Bundle Adjustment for 3-D Reconstruction: Implementation and Evaluation

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1 undle Adjustment or 3-D Reconstruction: Implementation and Evaluation Yuuki Iwamoto, Yasuyuki Sugaya 2 and Kenichi Kanatani We describe in detail the algorithm o bundle adjustment or 3-D reconstruction rom multiple images based on our latest research results The main ocus o this paper is the handling o camera rotations and the eiciency o computation and memory space usage when the number o eature points and the number o rames are large An appropriate consideration o these is the core o the implementation o bundle adjustment Doing experiments o undamental matrix computation rom two images and 3-D reconstruction rom multiple images, we evaluate the perormance o bundle adjustment ),6),2),2) Department o omputer Science, Okayama University 2 Department o omputer Science and Engineering, Toyohashi University o Technolgy (X, Y, Z) (x, y) X x y A P Z A P 3 4 (u, v ) t R 4) P = KR / u / I t, K / v / A (2) I K 3) () P X + P 2 Y + P 3 Z + P 4 x = P 3 X + P 32 Y + P 33 Z + P, y = P 2 X + P 22 Y + P 23 Z + P (3) P 3 X + P 32 Y + P 33 Z + P 34 P (ij) P ij N (X α, Y α, Z α) M (x α, y α ) ( =,, M, α =,, N) P (3) x, y 3) = 6 c 2 Inormation Processing Society o Japan

2 E = N M α= = [( pα I α x ) 2 ( α qα + y ) 2 ] α r α r α I α α p α = P X α + P 2 Y α + P 3 Z α + P 4, q α = P 2 X α + P 22 Y α + P 23 Z α + P 24 r α = P 3 X α + P 32 Y α + P 33 Z α + P 34 (5) (x α, y α), α =,, N, =,, M (4) 3 (X α, Y α, Z α ) P 5),6),2),2) 3 3 (X α, Y α, Z α ) P (4) E 3 ( X α, Y α, Z α) P (2), (u, v ) t = (t, t 2, t 3) R,, u, v, t, t 2, t 3 R R 9 R R (4) = I 3 3 R 3 R 3 R R R R + R R = I R R = O ( R R ) = R R R R ω, ω 2, ω 3 ω 3 ω 2 R R ω 3 ω A (6) ω 2 ω 3 ω, ω 2, ω 3 3 SO(3) so(3) 6) a T a T a T 7) (6) ω = (ω, ω 2, ω 3) 6) I ω I (a I)b = a b, (a I)T = a T (6) R R = ω R (7) t t dr /dt ω (7) 7) 7),9) (7) 4 E X α, Y α, Z α, α =,, N,, t, t 2, t 3, u, v, ω, ω 2, ω 3, =,, M 3N + 9M ξ, ξ 2,, ξ 3N+9M ξ k ξ k 2 / ξ k (4) E E N [( )( ) I α pα = 2 xα p α r α r ξ k rα 2 α p α r α ξ k ξ k α= = ( qα + y )( )] α q α r α r α q α r α ξ k ξ k 9) 2 N 2 E ξ k ξ l = 2 α= = ( + I α r 4 α r α q α ξ k [( )( ) p α r α p α r α r α p α r α p α ξ k ξ k ξ l ξ l )] ξ l )( r α q α r α q α r α q α ξ k ξ l ξ k E E/ ξ k 2 2 E/ ξ k ξ l p α, q α, r α p α / ξ k, q α / ξ k, r α / ξ k (8) (9) 2 c 2 Inormation Processing Society o Japan

