbert Space Kernel Method & Feature Map By applyng a kernel functon on 3D ponts K(x, x j ), a R 3 H feature map φ s mplctly nduced: K(x, x j ) = φ(x ), φ(x j ) (2) and H s called Hlbert space, whch s usually much hgher or possbly nfnte dmensonal: K(x, x j ) = e x x j 2 /2σ 2 φ(x ) f (ξ) = e ξ x 2 /2σ 2 (3) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

9 Rgd Transformaton n Hlbert Space Gaussan n Hlbert Space mean: µ (1) H = 1 l 1 µ (2) H = 1 l 2 l 1 =1 l 2 =1 φ(x (1) ) (4) φ(x (2) ) (5) covarance : C (1) H = 1 l 1 C (2) H = 1 l 2 l 1 =1 l 2 =1 ( ) ( ) φ(x (1) ) µ (1) H φ(x (1) ) µ (1) H (6) ( ) ( ) φ(x (2) ) µ (2) H φ(x (2) ) µ (2) H (7) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

10 Kernel PCA Rgd Transformaton n Hlbert Space Assume all ponts are already centralzed: C = 1 l l φ(x )φ(x ) (8) =1 the none-zero egenvalue λ k and correspondng egenvector u k of C should satsfy: λ k u k = Cu k (9) by substtutng (8) nto (9), we can have: u k = 1 λ k Cu k = l α k φ(x ) (10) =1 where α k = φ(x ) u k λ k l. Therefore, all egenvectors u k wth λ k 0 must le n the span of φ(x 1 ), φ(x 2 ),...φ(x l ), and (10) s referred to as the dual form of u k. Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

11 Kernel PCA, cont. Rgd Transformaton n Hlbert Space By left multplyng l j=1 φ(x j) on both sdes of equaton (9) : l j=1 φ(x j) λ k u k = l j=1 φ(x j) Cu k λ k l,j=1 αk φ(x ), φ(x j ) = 1 l l,j=1 αk φ(x ), φ(x j ) 2 λ k l,j=1 αk K(x, x j ) = 1 l l,j=1 αk K(x, x j ) 2 (11) lλ }{{} k α k = Kα k η k t can be seen that {η k, α k } s actually an egenvalue-egenvector par of matrx K. In ths way, the egenvector decomposton of blnear form C n H can be transformed to the decomposton of matrx K. Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

12 Kernel PCA, cont. Rgd Transformaton n Hlbert Space (a) (b) (c) (d) Fgure: (a) A pont cloud of table tenns racket; (b d) reconstructon usng the frst 1 3 prncpal components. For each pont n the boundng-box volume, the darkness s proportonal to the densty of the Gaussan n the feature space H. Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

13 Un-centralzed Case Rgd Transformaton n Hlbert Space u k = l =1 αk (φ(x ) µ) = [ l =1 αk φ(x ) 1 ] l l m=1 φ(x m) = φ(m) (I l 1 l 1 (12) l1 l ) α k }{{} I E (where φ(m) = [φ(x 1 ), φ(x 2 ),, φ(x l )], 1 l s a l dmenson vector wth all entry equal 1, I l s l l dentty matrx, α k = [ α1 k, αk 2,, ] αk l ) u k u h = 0, k h (13) u k 1 = φ(m 1 ) I E 1 α k (14) u k 2 = φ(m 2 ) I E 2 α k (15) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

14 Rgd Transformaton n Hlbert Space Rotaton n Hlbert Space Only D egenvectors are used to represent the covarance of hgh dmenson Gaussan dstrbuton of each pont cloud:: [ ] U 1 = u 1 1,, u k 1,, u D 1 [ ] U 2 = u 1 2,, u k 2,, u D 2 Algn U 1 wth U 2 : U 2 = R H U 1 U 2 = R H U 1 R H = U 2 U 1 = D U 2 U 1 = R HU 1 U 1 k=1 uk 2 uk 1 = φ(m 2 ) I E 2 ( D k=1 α k 2α k 1 ) I E 1 } {{ } γ φ(m 1 ) (16) (17) (18) (19) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

15 Rgd Transformaton n Hlbert Space Translaton n Hlbert Space f M 1 has already been centered,.e. µ (1) H = 0 b H = µ (2) H = 1 l 2 φ(m 2 ) 1 l2 (20) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

16 Rgd Transformaton n R 3 3.Rgd Transformaton n R 3 Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

17 Consstency Rgd Transformaton n R 3 consstency error: {R, b} x (1) t Rx (1) t + b φ φ φ(rx (1) t + b) }{{} Φ t φ(x (1) t ) {R H, b H } R H φ(x (1) t ) + b H }{{} Ψ t {R, b } = arg mn R,b 1 l 1 l 1 t=1 Ψ t Φ t 2 (21) Because Φ(x) 2 can preserve constant under any translaton b and rotaton R, and Ψ t s fxed, : {R, b } = arg max R,b 1 l 1 l 1 t=1 Φ t Ψ t (22) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

