#### DMCA

## Statistical efficiency of curve fitting algorithms (2004)

Venue: | Computational Statistics and Data Analysis |

Citations: | 28 - 2 self |

### Citations

306 | Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation
- Taubin
- 1991
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Citation Context ... (1.2) and improve its performance. One way to define weights wi results from a linear approximation to di: di ≈ |P(xi, yi; Θ)| ‖∇xP(xi, yi; Θ)‖ where ∇xP = (∂P/∂x, ∂P/∂y) is the gradient vector, see =-=[20]-=-. Then one minimizes the function n∑ [P(xi, yi; Θ)] F4(Θ) = 2 ‖∇xP(xi, yi; Θ)‖2 (1.4) i=1 This method is called the gradient weighted algebraic fit (GRAF). It is a particular case of (1.3) with wi = 1... |

283 |
Statistical Optimization for Geometric Computation: Theory and Practice
- Kanatani
- 1996
(Show Context)
Citation Context ...me that σ2 i = σ2 for all i, but note that our results can be easily generalized to arbitrary σ2 i > 0. Concerning the true points ¯xi, i = 1, . . ., n, two assumptions are possible. Many researchers =-=[6, 13, 14]-=- consider them as fixed, but unknown, points on the true curve. In this case their coordinates (¯xi, ¯yi) can be treated as additional parameters of the model (nuisance parameters). Chan [6] and other... |

144 | Direct least-squares fitting of algebraic surfaces
- PRATT
- 1987
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Citation Context ...ighly nonlinear problem, and in its exact form (4.4) is not used in practice. Instead, there are two modifications of GRAF popular among experimenters. One is due to Chernov and Ososkov [8] and Pratt =-=[17]-=-: F ′ 4 (a, b, R) = R−2 n∑ [(xi − a) 2 + (yi − b) 2 − R 2 ] 2 → min (4.6) i=1 (it is based on the approximation (xi − a) 2 + (yi − b) 2 ≈ R 2 ), and the other due to Agin [1] and Taubin [20]: F ′′ 4 (... |

111 | Least-Square Fitting of Circles and Ellipses
- Gander, Golub, et al.
- 1994
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Citation Context ...own parameters to be estimated. Typically, P is a polynomial in x and y, and its coefficients are unknown parameters (or functions of unknown parameters). For example, a number of recent publications =-=[2, 10, 11, 16, 19]-=- are devoted to the problem of fitting quadrics Ax 2 +Bxy+Cy 2 +Dx+Ey+F = 0, in which case Θ = (A, B, C, D, E, F) is the parameter vector. The problem of fitting circles, given by equation (x−a) 2 +(y... |

93 | P.: Heteroscedastic regression in computer vision: Problems with bilinear constraint
- Leedan, Meer
- 2000
(Show Context)
Citation Context ...own parameters to be estimated. Typically, P is a polynomial in x and y, and its coefficients are unknown parameters (or functions of unknown parameters). For example, a number of recent publications =-=[2, 10, 11, 16, 19]-=- are devoted to the problem of fitting quadrics Ax 2 +Bxy+Cy 2 +Dx+Ey+F = 0, in which case Θ = (A, B, C, D, E, F) is the parameter vector. The problem of fitting circles, given by equation (x−a) 2 +(y... |

48 |
Least-squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola, and parabola
- Ahn, Rauh, et al.
(Show Context)
Citation Context ...own parameters to be estimated. Typically, P is a polynomial in x and y, and its coefficients are unknown parameters (or functions of unknown parameters). For example, a number of recent publications =-=[2, 10, 11, 16, 19]-=- are devoted to the problem of fitting quadrics Ax 2 +Bxy+Cy 2 +Dx+Ey+F = 0, in which case Θ = (A, B, C, D, E, F) is the parameter vector. The problem of fitting circles, given by equation (x−a) 2 +(y... |

23 |
Computer Perception of Curved Objects Using a Television Camera
- Turner
- 1974
(Show Context)
Citation Context ...2 ‖∇xP(xi, yi; Θ)‖2 (1.4) i=1 This method is called the gradient weighted algebraic fit (GRAF). It is a particular case of (1.3) with wi = 1/‖∇xP(xi, yi; Θ)‖ 2 . The GRAF is known since at least 1974 =-=[21]-=- and recently became standard for polynomial curve fitting [20, 16, 10]. The computational cost of GRAF depends on the function P(x, y; Θ), but, generally, the GRAF is much faster than the OLSF. It is... |

21 |
On circular functional relationships
- Chan
- 1965
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Citation Context ...me that σ2 i = σ2 for all i, but note that our results can be easily generalized to arbitrary σ2 i > 0. Concerning the true points ¯xi, i = 1, . . ., n, two assumptions are possible. Many researchers =-=[6, 13, 14]-=- consider them as fixed, but unknown, points on the true curve. In this case their coordinates (¯xi, ¯yi) can be treated as additional parameters of the model (nuisance parameters). Chan [6] and other... |

21 | Estimation of a circular arc center and its radius - Landau - 1987 |

17 |
den Hengel, A.: Rationalising the Renormalisation Method of Kanatani
- Chojnacki, Brooks, et al.
- 2001
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Citation Context |

