IPSJ SIG Technical Report Vol.2009-CVIM-167 No /6/10 1 bundle 10) A 90 Bundle Adjustment TAKAYUKI OKATANI 1 Bundle adjustment is a general meth
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1 1 bundle 10) A 90 Bundle Adjustment TAKAYUKI OKATANI 1 Bundle adjustment is a general method for the problem of estimating parameters of a geometric model from images. It is characterized by that nonlinear optimization is numerically performed to attain high estimation accuracy, even if the size of the optimization is very large. In this article I present a tutorial on this method. Specifically, I describe basic numerical algorithms with their implementation techniques, theoretical grounds in terms of statistical inference, and gauge freedom associated with the choice of a coordinate system. Moreover, I present a method for recursive computation that realizes online estimation for time series observations. 1. bundle adjustment 4 6 Triggs 1 Graduate School of Information Sciences, Tohoku University 3 photogrammetry 1960 (1) (2) (3) Triggs, McLauchlan, Hartley, Fizgibbons 10) 10 Triggs 10) c 2009 Information Processing Society of Japan
2 5 9) R i t i?1 6?2 p i Hartley-Zisserman 3) p 1 ; : : : ; p m 3 q 1 ; : : : ; q n (2) x A H z ij D Œu ij ; v ij > I, J z ij (2) 2. (2) m n m 3 E.p 1 ; : : : ; p m ; q 1 ; : : : ; q n / D 1 X.u ij u.p i ; q j // 2 C.v ij v.p i ; q j // 2 (3) 2 i;j i D 1; : : : ; m i u.p i ; q j /, v.p i ; q j / P i (3 4 ) P i K i ŒR i j t i (1) u.p i ; q j / D.P i / 1 Œq > j 1 >.P K i R i, t i i / 3 Œq > ; v.p j 1 > i ; q j / D.P i / 2 Œq> j 1 >.P i / 3 Œq > : (4) j 1 > j D 1; : : : ; n j 3 3.P i / k P i k (3) E q j D ŒX j ; Y j ; Z j > i z ij D Œu ij ; v ij > " # " # zij qj / P i (2) 1 1 p 1 ; : : : ; p m, q 1 ; : : : ; q n reprojection error x (3) x 3) / E.x/ D E.p 1 ; : : : ; p m ; q 1 ; : : : ; q n / (5) (1) P i K i E x x self-calibration x x! x C x?1 3)?2 A.1 2 c 2009 Information Processing Society of Japan
3 ?1 3. E.x/ E.x/ (6) E.x/ E E.x/ D ke.x/k2 D 1 X e 2 2 k (6) k 1 3 sparse E.x/ (Newton) E.x/ 2 11),12) E.x/ x x de=dx D 0 de=dx D 0 x 6 x x x! x C x?1 DLT Direct Linear Transformation 3) 3 c 2009 Information Processing Society of Japan
4 x E.x/ E.x C x/ E.x/ C g > x C 1 2 x> H x (7) g E x H x g D de ˇ ; H D d 2 E dx ˇx dx 2 ˇ (8) ˇx g x H x x x (7) x steepest descent H x D g (9) H x D H 1 (13) x E.x C x/ E.x/ 10 (13) g x H E.x C x/ < E.x/ x x C x 0.1 positive-definite E.x/ x H H Gauss- x Newton (6) E.x/ H Trust-region Trust-region x k xk H A J > J (10) J e.x/ J D de dx (9) a g D J > e A, a x (9) A x D a (12) (11) Levenberg-Marquardt (12).A C I/ x D a (13) damping factor 0 D 0 D 0: H A kek 6 e 2 4 c 2009 Information Processing Society of Japan
