(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,

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1 [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 ] Yasuyuki SUGAYA, Member (Department of Information and Computer Sciences, Toyohashi University of Technology, Toyohashi-shi, Japan) and Kenichi KANATANI, Member (Graduate School of Natural Science and Technology, Okayama-shi, Japan). Vol. 92 No. 4 pp (b) 1(c) Canny (4),(5) 1(c) 1(d) 1(d) 2 CG 3. Ax 2 +2Bxy+Cy 2 +2(Dx+Ey)f 0 +F f 2 0 =0 (1) f 0 (1) (x α, y α ), α = 1,..., N (1) α = 1,..., N Ax 2 α+2bx α y α +Cy 2 α+2(dx α +Ey α )f 0 +Ff (2) A, B, C, D, E, F 3 A, B, C, D, E, F [II] 301

2 (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, y 2 α, 2f 0 x α, 2f 0 y α, f 2 0 ) (5) 2 (b) CG (2) (u, ξ α ) 0, α = 1,..., N (6) a, b (a, b) F = 1, A + B = 1, A 2 + B 2 + C 2 + D 2 + E 2 + F 2 = 1 (3) u = 1 (1) ( 1) ( 1) u (x α, y α ) x α 302 Vol.92, No.4, 2009

3 (1) (conic) (1) (6) ( 2) 4. (6) u N (u, ξ α) 2 (least squares) 6 6 M LS M LS ξ α ξ α (7) (u, ξ α ) 2 = (u, ξ α ξ α u) = (u, M LS u) (8) u u M LS (8) (3) F = 1 A + B = 1 (8) (2),(9),(10),(11) 5. (1) 0 (8) α (u,ξ α )=Ax 2 α+2bx α y α +Cy 2 α+2(dx α +Ey α )f 0 +Ff 2 0 (9) 0 (x α, y α ) x, y 0 ( x α, ȳ α ) x α = x α + ɛ α, y α = ȳ α + η α (10) ɛ α, η α 0 σ 2 (9) ( x α, ȳ α ) (9) 0 (u, ξ α ) = 2(A x α + Bȳ α + Df 0 )ɛ α +2(B x α + Cȳ α + Ef 0 )η α + (11) (u, ξ α ) 0 4 ( (A x α +Bȳ α +Df 0 ) 2 +(B x α +Cȳ α +Ef 0 ) 2) σ 2 (12) ( 3) u 4σ 2 (u, V 0 [ξ α ]u) (13) 6 6 V 0 [ξ α ] x 2 α x α ȳ α 0 f 0 x α 0 0 x α ȳ α x 2 α + ȳ 2 α x α ȳ α f 0 ȳ α f 0 x α 0 V 0 [ξ α ] 0 x α ȳ α ȳα 2 0 f 0 ȳ α 0 f 0 x α f 0 ȳ α 0 f (14) 0 f 0 x α f 0 ȳ α 0 f (u, ξ α ) 0 4σ 2 (u, V 0 [ξ α ]u) (u, ξ α )/ 4(u, V 0 [ξ α ]u) 0 σ 2 J = 1 4 (u, ξ α ) 2 (u, V 0 [ξ α ]u) (15) ( 2) AC B 2 > 0 (7) ( 3) c c 2 [II] 303

4 (maximum likelihood estimation) V 0 [ξ α ] (14) ( x α, ȳ α ) (x α, y α ) (15) u u (15) (1) ( 4) (6) (15) Chojnacki (12) FNS 6. (15) ξ α N (u, ξ) = 0 u N ξ α ξ α (Euclidean distance) (Mahalanobis distance) ξ α ξ α ( 4) ξ α ξ α ξ α V [ξ α ] J = (ξ α ξ α, V [ξ α ] 5 (ξ α ξ α )) (16) ξ α ξ α ( ) 5 5 ( 5) (16) (u, ξ α ) = 0, α = 1,..., N (16) (15) V 0 [ξ α ] V [ξ α ] () (14) ξ α 7. Taubin (15) FNS (1) 4. Taubin (15) V 0 [ξ α ] N TB 1 N V 0 [ξ α ] (17) (15) J TB J TB = N 4 = N 4 (u, ξ α ) 2 (u, N TB u) = N 4 (u, M LS u) (u, N TB u) (u, ξ α ξ α u) (u, N TB u) (18) M LS (7) J TB (Rayleigh quotient) u ( 6) (generalized eigenvalue problem) M LS u = λn TB u (19) (10) (14) V 0 [ξ α ] (17) N TB N TB ( 7) ξ α u, V 0 [ξ α ] ( ξ, u) = 0 4 ( 4) ξ α x α V 0 [ξ α ] 3 3 V 0 [x α ] ( 5) 5 0 (8) (1) ( 6) (19) N TB λ u ( 7) A 0 x (x, Ax) > 0 (8) 304 Vol.92, No.4, 2009