13 |
Large Sample Bias in Least Squares Estimators of a Circular Arc Center
- Berman
- 1989
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Citation Context ...d. He also showed that the covariance matrices of the OLSF and the GRAF attain, to the leading order in σ, his lower bound. We note, however, that in most cases the OLSF and algebraic fits are biased =-=[4, 5]-=-, hence the KCR lower bound, as it is derived in [13, 14], does not immediately apply to these methods. In this paper we extend the KCR lower bound to biased estimates, which include the OLSF and all ... |

12 |
The Statistical Behaviour of Some Least Squares Estimators of the Centre and Radius of a Circle
- Berman, Culpin
- 1986
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Citation Context ... T and ¯xi = (¯xi, ¯yi) T , for brevity. The random vectors ei = xi − ¯xi are assumed to be independent and have zero mean. Two specific assumptions on their probability distribution can be made, see =-=[4]-=-: Cartesian model: Each ei is a two-dimensional normal vector with covariance matrix σ 2 i I, where I is the identity matrix. Radial model: ei = ξini where ξi is a normal random variable N(0, σ 2 i ),... |

12 |
Orthogonal least squares fitting by conic sections, in
- Späth
- 1997
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Citation Context |

10 |
Fitting ellipses and general second-order curves
- Agin
- 1981
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Citation Context ...nd Ososkov [8] and Pratt [17]: F ′ 4 (a, b, R) = R−2 n∑ [(xi − a) 2 + (yi − b) 2 − R 2 ] 2 → min (4.6) i=1 (it is based on the approximation (xi − a) 2 + (yi − b) 2 ≈ R 2 ), and the other due to Agin =-=[1]-=- and Taubin [20]: F ′′ 4 (a, b, R) = 1 ∑ (xi − a) 2 + (yi − b) 2 n∑ [(xi − a) 2 + (yi − b) 2 − R 2 ] 2 → min (4.7) i=1 9(here one simply averages the denominator of (4.4) over 1 ≤ i ≤ n). We refer th... |

10 |
Effective algorithms for circle fitting
- Chernov, Ososkov
- 1984
(Show Context)
Citation Context ... = 0, in which case Θ = (A, B, C, D, E, F) is the parameter vector. The problem of fitting circles, given by equation (x−a) 2 +(y−b) 2 −R 2 = 0 with three parameters a, b, R, also attracted attention =-=[8, 14, 15, 18]-=-. We consider here the problem of fitting general curves given by implicit equations P(x, y; Θ) = 0 with Θ = (θ1, . . .,θk) being the parameter vector. Our goal is to investigate statistical propertie... |

9 |
Cramér-Rao lower bounds for estimation of a circular arc center and its radius
- Chan, Thomas
- 1995
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Citation Context ...fore, Dmin = ⎛ ⎜ ⎝ ∑ u 2 i ∑ uivi ∑ ui ∑ uivi ∑ v 2 i ∑ vi ∑ ⎞−1 ui ∑ ⎟ vi ⎠ n (4.5) where we denote, for brevity, ui = ¯xi − a R , vi = ¯yi − b R The above expression for Dmin was derived earlier in =-=[7, 14]-=-. Now, our Theorem in Section 3 shows that the weighted algebraic fit (4.3) is statistically efficient if and only if the weight function satisfies w(x, y; a, b, R) = c(a, b, R)/(4R 2 ). Since c(a, b,... |

5 |
Least-squares fitting by circles
- Späth
(Show Context)
Citation Context ... = 0, in which case Θ = (A, B, C, D, E, F) is the parameter vector. The problem of fitting circles, given by equation (x−a) 2 +(y−b) 2 −R 2 = 0 with three parameters a, b, R, also attracted attention =-=[8, 14, 15, 18]-=-. We consider here the problem of fitting general curves given by implicit equations P(x, y; Θ) = 0 with Θ = (θ1, . . .,θk) being the parameter vector. Our goal is to investigate statistical propertie... |

4 |
The circular structural model
- Anderson
- 1981
(Show Context)
Citation Context ...onsider them as fixed, but unknown, points on the true curve. In this case their coordinates (¯xi, ¯yi) can be treated as additional parameters of the model (nuisance parameters). Chan [6] and others =-=[3, 4]-=- call this assumption a functional model. Alternatively, one can assume that the true points ¯xi are sampled from the curve P(x, y; ¯ Θ) = 0 according to some probability distribution on it. This assu... |

3 |
Fitting circles and lines by least squares: theory and experiment, preprint, available at http://www.math.uab.edu/cl/cl1
- Chernov, Lesort
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Citation Context ...[20]: F ′′ 4 (a, b, R) = 1 ∑ (xi − a) 2 + (yi − b) 2 n∑ [(xi − a) 2 + (yi − b) 2 − R 2 ] 2 → min (4.7) i=1 9(here one simply averages the denominator of (4.4) over 1 ≤ i ≤ n). We refer the reader to =-=[9]-=- for a detailed analysis of these and other circle fitting algorithms, including their numerical implementations. We have tested experimentally the efficiency of four circle fitting algorithms: the OL... |