5 A x D a x A (a) (b) (c) A LU (a) J (b) H (c) H A LU 4 A A x D a 1. A L A! LL > (14) 2. Ly D a y 0 J.i; j / e x 3. L > x D y i =@x j e i x j 0 A 1 x D A 1 a A 1 SFM e i x e i x i =@x k J H 1 A 6 A A x D a kjıx C ek 2! min. 2 A D J > J 1.A C I/ x D a " # " # J e 2 H 1=2 x C (15) I 0 1(c) 2 QR Householder Givens 2 time [sec.] sparse dense J H A sparse matrix 0 points 5 c 2009 Information Processing Society of Japan
6 permutation A P 2 PAP > LL > x C x! x A x D a " # " # " # A11 A 12 x1 a1 fill-in D (16) A 21 A 22 x 2 a 2 NP 2 A 11 x 1 C A 12 x 2 D a 1 A 21 x 1 C A 22 x 2 D a 2 x 1 x 2 D A J H A 21 x 1 C a 2 / x 1 arrow-head Triggs 10) A.A 11 A 12 A22 1 A 21/ x 1 D.I A 12 A22 1 /a 2 (17a) x A 22 x 2 D a 2 A 21 x 1 (17b) fill-in 2 x D Œ x > 1 ; x> 2 > A 11 A 12 A 22 1A 21 A 11 (Schur s complement) A A 1 A x 1 x 2 SFM SFM 2?? x x 1 x 2 D 0 x C x! x A 22 x 2 D 0 x 2 x 2 A 1 D 0 x 1 1 resection-intersection C/C++ Fortran x 2 D 0 F.x 1 /x 2 D 0 x 2 x 2 x 1 x 2 D Ox 2.x 1 / E x 1 E 0.x 1 / E.x 1 ; Ox 2.x 1 // E 0 MATLAB Optimization Toolbox.x 1 / lsqnonlin Trust-region x?1 Trust-region Wiberg Z?1 2 Z! MS Z 7) 6 c 2009 Information Processing Society of Japan
7 2 e J SFM J lemvar MATLAB 4. BLAS BLAS API CPU PC CPU SIMD Intel GotoBLAS 2) QR 4.1.u; v/ SIFT KLT LAPACK BLAS CHOLMOD?1 MATLAB 7.2?2 UMFPACK?3 MATLAB z i D Œu i ; v i > Nz i D Œ Nu i ; Nv i > PARDISO?4 Intel Math Kernel Library u i D Nu i C " fill-in?5 v i D Nv i C " 0 Math Kernel Library BLAS MINPACK lmdif, lmder ", " 0 0 sba/lemvar?6 sba SFM " " 0 0?1 n?3 Minimum degree Nested dissection?6 lourakis/sba/ ", " 0.u i ; v i /.u j ; v j / (18a) (18b) N.0; 2 / 2 Œ"; " 0 > N.0; 2 I/ " " V N.0; V/ 7 c 2009 Information Processing Society of Japan
8 Nz i SIFT KLT, 4.2. z i.i D 1; : : : ; n/ 1 z i (19) Nz i Nz i SIFT KLT f c.nz i ; / D 0; i D 1; : : : ; n: (21) outlier (20) Ÿ i Nz i d d z i 4.2 (21) Nz i Cauchy d 4.2 z i Ÿ i 4.1 multi-linear constraint 1 z i 1 Ÿ i 4.2 Nz i Nz i Ÿ i 1 3 trifocal tensor 2.1 SFM Nz i Ÿ i nuisance parameter latent parameter 4.1. z i.i D 1; : : : ; n/ 1 z i Nz i i z i D Nz i C i (19) n.u i ; v i /.i D 1; : : : ; n/ i i D 1; : : : ; n 0 N.0; V/ 1 v D au C b. Nu i ; Nv i / Nz i i Ÿ i u i D Nu i C " i (22a) v i D Nv i C " 0 i (22b) Nz i D f.ÿ i ; /; i D 1; : : : ; n (20) 1 ; : : : ; n " i, " 0 i N.0; 2 /. Nu i ; Nv i / Nv i D a Nu i C b (21) 8 c 2009 Information Processing Society of Japan