3 5 3 (5) p α, q α, r α (X β, Y β, Z β ) δ αβ p α, X β = δ αβ P q α X β = δ αβ P 2, r α X β = δ αβ P 3, p α, Y β = δ αβ P 2 q α 6 Y β = δ αβ P 22, r α Y β = δ αβ P 32, p α = δ αβ P 3 Z β q α Z β = δ αβ P 23 r α Z β = δ αβ P 33 () (2) P P AR I t AK KR I t A u v / AP = u v / AP / = P u P 3 / P 2 u P 32 / P 3 u P 33 / P 4 u P 34 P 2 v P 3 / P 22 v P 32 / P 23 v P 33 / P 24 v P 34 / A () p α, q α, r α λ p α = δ ( λ p α u ) q α r α, = δ ( λ q α v ) r α, λ λ 7 r α λ = (2) (2) P u ( ) ( ) P AR I t AK KR I t u A u v / AP = P 3 P 32 P 33 P A (3) / v ( ) P AR I t P v 3 P 32 P 33 P 34 A (4) p α, q α, r α (u λ, v λ ) p α = δ λr α q α r α, =, =, u λ u λ u λ p α q α =, = δ λr α r α, = (5) v λ v λ u λ 8 t (2) P 4 P 4 (R + u R 3 )t + (R 2 + u R 23 )t 2 + (R 3 + u R 33 )t P 24 A = KR t (R 2 + v R 3 )t + (R 22 + v R 23 )t 2 + (R 32 + v R 33 )t 3 A P 34 (R 3 t + R 23 t 2 + R 33 t 3 ) (6) P P t 24 A t 3 P 34 P P 24 P 34 A R + u R 3 R 2 + v R 3 R 3 R 3 + u R 33 R 32 + v R 33 R 33 t 2 P 4 P 24 P 34 A R 2 + u R 23 R 22 + v R 23 R 23 A A (7) (t λ, t λ2, t λ3 ) tλ (5) tλ p α = δ λ ( r + u r 3 ), tλ p α = δ λ ( r 2 + v r 3 ), tλ p α = δ λ r 3 r, r 2, r 3 r = R R 2 R 3 9, r 2 = R 2 R 22 R 32, r 3 = R 3 R 23 R 33 (2) P (8) (9) 3 c 2 Inormation Processing Society o Japan

4 P = K(ω R) ω 3 ω 2 ω 2 t 3 ω 3 t 2 I t = ω 3 ω ω 3 t ω t 3 A (2) ω 2 ω ω t 2 ω 2 t (ω R) = R (ω I) (ω I)t = ω t 7) P / ω, P / ω 2, P / ω 3 R P 3 u R 33 R 2 +u R 23 (t 2 R 3 t 3 R 2 )+u (t 2 R 33 t 3 R 23 ) R ω 32 v R 33 R 22 +v R 23 (t 2 R 32 t 3 R 22 )+v (t 2 R 33 t 3 R 23 ) A, R 33 R 23 (t 2 R 33 t 3 R 23 ) R P 3 +u R 33 R u R 3 (t 3 R t R 3 )+u (t 3 R 3 t R 33 ) R ω 32 +v R 33 R 2 v R 3 (t 3 R 2 t R 32 )+v (t 3 R 3 t R 33 ) A, 2 R 33 R 3 (t 3 R 3 t R 33 ) R P 2 u R 23 R +u R 3 (t R 2 t 2 R )+u (t R 23 t 2 R 3 ) R ω 22 v R 23 R 2 +v R 3 (t R 22 t 2 R 2 )+v (t R 23 t 2 R 3 ) A (2) 3 R 23 R 3 (t R 23 t 2 R 3 ) (ω λ, ω λ2, ω λ3 ) ωλ (5) ωλ p α = δ λ ( r + u r 3 ) (X α t ), ωλ q α = δ λ ( r 2 + v r 3 ) (X α t ), ωλ r α = δ λ r 3 (X α t ) (22) X α = (X α, Y α, Z α ) E 2 E (LM) 9) ( ) X α,, (u, v ), t, R E c = ( 2 ) 2 E/ ξ k, 2 E/ ξ k ξ l, k, l =,, 3N + 9M ( 3 ) ξ k, k =,, 3N + 9M ( + c) 2 E/ ξ 2 2 E/ ξ ξ 2 2 E/ ξ ξ 3N+9M 2 E/ ξ 2 ξ ( + c) 2 E/ ξ2 2 2 E/ ξ 2 ξ A 2 E/ ξ 3N+9M ξ 2 E/ ξ 3N+9M ξ 2 ( + c) 2 E/ ξ 3N+9M 2 ξ E/ ξ ξ 2 E/ ξ ξ 3N+9M = E/ ξ 3N+9M ( 4 ) X α,, (u, v ), t, R A (23) X α X α + X α, +, (ũ, ṽ ) (u, v ), t t + t, R R(ω )R (24) R(ω ) N [ω ] ω 8) ( 5 ) X α,, (ũ, ṽ ), t, R Ẽ Ẽ > E c c (3) ( 6 ) X α X α,, (u, v ) (ũ, ṽ ), t t, R R (25) Ẽ E δ δ E Ẽ, c c/ (2) (23) c = ξ k 3 R = I, t =, t 22 = (26) 3 2 Y t 2 = 2 Y X Z t 2 = t 23 = (23) ω, ω 2, ω 3, t, t 2, t 3, t 22 3N + 9M 7 (23) LM X α,, (u, v ), t, R (26) (26) X α, t, R X α, t, R X α = ) s R (X α t, R = R R, t = s R (t t ) (27) s = (j, R (t 2 t )) j = (,, ) exp(ω I) so(3) Lie SO(3) 6) 4 c 2 Inormation Processing Society o Japan

