## Direct least Square Fitting of Ellipses (1998)

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Citations: | 273 - 3 self |

### BibTeX

@MISC{Fitzgibbon98directleast,

author = {Andrew Fitzgibbon and Maurizio Pilu and Robert B. Fisher},

title = {Direct least Square Fitting of Ellipses},

year = {1998}

}

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### Abstract

This work presents a new efficient method for fitting ellipses to scattered data. Previous algorithms either fitted general conics or were computationally expensive. By minimizing the algebraic distance subject to the constraint 4ac - b² = 1 the new method incorporates the ellipticity constraint into the normalization factor. The proposed method combines several advantages: (i) It is ellipse-specific so that even bad data will always return an ellipse; (ii) It can be solved naturally by a generalized eigensystem and (iii) it is extremely robust, efficient and easy to implement.

### Citations

1091 |
The Algebraic Eigenvalue Problems
- Wilkinson
- 1965
(Show Context)
Citation Context ... lCu as Q 2 u = lCu. Now, substituting v = Qu and premultiplying by Q -1 gives v = lQ -1 CQ -1 v so that s(S, C) = s(Q -1 CQ -1 ) -1 and thus i(S, C) = i(Q -1 CQ -1 ). From Sylvester’s Law of Inerti=-=a [18]-=-, we have that for any symmetric and nonsingular X, i(S) = i(X T SX). Therefore, substituting X = X T = Q -1 , we have i(C) = i(Q -1 CQ -1 ) = i(S, C). o We can now state Theorem 1. THEOREM 1. The sol... |

448 |
Computer and Robot Vision
- Haralick, Shapiro
- 1992
(Show Context)
Citation Context ..., y) to the conic F(a; x) = 0. The fitting of a general conic may be approached by minimizing the sum of squared algebraic distances 16 2 7 'A a N = Ê F xi i= 2 1 of the curve to the N data points xi=-= [7]-=-. In order to avoid the trivial solution a = 06, and recognizing that any multiple of a solution a represents the same conic, the parameter vector a is constrained in some way. Many of the published a... |

227 |
Estimation of Planar Curves, Surfaces, and Nonplanar Space Curves De ned by Implicit Equations with Applications to Edge and Range Image Segmentation
- Taubin
- 1991
(Show Context)
Citation Context ...aint applied to the parameters. For instance, many authors suggest �a� 2 = 1, Rosin [14] and Gander [5] impose a + c = 1 while Rosin also investigates f = 1 [14]. Taubin’s approximate square dis=-=tance [17] may a-=-lso be viewed as the quadratic constraint �Na� 2 = 1 where N is the Jacobian [¶F(a; x1) ¡ ¶F(a; xN)] T . Note that these constraints are all either linear, of the form c a = 1 or quadratic, co... |

189 | Recovery of parametric models from range images: the case for superquadrics with global deformations - Solina, Bajcsy - 1990 |

129 |
Fitting Conic Sections to Scattered Data
- Bookstein
- 1979
(Show Context)
Citation Context ... x1) ¡ ¶F(a; xN)] T . Note that these constraints are all either linear, of the form c a = 1 or quadratic, constraining a T Ca = 1 where C is a 6 ™ 6 constraint matrix. In a seminal work, Bookste=-=in [1]-=- showed that if a quadratic constraint is set on the parameters (e.g., to avoid the trivial solution a = 06) the minimization (2) can be solved by considering rankdeficient generalized eigenvalue syst... |

114 |
Methods for Statistical Data Analysis of Multivariate Observations
- Gnanadesikan
- 1977
(Show Context)
Citation Context ...s of nonellipticity and illustrate the potential of the method for coarse bounding of generic 2D data. 4.2 Low-Eccentricity Bias Fig. 1b shows three experiments designed after Sampson [16] (following =-=[6]-=-) and basically consists of the same parabolic data but with different realizations of added isotropic Gaussian noise (s = Fig. 4. Simple six-line Matlab implementation of the ellipse fitting method. ... |

103 |
Fitting Conic Sections to Very Scattered Data: An Iterative Refinement of the Bookstein Algorithm
- Sampson
- 1982
(Show Context)
Citation Context ...sion or noise—they often yield unbounded fits to hyperbolae. In a situation where ellipses are specifically desired, such fits must be rejected as useless. A number of iterative refinement procedure=-=s [16]-=-, [8], [12] alleviate this problem, but do not eliminate it. In addition, these techniques often increase the computational burden unacceptably. This paper introduces a new fitting method that combine... |

89 | Computer description of curved objects - Agin, Binford - 1976 |

83 |
Optimization Theory and Application
- Rao
- 1984
(Show Context)
Citation Context ...llipse. The appropriate constraint is well known, namely, that the discriminant b 2 - 4ac be negative. However, this constrained problem is difficult to solve in general as the Kuhn-Tucker conditions =-=[13]-=- do not guarantee a solution. In fact, we have not been able to locate any reference regarding the minimization of a quadratic form subject to such a nonconvex inequality. Although the imposition of t... |

67 | Describing complicated objects by implicit polynomials - Keren, Cooper, et al. - 1994 |