9 i Œ Nu i ; Nv i > D Œ i ; a i C b > (20) f. i ; Œa; b > / 4.2. C W u 2 =a 2 C v 2 =b 2 D 1 n.u i ; v i /.i D 1; : : : ; n/ C. Nu i ; Nv i / (22) i, 0 i N.0; 2 /. Nu i ; Nv i / C n.u i ; v i / $.u 0 i ; v0 i / Nu 2 i =a2 C Nv i 2=b2 D 1 (21).u i ; v i / i Œ Nu i ; Nv i > D Œa cos i ; b sin i >.u 0 i ; v0 i /. Nu0 i ; Nv0 i / Œ Nu0 i ; Nv0 i ; 1 > / HŒu i ; v i ; 1 > H (20) u i ; v i / n.u i ; v i / $.u 0 i ; v0 i / 2.u 0 i ; v0 i / H H. Nu i ; Nv i / $. Nu 0 i ; Nv0 i / H Nu 0 i Nu Nv i / H i 6 4 Nv i 7 5 (23) Z p.zi / Z 1 1 Z L. I Z/ p.zi / Z likelihood H H.3; 3/ 1 8 i Œ Nu i ; Nv i > Nu 0 i, Nv0 i z i D Nz i C i i N.0; V/ V D 2 I i V z i N.Nz i ; 2 I/ SFM. 2.1 SFM 2 Nz i D f.ÿ i ; / z i Ÿ i ; 1 z i i p.z i I Ÿ i ; / D 1 z i Œu i1 ; v i1 ; u i2 ; v i2 ; : : : > Ÿ i p i (4) 2 2 exp z i Nz i / 2 (24) D Œq > 1 ; : : : ; q> m > Z D Œz > 1 ; : : : ; z> n > Z 1 1 j i D 1; : : : z j Œu 1j ; v 1j ; u 2j ; v 2j ; : : : > ny Ÿ j q j D ŒX j ; Y j ; Z j > (4) 1 z i 1 Ÿ i p.zi x/ D D Œp > 1 ; : : : ; p> m > x Œ > ; Ÿ > 1 ; : : : ; Ÿ> n > id1 p.z i I Ÿ i ; / (25) 9 c 2009 Information Processing Society of Japan
10 Z L.xI Z/ p.zi x/ x x Ox D t.z/ E x ŒOx D x L l.xi Z/ log L Cov x.ox/ D Cov x.t.z// I 1.x/ (29) l.xi Z/ D E.x/ nx.z i f.ÿ i ; // 2 n log 2 2 (26) id1 E.x/ 1 nx.z i f.ÿ 2 i ; // 2 (27) id1 Cov x x A, B A B A B semi-positive definite I.x/ Fisher > I.x/ D l (26) l.xi Z/ D log p.zi x/ E x Œ x p.zi x/ l D E= 2 n log E x E x Œ D EŒ x 2 (3) E f c.nz i ; / D 0 Nz i ! i. Nz i / Minimize E.x/ D 1 nx.z i i / 2 (28a) 2 id1 0 subject to f. i ; / D 0; i D 1; : : : ; n (28b) 2! 0 x 13) z 1 ; : : : ; z n n n! 1 1 (Neyman-Scott) 6) unbiasedness 2 4) z i Nz i Nz i m m z i1 ; : : : ; z im Cramèr-Rao x (30) 10 c 2009 Information Processing Society of Japan
11 N.Nz i ; 2 I/ m I.Ox/ E x D Ox H A m n! 1 m! 1 2 H 1 2 A 1 I 1.Ox/ m m! 1 1 2! 0 J 1.Ox/ 1) I 1.Ox/ E E.Ox/ E 4.1?1 V D 2 I z i n z Ÿ i n i.d z i Nz i / 2 EŒk i k 2 D 2.n z n /?2 E.x/ D.1=2/ P i kz i Nz i k 2 2 O 2 D 2E.Ox/=.n.n z n // n x x 4.5 m! 1 2! 0 n! 1 2 H 1 2 A I 1.x/ I.x/ Ox x I.Ox/ SFM observed Fisher s information J.Ox/ 2 ˇ (31) ˇxDOx J 1 gauge.ox/ E H 7 3 1?3 J H A A D J > J J D.1= 2 /H I D.1= 2 /A J.Ox/?4?1 m n? ?2 4),8)?4 x 0 D t.x/ t E D kek 2 =2 11 c 2009 Information Processing Society of Japan
