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## Ellipse Fitting with Hyperaccuracy

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Citations: | 12 - 7 self |

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430 | Direct least-squares fitting of ellipses,”
- Fitzgibbon, Pilu
- 1999
(Show Context)
Citation Context ... if the points {(xα, yα)} are sampled from an ellipse, the fitted equation may define a hyperbola or other curves in the presence of large noise, and a technique for preventing this has been proposed =-=[8]-=-. Here, we do not impose any constraints to prevent non-ellipses, assuming that noise is sufficiently small. 1 The program is available at http://www.suri.it.okayama-u.ac.jp 2 One can set f0 = 1 unles... |

314 | Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation.
- Taubin
- 1991
(Show Context)
Citation Context ...). They have the form ∆ ∗ 1M = −2 ∆ ∗ 2M = −2 N� α=1 N� α=1 ((∆1u, V0[ξ α]u) + O(σ 2 )) ¯ ξ α ¯ ξ ⊤ α (u, V0[ξ α]u) 2 , (20) ((∆1u, V0[ξ α]u) + O(σ 2 ))(∆ξ α ¯ ξ ⊤ α + ¯ ξ α∆ξ ⊤ α ) (u, V0[ξ α]u) 2 . =-=(21)-=- Equating terms of O(1), O(σ), and O(σ 2 ) on both sides of eq. (19), we obtain the following expressions (we omit the derivation): ∆1u = − ¯ M − ∆1Mu (22) ∆2u = − ¯ M − ∆2Mu + ¯ M − ∆1M ¯ M − ∆1Mu + ... |

288 |
Statistical Optimization for Geometric Computation: Theory and Practice,
- Kanatani
- 1996
(Show Context)
Citation Context ...7, 19, 23]. However, their major concern is the consistency and efficiency of the estimator in the asymptotic limit as the number of points increases. A contrasting approach was presented by Kanatani =-=[11]-=-, who generalized ellipse fitting into an abstract framework, which he called geometric fitting. Having actual image processing in mind, he pursued fitting schemes whose accuracy rapidly increases as ... |

195 |
Geometric Computation for Machine Vision,
- Kanatani
- 1993
(Show Context)
Citation Context ... lower bound. 1 Introduction Circular and spherical objects in the scene are generally projected onto ellipses on the image plane, and their 3-D shapes and positions can be computed from their images =-=[9]-=-. For this reason, fitting ellipses (including circles) to a point sequence is one of the first steps of various vision applications, and numerous papers have been written on this subject. They are cl... |

164 |
Fitting conic sections to scattered data.
- Bookstein
- 1979
(Show Context)
Citation Context ...+ Cy 2 + 2f0(Dx + Ey) + F f 2 0 = 0, (1) where f0 is an arbitrary scaling constant 2 . If we define u = � A B C D E F �⊤ � 2 2 , ξ = x 2xy y 2f0x 2f0y f 2 �⊤ 0 , (2) eq. (1) is written as (u, ξ) = 0. =-=(3)-=- Throughout this paper, we denote the inner product of vectors a and b by (a, b). Since the magnitude of the vector u is indeterminate, we adopt normalization �u� = 1. Geometrically, eq. (3) describes... |

134 |
Fitting conic sections to “very scattered” data: An iterative refinement of the Bookstein algorithm. Comput. Graphics Image Process.
- Sampson
- 1982
(Show Context)
Citation Context ...[ξ α]u) in ¯ M and ∆1M by û (the corresponding perturbation of ∆2M is of O(σ 3 )). They have the form ∆ ∗ 1M = −2 ∆ ∗ 2M = −2 N� α=1 N� α=1 ((∆1u, V0[ξ α]u) + O(σ 2 )) ¯ ξ α ¯ ξ ⊤ α (u, V0[ξ α]u) 2 , =-=(20)-=- ((∆1u, V0[ξ α]u) + O(σ 2 ))(∆ξ α ¯ ξ ⊤ α + ¯ ξ α∆ξ ⊤ α ) (u, V0[ξ α]u) 2 . (21) Equating terms of O(1), O(σ), and O(σ 2 ) on both sides of eq. (19), we obtain the following expressions (we omit the d... |