5 2 (8), (9) N MN α= = E/ ξ k (8) ξ k β X β () δ αβ N α= α = β ξ k λ λ, (u λ, v λ ), t λ R λ (2), (5), (8), (22) δ λ = = λ (8) N α= = α α α 2 E/ ξ k ξ l (9) ξ k, ξ l (9) N α= ξ k, ξ l (9) = ξ k, ξ l N α= = E/ ξ k, 2 E/ ξ k ξ l 2 2 E/ ξ k ξ l H(k, l) (3N + 9M 7) (3N + 9M 7) N, M 3N 3 E, 3N 9M F 9M 9 G E 2 E/ X 2 α, 2 E/ X α Y α 2 F 2 E/ X α, 2 E/ X α u 2 G 2 E/ 2, 2 E/ u 2 27NM + 9N + 8M H(k, l) (k, l) 3 3N +9M 7 (23) (3N +9M 7) (3N +9M 7) N, M LU 2) 2 27NM + 6N + 4M I α = 3 (23) (23) E (c) F ( ) ( ) ξ P d P E (c) N F N ξ F F F N G (c) ξ P ξ 3 3N ξ F 9M 7 d P, d F (23) 3N 9M 7 E (c) α, α =,, N α (X α, Y α, Z α) E (c) ( + c) F α α (X α, Y α, Z α ) α E 2 3 (9M 7) G (c) E 2 (9M 7) (9M 7) ( + c) (28) E (c) E (c) N ξ P + F F N = ξ F = d P, d F ( F (28) ) F N ξ P + G (c) ξ F = d F ξ P 2 ξ F 9M 7 ( N G (c) F α E (c) α F α ) ξ F = α= N α= F α E (c) α α E d F, α (29) E/ X α E/ Y α A (3) E/ Z α ξ F (29) 2 ξ P α X Y α A = E α (c) (F α ξ F + α E) (3) Z α 4 5 c 2 Inormation Processing Society o Japan

6 2 4) σ = LM ɛ LM δ δ = nɛ 2 / 2 n = N α= = Iα ɛ = = = 6 x, y (Hartley 8 3) ) 22),, ) (u, v ) (4) e N = 9 E e = (32) N 7 N 7 7 7) σ e 2 / 2 σ 2 N 7 χ 2 N 7 (32) e σ 4) 4) xyx y 3 4 EFNS 3) ) 22) 3 22) Oxord 2(a) P 5266 (23) (b) % P (u, v ) t R A 3 A2 n = N Iα 6432 (4) α= = (32) E e = 2n (3N + 9M 7) (33) 6 c 2 Inormation Processing Society o Japan