12 J H?1 4 4 q kqk 2 D 1 7 SFM 1 J H 2 x I H x D g H x Œ1; 0; 0 1 z xy H II I II 2 x 2 I 1 trivial gauge 10) SFM 1 0 P 1 D K 1 ŒI j 0 II x x 2 32 x 3 J J H H H H H x D g k xk 2 H x D H g x x x C x! x x M kmk 2 D Œ0; 0; 0 I?2 II x SFM e.x/ D e.t.x// J, H?1 critical configuration 3 J H x?2 I 12 c 2009 Information Processing Society of Japan
13 P3 P2 P1 E.x/ 2 I II P1,P2,P inner gauge 3 3 SFM 3 m m C 1 1?1 1 m E 1Wm P m kd1 E k m C 1 1 E mc1 E 1WmC1 D E 1Wm C E mc1 m x 1 m C 1 x x 1 McLauchlan VSDF(Variable State Dimension Filter) 5) SLAM UI Triggs 10) H E 1WmC1.x 1 ; x 2 / D E 1Wm.x 1 / C E mc1.x 1 ; x 2 / (32) 6.3 x 1 3 A x D a E 1WmC1.x/ D E 1Wm.x/ C E mc1.x/ m E 1Wm.x/?1 13 c 2009 Information Processing Society of Japan
14 x D x x A A 2 OE 1Wm.x/ A 11 E 1Wm.x 1 / D.1=2/.x 1 x 1 /> A 11.x 1 x 1 / E mc1 OE 1Wm.x/ 1 2.x x / > A.x x / (33) " # " # " # A 0 11 A 0 x 12 x1 a 0 E 1Wm A 0 21 A 0 D 1 22 x 2 a 0 (36) 2 m C 1 E 1WmC1 D E 1Wm C E mc1 1 E 1Wm OE 1Wm.x/ " # " # " A 11 C A 0 11 A 0 12 x1 A E 1WmC1 OE 1Wm C E mc1 (34) A 0 21 A 0 D 11.x 1 x 1 / C # a x 2 a 0 (37) 2 x x A OE 1Wm.x/ E 1Wm.x/ E 1WmC1 E 1WmC1 D E 1Wm C E mc1 E mc1 E mc1 E 1WmC1 D E 1Wm wc1 C E m wc2wmc1 w A 0 x D a 0 (34) 2 w.a C A 0 / x D A.x x / C a 0 (35) =2 w x x C x! x A 0 w x 6.4 A 0 (35) A E 1Wm?1 A LL > A 0 LL > C A 0 A C A 0 Triggs 10) 59 B.5 3 x 1 A 0 x Ox A 0 A CA 0! A Ox! x?1 w E (32) 1Wm w E m wc1wm E 1Wm x 1 D x 1 14 c 2009 Information Processing Society of Japan
15 OE 1Wm OE 1Wm D 1 2 x> 1.A 11 A 12 A22 1 A 21/ x 1 (41) (38).x 1 ; x 2 ; x 3 / E 1Wm x E 1Wm x x D Œx > 1 ; x> 2 > 2 x 2 x 1 x 2 E new x 2 (38) x 1 x 2 E new (40) x 2 D x 2 A22 1A 21.x 1 x 1 / x 2 E 1Wm.x 1 ; x 2 / C E new.x 1 ; x 3 / (38) x 2 D A 1 22 A 21 x 1 (40)?1 A 11 (41) E new.x 1 ; x 3 / x 2 E new x 3 " # " # " # A 0 11 A 0 13 x1 a 0 (38) E new.x 1 ; x 3 / x 1 x 2 A 0 31 A 0 D 1 33 x 3 a 0 (42) 3 x 1 E 1Wm.x 1 ; x 2 / x 2 x 1 E 1Wm.x 1 ; x 2 / " # " # " # 2 x 2 A 11 C A 0 11 A 0 13 x1 A x 1 A 0 31 A 0 D 11.x 1 x 1 C/a x 3 a 0 (43) 3 E 1Wm x D Œx > 1 ; x> 2 > E 1Wm 2?1 OE 1Wm.x 1 C x 1; x 2 C x 2/ D 1 h i " # " # A 11 D A 11 A 12 A22 1 A 21 (44) x > 2 1 x > A 11 A 12 x1 (39) 2 A 21 A 22 x 2 x 2 x 2.x 1 ; x 2 / (38) OE 1Wm.x 1 ; x 2 / C A A 11 E new.x 1 ; x 3 /.x 1 ; x 2 ; x 3 / x 1 x 1 1 OE 1Wm.x 1 ; x 2 / x 2 x 2 D Ox 2.x 1 / x 1 E 1Wm E new E 1Wm 2 E 1Wm OE 1Wm.x 1 ; x 2 / x 2 OE 1Wm D OE 1Wm.x 1 ; Ox 2.x 1 // E 1Wm wc1 E m wc2wm 2 E 1Wm wc1 2 x OE 1Wm =@ x 2 D 0 15 c 2009 Information Processing Society of Japan