96 | Heteroscedastic regression in computer vision: problems with bilinear constraint
- Leedan, Meer
- 2000
(Show Context)
Citation Context ...an attain that bound except for higher order terms in the noise level [6, 11, 13]. It has turned out that all existing iterative linear computing schemes, such as renormalization 1 [10, 11, 14], HEIV =-=[18]-=-, and FNS [7], has accuracy equivalent to ML [13]. It has been experimentally confirmed that these methods indeed attain high accuracy very close to the KCR lower bound. We say that an estimation meth... |

82 |
Describing Complicated Objects by Implicit Polynomials",
- Keren, Cooper, et al.
- 1994
(Show Context)
Citation Context ...) with the same accuracy as the FNS and the HEIV [13].s488 K. Kanatani 4 Error Analysis of ML Substituting ξ α = ¯ ξ α + ∆ξ α in the matrix M in eqs. (14), we obtain ∆1M = N� α=1 M = ¯ M + ∆1M + ∆2M, =-=(15)-=- ∆ξ ¯⊤ αξα + ¯ ξα∆ξ ⊤ α , ∆2M = (u, V0[ξα]u) N� α=1 ∆ξα∆ξ ⊤ α , (16) (u, V0[ξα]u) where ¯ M is the value of the matrix M defined by the true values { ¯ ξ α} of {ξ α}. The matrix L in eqs. (14) is writ... |

78 | On the fitting of surfaces to data with covariances”,
- Chojnacki, Brooks, et al.
- 2000
(Show Context)
Citation Context ... bound except for higher order terms in the noise level [6, 11, 13]. It has turned out that all existing iterative linear computing schemes, such as renormalization 1 [10, 11, 14], HEIV [18], and FNS =-=[7]-=-, has accuracy equivalent to ML [13]. It has been experimentally confirmed that these methods indeed attain high accuracy very close to the KCR lower bound. We say that an estimation method has hypera... |

58 |
Parametrized families of polynomials for bounded algebraic curve and surface fitting”,
- Taubin, P, et al.
- 1994
(Show Context)
Citation Context ... α ¯ ξ ⊤ α + ¯ ξ α∆ξ ⊤ α ) (u, V0[ξ α]u) 2 . (21) Equating terms of O(1), O(σ), and O(σ 2 ) on both sides of eq. (19), we obtain the following expressions (we omit the derivation): ∆1u = − ¯ M − ∆1Mu =-=(22)-=- ∆2u = − ¯ M − ∆2Mu + ¯ M − ∆1M ¯ M − ∆1Mu + ¯ M − ∆ ∗ 1M ¯ M − ∆1Mu − ¯ M − ∆ ∗ 2Mu + ¯ M − ∆2Lu − � ¯ M − ∆1Mu� 2 u. (23) From the first of eqs. (14), we have ¯ Mu = 0 and hence ¯ M − u = 0. It foll... |

40 |
Statistical bias of conic fitting and renormalization.
- Kanatani
- 1994
(Show Context)
Citation Context ...ximum likelihood) can attain that bound except for higher order terms in the noise level [6, 11, 13]. It has turned out that all existing iterative linear computing schemes, such as renormalization 1 =-=[10, 11, 14]-=-, HEIV [18], and FNS [7], has accuracy equivalent to ML [13]. It has been experimentally confirmed that these methods indeed attain high accuracy very close to the KCR lower bound. We say that an esti... |

28 | Statistical efficiency of curve fitting algorithms.
- Chernov, Lesort
- 2004
(Show Context)
Citation Context ...006. c Springer-Verlag Berlin Heidelberg 2006sEllipse Fitting with Hyperaccuracy 485 In his framework, a lower bound on the covariance matrix of the estimator is obtained [11, 12]. Chernov and Lesort =-=[6]-=- called it the KCR (Kanatani-CramerRao) lower bound and showed that it can be derived under a weaker assumption. It can be shown that ML (maximum likelihood) can attain that bound except for higher or... |

21 |
On circular functional relationships
- Chan
- 1965
(Show Context)
Citation Context ...stics combining voting and least squares in many different forms [3, 4, 15, 20–22], but there are also theoretical treatments, mainly by statisticians, regarding the problem as statistical estimation =-=[1, 2, 5, 16, 17, 19, 23]-=-. However, their major concern is the consistency and efficiency of the estimator in the asymptotic limit as the number of points increases. A contrasting approach was presented by Kanatani [11], who ... |