5 ξ α = ( V 0 [ξ α ] = z α f0 2 ( ), u = V 0 [z α ] M LS Ñ TB M LS z α z α, z α z α z α, ) ( Ñ TB z 1 N v F ) (20) V 0 [z α ] (21) z α (22) (19) M LS v = λñ TBv, (v, z) + f 2 0 F = 0 (23) 1 (19) Ñ TB v 2 F Taubin 5 Taubin Taubin (13) Taubin 1990 (14) (16) 8. (1) u u û u û u ( 6) û V [û] = E[ u u ] (24) E[ ] KCR (KCR lower bound)) V [û] 4σ 2 M 5 (25) ( 8) σ 2 (a) u u O u^ u (b) 5 Taubin (11) 6 u û u u ( 8) A x (x, Ax) 0 (8) [II] 305

6 (a) (b) 7 (9) M M ξ α ξ α (u, V 0 [ ξ α ]u) (26) ξ α V 0 [ ξ α ] ξ α V 0 [ξ α ] (x α, y α ) u (25) O(σ 4 ) KCR 7 (9) 9. Taubin 2 (1),, [I], vol.92, no.3, pp , March (2) K. Kanatani, Geometric Computation for Machine Vision, Oxford University Press, Oxford, U.K (3),,,, (4),,,, (D-II), vol.j85-d-ii, no.12, pp , Dec (5),, 14 (SSII08), IN1-14, pp.1 6 June (6) A. Fitzgibbon, M. Pilu and P. B. Fisher, Direct least square fitting of ellipses, IEEE Trans. Pattern Anal. Mach. Intell., vol.21, no.5, pp , May (7),,, (8),,, (9) Y. Kanazawa and K. Kanatani, Optimal conic fitting and reliability evaluation, IEICE Trans. Inf. & Syst., vol.e79-d, no.9, pp , Sept (10) K. Kanatani, Statistical Optimization for Geometric Computation: Theory and Practice, Elsevier Science, Amsterdam, The Netherlands, 1996; Dover, New York, (11),,,,, no.2006-cvim , pp , May (12) W. Chojnacki, M.J. Brooks, A. van den Hengel and D. Gawley, On the fitting of surfaces to data with covariances, IEEE Trans. Pattern Anal. Mach. Intell., vol.22, no.11, pp , Nov (13) G. Taubin, Estimation of planar curves, surfaces and, non-planar space curves defined by implicit equations with applications to edge and range image segmentation, IEEE Trans. Pattern Anal. Mach. Intell., vol.13, no.11, pp , Nov (14) N. Tagawa, T. Toriu, and T. Endoh, Un-biased linear algorithm for recovering three-dimensional motion from optical flow, IEICE Trans. Inf. & Syst., vol.e76-d, no.10, pp , Oct (15) N. Tagawa, T. Toriu, and T. Endoh, Estimation of 3-D motion from optical flow with unbiased objective function, IEICE Trans. Inf. & Syst., vol.e77-d, no.10, pp , Oct (16) N. Tagawa, T. Toriu, and T. Endoh, 3-D motion estimation from optical flow with low computational cost and small variance, IEICE Trans. Inf. & Syst., vol.e79-d, no.3, pp , March IEEE 306 Vol.92, No.4, 2009

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