16 z 1 ; : : : ; z m x! x C x x 1 ; : : : ; x m x q kqk 2 D 1 1 J H z k x k R R R information filter 7. 1) Efron, B. and Hinkley, D.V.: Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher Information, Biometrika, Vol.65, No.3, pp (1978). 2) Goto, K. and vande Geijn, R.A.: High-performance implementation of the level-3 BLAS, Triggs ACM Trans. Math. Softw., Vol.35, No.1 (2008). 3) Hartley, R. and Zisserman, A.: Multi-View Geometry in Computer Vision, Cambridge University Press, 2nd edition (2003). 4) Kanatani, K.: Statistical optimization for geometric fitting: Theoretical accuracy bound and high order error analysis, International Journal of Computer Vision, Vol.80, No.2, pp.167 A (2007). 3 5) McLauchlan, P. F.: A Batch/Recursive Algorithm for 3D Scene Reconstruction, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. (a) a kak 2 D 1 D a (2000). (b) quaternion q D Œcos.=2/; sin.=2/ 6) Neyman, J. and Scott, E.: Consistent estimates based on partially consistent observations, (c).; ; / z, x, z R D R z. /R x./r z./ Econometrica, Vol.16, No.1, pp.1 32 (1948). (a) Rodrigues 7) Okatani, T. and Deguchi, K.: On the Wiberg Algorithm for Matrix Factorization in the Presence of Missing Components, International Journal of Computer Vision, Vol.72, No.3, pp. R D I C sin Œa C.1 cos /Œa 2 (45) (2007). 8) Okatani, T. and Deguchi, K.: On bias correction for geometric parameter estimation in computer vision, Proceedings of IEEE Computer Society Conference on Computer Vision and a D Œa 1 ; a 2 ; a 3 > kak 2 D Pattern Recognition (2009). 0 a 3 a 2 9) Pollefeys, M., Koch, R. and Gool, L.V.: Self-calibration and metric reconstruction inspite of Œa D 6 4 a 3 0 a (46) varying and unknown intrinsic camera parameters, International Journal of Computer Vision, a 2 a 1 0 Vol.32, No.1, pp.7 25 (1999). 10) Triggs, B., McLauchlan, P., Hartley, R. and Fitzgibbon, A.: Bundle Adjustment A Modern Synthesis, Vision Algorithm: Theory & Practice (Triggs, B., Zisserman, A. and Szeliski, (gimbal lock) R., eds.), Springer-Verlag LNCS 1883 (2000). 11) (1994). R R R R q 12) (1984). 4 kqk 2 13) (1991). D 1 E 16 c 2009 Information Processing Society of Japan
(a) (b) (c) Canny (d) 1 ( x α, y α ) 3 (x α, y α ) (a) A 2 + B 2 + C 2 + D 2 + E 2 + F 2 = 1 (3) u ξ α u (A, B, C, D, E, F ) (4) ξ α (x 2 α, 2x α y α,
[II] Optimization Computation for 3-D Understanding of Images [II]: Ellipse Fitting 1. (1) 2. (2) (edge detection) (edge) (zero-crossing) Canny (Canny operator) (3) 1(a) [I] [II] [III] [IV ] E-mail sugaya@iim.ics.tut.ac.jp
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