4 2 (a) (b) (c) 327796573463469 2378732275724 4 626388763577 2 7678668765 4 626973343624 3 7232393526 42 62679434579 4 6984294963539 43 6264995753774 5 684648452468 44 626262285242 6 67536625569 45

e 2(c) 2(d) 49 2 5 ++ PU Intel ore2duo E675, 266GHz 4G OS Windows Vista 2(e) 3 (e) 3 2 4) Oxord 3 3 9) 3 3 2),5),),2) ),),8) : ( 2572) ),,,,, 27-VIM-6- (27-9), 63 7 2),,,, 3,, 25-VIM-5-2 (25-), 45 52

7 4 2 (a) (b) (c) (d) (a) 36 (b) (c) e (d) e (e) 3 e = e = e 2(c) 2(d) PU Intel ore2duo E675, 266GHz 4G OS Windows Vista 2(e) 3 (e) 3 2 4) Oxord 3 3 9) 3 3 2),5),),2) ),),8) : ( 2572) ),,,,, 27-VIM-6- (27-9), ),,,, 3,, 25-VIM-5-2 (25-), ) R I Hartley, In deense o the eight-point algorithm, IEEE Trans Patt Anal Mach Intell, 9-6 (997-6), ) R Hartley and A Zisserman, Multiple View Geometry in omputer Vision, 2nd ed, ambridge University Press, ambridge, UK, 24 5),,, D-II, J74-D-II-8 (993-8), ) K Kanatani, Group-Theoretical Methods in Image Understanding, Springer, erlin, Germany, 99 7) K Kanatani, Statistical Optimization or Geometric omputation: Theory and Practice Elsevier, Amsterdam, the Netherlands, 996; reprinted, Dover, York, NY, USA, 25 8), AD,, c 2 Inormation Processing Society o Japan

8 9),,, 25 ) ( ),, 2,, 29, pp ),,,,,, 25-VIM-5-6 (25-9), ),,,, PRMU23-8 (23-), ),, FNS,, 27-VIM-58-4, (27-3), ) K Kanatani and Y Sugaya, ompact undamental matrix computation, IPSJ Trans omput Vis Appl 2 (2-3), ) M I A Lourakis and A A Argyros, Is Levenberg-Marquardt the most eicient optimization algorithm or implementing bundle adjustment?, Proc th Int on omput Vis, Vol 2, October 25, eijing, hina, pp ) M I A Lourakis and A A Argyros, SA: A sotware package or generic sparse bundle adjustment, AM Trans Math Sotware, 36- (29-3), 2: 3 7),,, 3,, 44- (23-), ),,, :,, 27-VIM-57-5 (27-), 9 6 9),,,,, 4-SIG3 (24-2), ),,, 29-VIM (29-6), 6 2) Triggs, P F McLauchlan, R I Hartley, and A Fitzgibbon, undle adjustment A modern synthesis, in Triggs, A Zisserman, and R Szeliski, (eds), Vision Algorithms: Theory and Practice, Springer, erlin, 2, pp ),,,, 3,, 29-VIM-68-5 (29-9), 8 A ( P = Q ) q P 3 3 Q 4 q () P det Q < Q q Q = ckr, q = ckr t (34) c t t = Q q (35) R R R = I (34) QQ = c 2 KR RK = c 2 KK (36) (QQ ) = c 2 (K ) (K ) (37) (QQ ) (QQ ) = (38) (37), (38) = c K = ck (39) (34) (39) Q = R (4) R R = (Q) (4) K (39) (3,3) A2 3 (3) xp 3 X + xp 32 Y + xp 33 Z + xp 34 = P X + P 2 Y + P 3 Z + P 4 yp 3 X + yp 32 Y + yp 33 Z + yp 34 = P 2 X + P 22 Y + P 23 Z + P 24 (42) p α n α (= I = α) p α 3 X α 2n α x α P 3 P x α P 32 P 2 x α P 33 y α P 3 P 2 y α P 32 P 22 y α P 33 P 23 P 3 X α Y α Z α x α P 34 P 4 = y α P 34 P 24 9) (43) 8 c 2 Inormation Processing Society o Japan

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