IPSJ SIG Technical Report Taubin Ellipse Fitting by Hyperaccurate Least Squares Yuuki Iwamoto, 1 Prasanna Rangarajan 2 and Kenichi Kanatani
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1 Taubin Ellipse Fitting by Hyperaccurate Least Squares Yuuki Iwamoto, 1 Prasanna Rangarajan 2 and Kenichi Kanatani 1 This paper presents a new method for fitting an ellipse to a point sequence extracted from images. The basic principle is the least squares, minimizing the algebraic distance. Exploiting the fact that the least-squares solution depends on the way the scale is normalized, we analyze the accuracy to high order terms with the scale normalization weight unspecified and determine the weight so that the second order bias is zero. We demonstrate by experiments that our method is superior to the Taubin method, which is also noniterative and known to be highly accurate. Although the highest accuracy is achieved by maximum likelihood, it requires iterations, which may not converge in the presence of large noise. In contrast, our method analytically computes a solution without iterations. 1 Department of Computer Science, Okayama University, Japan 2 Department of Electrical Engineering, Southern Methodist University, U.S.A ,18 17,24,25 22 FS 4 HEIV KCR ,15 2. Ax 2 + 2Bxy + Cy 2 + 2f Dx + Ey + f 2 F = 1 f x, y 1 21,22 x α, y α, α = 1,..., 1 A,..., F 1 1 f = 6 1 c 29 Information Processing Society of Japan
2 J A,..., F J = 1 2 Ax 2 α + 2Bx α y α + Cyα 2 + 2f Dx α + Ey α + f 2 F 2 2 A = = F = F = 1 3 A + C = 1 4 A 2 + B 2 + C 2 + D 2 + E 2 + F 2 = 1 5 A 2 + B 2 + C 2 + D 2 + E 2 = 1 6 A 2 + 2B 2 + C 2 = 1 7 AC B 2 = ,5,2, 4 1 7, u = A B C D E F 9 2 u, u = 1 1 Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 1 A 2 + 4B 2 + C 2 + 4f 2 D2 + E 2 + f 4 F 2 = 2 Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 1 A 2 + 4B 2 + C 2 + 4f 2 D2 + E 2 = 3 4 AC B 2 = AC B 2 < 1 a, b a, b u 6 ξ ξ = x 2 2xy y 2 2f x 2f y f u, ξ = 12 x α, y α ξ ξ α 1 J = 1 u, ξ α 2 = 1 u ξ α ξ α u = u, Mu M M = 1 ξ α ξ α u 2 1 u Mu = λu 15 λ λ u λ Mu λ 15 λ u 6 u u u = u, u u u, u > u, u < 6 u, u > 6 M u, Mu > u u, u < λ < u u, u λ λ 2 c 29 Information Processing Society of Japan
3 u u 5 I 15 Mu = λu 16 M Taubin Taubin 23 TB 22 x 2 1 α x α y α f x α TB = 4 x α y α x 2 α + yα 2 x α y α f y α f x α x α y α yα 2 f y α f B x α f y α f 2 f x α f y α f 2 C A x α, y α x α, ȳ α x α, y α ξ α ξ α = ξ α + 1ξ α + 2ξ α 18 ξ α, 1ξ α, 2ξ α x α, y α x 2 1 α 2 x α ȳ α ȳ ξ α = 2 α, 1 ξ 2f B x α α = 2f ȳ α A f 2 2 x α x α 2 x α y α + 2ȳ α x α 2ȳ α y α 2f x α 2f y α 1, 2 ξ α = C B x 2 α 2 x α y α yα 2 14,15 2 ξ α 2 ξ α V [ξ α ] = E[ 1ξ α 1ξ α ] E[ ] x α, y α σ E[ x α ] = E[ y α ] =, E[ x 2 α] = E[ y 2 α] = σ 2, E[ x α y α ] = V [ξ α ] = E[ 1ξ α 1ξ α ] = σ 2 V [ξ α ] 2 1 C A 19 x 2 1 α x αȳ α f x α x αȳ α x 2 α + ȳα 