20 |
Cramér-Rao lower bounds for curve fitting
- Kanatani
- 1998
(Show Context)
Citation Context ...ct to u, we have ∇uJ = N� α=1 2(ξ α, u)ξ α (u, V0[ξ α]u) − (u, ξα) 2 . (11) (u, V0[ξα]u) N� α=1 The ML estimator û is obtained by solving ∇uJ = 0, or M = N� α=1 2(ξ α, u) 2 V0[ξ α]u (u, V0[ξ α]u) 2 . =-=(12)-=- Mu = Lu, (13) ξαξ ⊤ α , L = (u, V0[ξα]u) N� α=1 (ξα, u) 2V0[ξα] . (14) (u, V0[ξ 2 α]u) The FNS of Chojnacki et al. [7] solves eq. (13) by iteratively computing eigenvalue problems; the HEIV of Leedan... |

19 | P.: Unbiased estimation of ellipses by bootstrapping
- Cabrera, Meer
- 1996
(Show Context)
Citation Context ...itting higher order terms 3 in σ, by ⎛ ¯x ⎜ V0[ξα] = ⎜ ⎝ 2 α ¯xα¯yα 0 f0¯xα 0 0 ¯xα¯yα ¯x 2 α + ¯y 2 α ¯xα¯yα f0¯yα f0¯xα 0 0 ¯xα¯yα ¯y 2 α 0 f0¯yα 0 f0¯xα f0¯yα 0 f 2 0 0 0 0 f0¯xα f0¯yα 0 f 2 ⎞ ⎟ . =-=(4)-=- ⎟ 0 0 ⎠ 0 0 0 0 0 0 Since ξ α has only 2 degrees of freedom (i.e., xα and yα), V0[ξ α] has rank 2. Let û be an estimator of u obtained by some means. Its accuracy is measured by the following covaria... |

14 |
Structural
- Anderson, Gerbing
- 1988
(Show Context)
Citation Context ...stics combining voting and least squares in many different forms [3, 4, 15, 20–22], but there are also theoretical treatments, mainly by statisticians, regarding the problem as statistical estimation =-=[1, 2, 5, 16, 17, 19, 23]-=-. However, their major concern is the consistency and efficiency of the estimator in the asymptotic limit as the number of points increases. A contrasting approach was presented by Kanatani [11], who ... |

12 |
The Statistical Behaviour of Some Least Squares Estimators of the Centre and Radius of a Circle
- Berman, Culpin
- 1986
(Show Context)
Citation Context ...stics combining voting and least squares in many different forms [3, 4, 15, 20–22], but there are also theoretical treatments, mainly by statisticians, regarding the problem as statistical estimation =-=[1, 2, 5, 16, 17, 19, 23]-=-. However, their major concern is the consistency and efficiency of the estimator in the asymptotic limit as the number of points increases. A contrasting approach was presented by Kanatani [11], who ... |

12 |
Optimal Conic Fitting and Reliability Evaluation,”
- Kanazawa, Kanatani
- 1996
(Show Context)
Citation Context ...ximum likelihood) can attain that bound except for higher order terms in the noise level [6, 11, 13]. It has turned out that all existing iterative linear computing schemes, such as renormalization 1 =-=[10, 11, 14]-=-, HEIV [18], and FNS [7], has accuracy equivalent to ML [13]. It has been experimentally confirmed that these methods indeed attain high accuracy very close to the KCR lower bound. We say that an esti... |

7 | Unbiased errors-in-variables estimation using generalized eigensystem analysis,
- Muhlich, Mester
- 2004
(Show Context)
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6 | 2005 Further improving geometric fitting
- Kanatani
(Show Context)
Citation Context ...whose accuracy rapidly increases as the noise level decreases for a fixed number of points. He asserted that such methods can tolerate larger image processing uncertainty for a desired accuracy level =-=[13]-=-. A. Leonardis, H. Bischof, and A. Prinz (Eds.): ECCV 2006, Part I, LNCS 3951, pp. 484-495, 2006. c Springer-Verlag Berlin Heidelberg 2006sEllipse Fitting with Hyperaccuracy 485 In his framework, a lo... |

6 |
Huffel, Consistent estimation in an implicit quadratic measurement error model
- Kukush, Markovsky, et al.
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1 |
The efficiency of adjusted least squares in the linear functional relationship
- Kukush, Maschlce
(Show Context)
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1 |
A Bayesian method for filtering parametric and nonparametric models to noisy data
- Werman, Keren
(Show Context)
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