18 Objectve Functon Rgd Transformaton n R 3 O = 1 l 1 l1 t=1 = 1 l1 (1) l 1 t=1 K(Rx {R, b } = arg max R,b Φ t φ(m 2) γφ(m 1 ) }{{} R H ( t + b, M 2 ) [γ 1 l 1 l 1 t=1 Φ t Ψ t } {{ } O (1) φ(x t (23) ) 1 φ(m 1 ) 1 l1 l } φ(m 2 ) 1 l2 l {{}} 2 {{} µ 1 µ 2 K(x (1) t, M 1 ) 1 ) K 1 1 l1 + 1 ] 1 l2 k 1 l 2 }{{} ρ t = 1 l1 l2 (1) l 1 t=1 =1 K(Rx t + b, x (2) )ρ t, (24) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

19 Rgd Transformaton n R 3 Smplfed Objectve Functon only a small number of ponts D + 1 s enough: {R, b } = arg max R,b 1 1 D + 1 D+1,l 2 l 2 t=1,=1 where S denotes the randomly selected subset of M 1 K(Rx (1) S t + b, x (2) )ρ t, (25) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

20 Rgd Transformaton n R 3 Implct Correspondence (a) (b) Fgure: (a) Two dentcal pont clouds wth exactly the same pont permutaton. (b) Vsualzaton of ρ t computed for all pars of ponts. Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

21 Rgd Transformaton n R 3 Relaton to Other Approaches our method: {R, b } = arg max R,b SoftAssgn /EM-ICP 1 1 D + 1 D+1,l 2 l 2 t=1,=1 K(Rx (1) S t + b, x (2) )ρ t, (26) {R, b } = arg mn R,b 1 l 1 l 1 l 2 t=1 =1 log K(Rx (1) t + b, x (2) )w t, (27) Gaussan Mxtures {R, b } = arg max R,b 1 l 1 l 1 l 2 t=1 =1 K(Rx (1) t + b, x (2) ) (28) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

22 Rgd Transformaton n R 3 Relaton to Other Approaches, cont. pseudo Gaussan mxture algnment: Remark: β k {}}{ u k 1 = φ(m 1 ) I E 1 α k = l 1 =1 β k (1) φ(x = l 1 =1 β k ) N (ξ; x (1), σ) pseudo Gaussan mxture: β k can be negatve; D pseudo Gaussan mxtures are algned smultaneously. (29) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

23 Qualtatve Experments Experment Results (a) (b) Fgure: Test of the proposed algorthm n typcal challengng crcumstances for regstraton: (a) large moton; (b) outlers; (c) nonrgd transformaton (c) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

24 Experment Results Qualtatve Experments,cont. Fgure: More test results on KIT 3D object database Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

25 Experment Results Quanttatve Experments Accuracy and Robustness (a) (b) Fgure: Test of four regstraton algorthm on (a) dfferent scales of motons; (b) dfferent porton of outlers added. Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

26 Experment Results Quanttatve Experments, cont. Effcency Pont cloud sze n complexty Our method n log n ICP[BM92] n log n GaussanMxtures[JV11] n SoftAssgn[GRLM97] n Table: Average executon tme (seconds) Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

27 Concluson Concluson kernel feature map pont cloud to Hlbert space; algn projectons of pont clouds n Hlbert space; project algnment back to R 3 ; accurate and robust to large moton and outlers; much faster than state-of-the-art methods; Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

28 Concluson END Questons are welcome! Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

29 Concluson P. J. Besl and H. D. Mckay. A Method for Regstraton of 3-D Shapes. PAMI, 14(2): , Steven Gold, Anand Rangarajan, Chenpng Lu, and Erc Mjolsness. New Algorthms for 2D and 3D Pont Matchng: Pose Estmaton and Correspondence. Pattern Recognton, 31: , Bng Jan and Baba C. Vemur. Robust Pont Set Regstraton Usng Gaussan Mxture Models. PAMI, 33(8): , Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29

30 Concluson Computaton Complexty Reducton Φ t, Ψ t = φ(px (1) t = ) ( D ũ k 2ũk 1 ( φ(x (1) t ) µ 1 ) + µ 2 ) k=1 D ũ k 2, φ(px(1) t ) ũ k 1, φ(x(1) t ) µ 1 + µ 2, φ(px (1) t ) k=1 (30) = D ũ k 2, φ(px(1) t ) ũ k 2, R Hφ(x (1) t ) µ 1 + µ 2, φ(px (1) t ) k=1 where we can see that φ(px (1) t ) and R H φ(x (1) t ) µ 1 are projected onto D egenvectors { ũ k D 2} respectvely, and an addtonal projecton of k=1 φ(px (1) t ) onto µ 2. Therefore, the computaton of the objectve functon s actually done n a space spanned by D egenvectors { ũ k D 2} k=1 and one µ 2, whch s a subspace of H. Hanchen Xong (Un.Innsbruck) 3D Regstraton 30 May / 29