2 x αȳ α f ȳ α f x α x V [ξ α ] = 4 αȳ α ȳ 2 α f ȳ α f B x α f ȳ α f 2 f x α f ȳ α f 2 C A V [ξ α ] σ 2 V [ξ α ] Taubin TB = 1 V [ξ α ] 22 x α, ȳ α x α, y α M = 1 ξ α + 1 ξ α + 2 ξ α ξ α + 1 ξ α + 2 ξ α = + 1 M + 2 M , 1M, 2M = 1 1M = 1 2 M = 1 ξ α ξ α, 24 ξα 1ξ α + 1ξ α ξ α, 25 ξα 2 ξ α + 1ξ α 1 ξ α + 2ξ α ξ α 16 u, λ u = ū + 1 u + 2 u +, λ = λ + 1 λ + 2 λ , , M + 2 M + ū + 1 u + 2 u + = λ + 1 λ + 2 λ + ū + 1 u + 2 u ū = λ ū 29 1 u + 1 M ū = λ 1 u + 1 λ ū c 29 Information Processing Society of Japan
4 2u + 1M 1u + 2M ū = λ 2u + 1λ 1u + 2λ ū ξ α, ū = 24 ū = 29 λ = 25 ū, 1 M ū = 3 ū 1λ = 3 1 u = 1 M ū 32 ū = ū P ū ū θ+ 1θ+ 2θ + 2 = 1 1 θ, 1 θ = 1 θ θ λ 2λ = ū, 2M ū ū, 1 M 1 M ū ū, T ū = 33 ū, ū ū, ū T = 2M 1M 1M u ū ū 2 u ū 2 u = P ū 2 u = 2 u u = 2λ ū + 1M 1M ū 2M ū ū, T ū = ū, ū ū T ū u V [u] = E[ 1u 1u ] = E[ 1Mu 1Mu ] = 1 [ ] E ξ 2 α, u ξ α ξ β, u ξ β β=1 = 1 u, E[ ξ 2 α ξ β ]u ξ α ξ β 2 u, V [ξ α ]u ξ α ξ α, 37 = 1 ū, V [ξ α ]u ξ α ξ α ξ α α E[ 1 ξ α 1 ξ β ] = δ αβ σ 2 V [ξ α ] δ αβ 37 V [u] Taubin V [u] 1 E[ 1 u] = 33 2 E[ 2u ] 34 T 26 E[ 2 M ] E[ 2 M ] = 1 ξα E[ 2 ξ α ] + E[ 1 ξ α 1 ξ α ] + E[ 2 ξ α ] ξ α ξα e 13 + V [ξ α ] + e 13 ξ α = σ 2 TB + 2S[ ξ c e 13] 2 22 ξ c, e 13 ξ c = 1 ξ α, e 13 = S[ ] S[A] = A + A /2 E[ 1 M 1 M ] E[ 1M 1M ] tr[ V 2 [ξ α ]] ξ α ξ α + ξ α, ξα V [ξ α ] + 2S[V [ξ α ] ξα ξ α ] tr[ ] 39, T E[T ] = σ 2 TB + 2S[ ξ c e 13] S[V [ξ α ] ξα ξ α ] tr[ V [ξ α ]] ξ α ξ α + ξ α, ξα V [ξ α ] 42 4 c 29 Information Processing Society of Japan
5 36 2u E[ 2 u ] = ū, E[T ]ū ū, ū ū E[T ]ū ξ c, ū =, ξ α, ū = E[T ]ū E[T ]ū = σ 2 TB ū + A + C ξ c 1 ξ 2 α, ξα V [ξ α ]ū +ū, V [ξ α ] ξα ξ α = I 2 E[ 2u ] E[ 2u ]= ū,e[t ]ūū E[T ]ū = I ūū E[T ]ū = E[T ]ū 45 I ūū = P ū = = 46 44, 45 2 E[ 2 u ] = σ 2 TB ū + A + C ξ c 1 ξ 2 α, ξα V [ξ α ]ū +ū, V [ξ α ] ξα ξ α 8. Taubin 44 ξ c, ū =, ξ α, ū = ū, E[T ]ū ū, E[T ]ū = σ 2 ū, TBū 1 ξ 2 α, ξα ū, V [ξ α ]ū = σ 2 ū, TB ū 1 2 tr[ ξα ξ α ]ū, V [ξ α ]ū = σ 2 ū, TB ū 1 tr[ ū, V 2 [ξ α ]ū ξ α ξ α ] = σ 2 ū, TB ū σ2 tr[ ] Taubin = TB Taubin 2 E[ 2 u ] = σ 2 q TB ū + A + C ξ c 1 2 +ū, V [ξ α ] ξα ξ α ξ α, ξα V [ξ α ]ū q = 1 tr[ ] 5 ū, TB ū TBū q TBū 5 q < 1 Taubin 9. = TB + 2S[ ξ c e 13] 1 tr[ V [ξ 2 α ]] ξ α ξ α + ξ α, ξα V [ξ α ] +2S[V [ξ α ] ξα ξ α ] 42 E[T ] = σ 2 43 E[ 2 u ] = σ 2 ū, ū ū, ū ū = ξ α, xα, ȳ α x α, y α Oσ 2 1, 2 43 E[ 2u ] Oσ a x, y σ Taubin c 29 Information Processing Society of Japan
6 u u u a b O 1 a 31 b σ = Taubin Chojnacki 4 FS 25 1b σ =.5 u ū u u ū u = P ūu 53 2a P ū I ūū ū 2b, c σ 1 B D B = 1 1 u a, D = 1 a= a=1 u a 2 54 u a a 2c KCR 12,14,15 D KCR = σ ξ ] tr[ α ξ α 55 ū, V [ξ α ]ū Taubin TB u = 1/λMu M 2 λ a b c a u ū u b, c 1a a b σ Taubin c KCR 5 λ λ u 2b Taubin 38 Taubin 2c Taubin 2 2b c 2b, c Taubin Taubin Taubin 6 c 29 Information Processing Society of Japan
7 3 155 Taubin Taubin ,15 2 Taubin : C o A. Albano, Representation of digitized contours in terms of conics and straight-line segments, Comput. Graphics Image Process., , ,,,, 29-CVIM , F. J. Bookstein, Fitting conic sections to scattered data, Comput. Graphics Image Process., , W. Chojnacki, M. J. Brooks, A. van den Hengel and D. Gawley, On the fitting of surfaces to data with covariances, IEEE Trans. Patt. Anal. Mach. Intell., , D. B. Cooper and. Yalabik, On the computational cost of approximating and recognizing noise-perturbed straight lines and quadratic arcs in the plane, IEEE Trans. Computers, , A. Fitzgibbon, M. Pilu and R. B. Fisher, Direct least square fitting of ellipses, IEEE Trans. Patt. Anal. Mach. Intell., , W. Gander, H. Golub, and R. Strebel, Least-squares fitting of circles and ellipses, BIT, , R. Gnanadesikan, Methods for Statistical Data Analysis of Multivariable Observations, Wiley, ew Yori,.Y., U.S.A K. Kanatani, Geometric Computation for Machine Vision, Oxford University Press, Oxford, U.K., ,,, ,,, , K. Kanatani, Statistical Optimization for Geometric Computation: Theory and Practice, Elsevier Science, Amsterdam, The etherlands, 1996; Dover, ew York, , KCR,, 25-CVIM , ,,, 25-CVIM , K. Kanatani, Statistical optimization for geometric fitting: Theoretical accuracy analysis and high order error analysis, Int. J. Comp. Vis , Y. Leedan and P. Meer, Heteroscedastic regression in computer vision: Problems with bilinear constraint, Int. J. Comput. Vision., , ,,,,, 28-CVIM , ,,,, D-II, J85-D-II , K. A. Paton, Conic sections in chromosome analysis, Patt. Recog., , P. L. Rosin, A note on the least squares fitting of ellipses, Patt. Recog. Lett., , [I],, , c 29 Information Processing Society of Japan
8 22 [II],, , 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. Patt. Anal. Mach. Intell., , ,,, 25-CVIM , ,,,, 26-CVIM , E[ 1M 1M ] E[ 1M 1M ] = E[ 1 ξα 1 ξ 1 α + 1 ξ α ξ α = 1 2 = ξ α ξ α = 1 2 ξβ 1 ξ β + 1 ξ β ξ β ] β=1 E[ ξ α 1 ξ α + 1 ξ α ξ α ξ β 1 ξ β + 1 ξ β ξ β ] E[ ξ α 1 ξ ξβ α 1 ξ β + ξ α 1 ξ α 1 ξ β ξ β + 1 ξ ξβ α ξ α 1 ξ β 1 ξ β ξ β ] E[ ξ α 1 ξ α, ξβ 1 ξ β + ξ α 1 ξ α, 1 ξ β ξ β + 1 ξ α ξ α, ξβ 1 ξ β + 1 ξ α ξ α, 1 ξ β ξ β ] = 1 E[ 1ξ 2 α, ξβ ξ α 1ξ β + 1ξ α, 1ξ β ξ α ξ β + ξ α, ξβ 1 ξ α 1 ξ β + 1 ξ α 1 ξ β, ξ α ξ β ] = 1 E[ ξ 2 α ξβ 1 ξ α 1 ξ β + tr[ 1 ξ β 1 ξ α ] ξ α ξ β = ξ α, ξβ 1ξ α 1ξ β + 1ξ α 1ξ ξα β ξ β ] ξα ξ β E[ 1 ξ α 1 ξ β ] + tr[ E[ 1 ξ β 1 ξ α ]] ξ α ξ β + ξ α, ξβ E[ 1 ξ α 1 ξ β ] + E[ 1ξ α 1 ξ β ] ξα ξ β 2 +δ αβ V [ξ α ] 2 2 ξα ξ β ξα ξ α δ αβ V [ξ α ]+tr[ δ αβ V [ξ α ]] ξ α ξ β + ξ α, ξβ δ αβ V [ξ α ] ξα ξ β V [ξ α ] + tr[ V [ξ α ]] ξ α ξ α + ξ α, ξα V [ξ α ] +V [ξ α ] ξα ξ α tr[ V [ξ α ]] ξ α ξ α + ξ α, ξα V [ξ α ] + 2S[V [ξ α ] ξα ξ α ] 57 8 c 29 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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