65 |
Non-parametric segmentation of curves into various representations
- Rosin, West
- 1995
(Show Context)
Citation Context ...to {p 2 , 2pq, q 2 + r 2 } so as to keep the conic discriminant always negative. A nonlinear minimization of the algebraic error over the space {p, q, r, d, e, f} is performed. In this journal, Rosin =-=[15]-=- reiterated this problem by stating that (2) ellipse-specific fitting is essentially a nonlinear problem and iterative methods must always be employed for this purpose. In the following section, we sh... |

62 | Least squares fitting of circles and ellipses
- Gander, Golub, et al.
- 1994
(Show Context)
Citation Context ...of the circle, is of great importance for many industrial applications. Despite its importance, however, there has been until now no computationally efficient ellipse-specific fitting algorithm [14], =-=[5]-=-. In this paper, we introduce a new method for fitting ellipses, rather than general conics, to segmented data. As we shall see in the next section, current methods are either computationally expensiv... |

50 |
Shape Detection in Computer Vision using the Hough Transform
- Leavers, F
- 1992
(Show Context)
Citation Context ... conclude by presenting some possible extensions. 2 PREVIOUS METHODS AND THEIR LIMITATIONS The literature on ellipse fitting divides into two broad techniques: clustering (such as Hough-based methods =-=[9]-=-, [19]) and leastsquares fitting. Least-squares techniques center on finding the set of parameters that minimize some distance measure between the data points and the ellipse. In this section, we brie... |

49 | R.B.: A Buyer’s Guide to Conic Fitting
- Fitzgibbon, Fisher
- 1995
(Show Context)
Citation Context ...ifficult in general, in this case we have the freedom to arbitrarily scale the parameters so we may simply incorporate the scaling into the constraint and impose the equality constraint 4ac - b 2 = 1 =-=[4]-=-. This is a quadratic constraint which may be expressed in the matrix form a T Ca = 1 as 0 0 2 0 0 0 0 -1 0 0 0 0 T 2 0 0 0 0 0# a 0 0 0 0 0 0# a = 1. (4) 0 0 0 0 0 0# 0 0 0 0 0 0# ! Now, following Bo... |

37 | Ellipses and Predicting Confidence Envelopes using a Bias Corrected Kalman Filter
- Porrill
- 1989
(Show Context)
Citation Context ...se—they often yield unbounded fits to hyperbolae. In a situation where ellipses are specifically desired, such fits must be rejected as useless. A number of iterative refinement procedures [16], [8]=-=, [12]-=- alleviate this problem, but do not eliminate it. In addition, these techniques often increase the computational burden unacceptably. This paper introduces a new fitting method that combines the follo... |

35 | Detecting partially occluded ellipses using the Hough transform - Yuen, Illingworth, et al. - 1989 |

30 |
Statistical bias of conic fitting and renormalization
- Kanatani
- 1994
(Show Context)
Citation Context ...r noise—they often yield unbounded fits to hyperbolae. In a situation where ellipses are specifically desired, such fits must be rejected as useless. A number of iterative refinement procedures [16]=-=, [8]-=-, [12] alleviate this problem, but do not eliminate it. In addition, these techniques often increase the computational burden unacceptably. This paper introduces a new fitting method that combines the... |

27 | P.L.: "A Note on the Least Squares Fitting of Ellipses
- Rosin
- 1993
(Show Context)
Citation Context ...ction of the circle, is of great importance for many industrial applications. Despite its importance, however, there has been until now no computationally efficient ellipse-specific fitting algorithm =-=[14]-=-, [5]. In this paper, we introduce a new method for fitting ellipses, rather than general conics, to segmented data. As we shall see in the next section, current methods are either computationally exp... |

17 | Ellipse detection and matching with uncertainty
- Ellis, Abbod, et al.
- 1992
(Show Context)
Citation Context ...nsolved. If ellipse fitting was needed, one had to rely either on generic conic fitting or on iterative methods to “nudge” the estimation towards ellipticity. For instance, Porrill [12], Ellis et =-=al. [2]-=-, and Rosin [14] use conic fitting to initialize a Kalman filter that iteratively minimizes some error metric in order to gather new image evidence and to reject nonellipse fits by testing the discrim... |

15 | Ellipse fitting by accumulating five-point fits - Rosin - 1993 |

6 | Ellipse-specific least-squares fitting - Pilu, Fitzgibbon, et al. - 1996 |

5 |
Part-based Grouping and Recogntion: A Model-Guided Approach
- Pilu
- 1996
(Show Context)
Citation Context ... have positive generalized eigenvalues. Now we show that the minimization of �Da� 2 subject to 4ac - b 2 = 1 yields exactly one solution, which corresponds, by virtue of the constraint, to an elli=-=pse [11]-=-. For the demonstration, we will require Lemma 1. 1. Note that the method of Lagrange multipliers is not valid when the gradient of the constraint function becomes zero. In (5), this means Ca = 0, but... |

4 | Stable Segmentation of 2D Curves
- Fitzgibbon
- 1998
(Show Context)
Citation Context ...oise performance compared to some of the least-squares fitting method reviewed in Section 2. In this short paper, we are not able to present a large body of results—which can be found in abundance i=-=n [3]��-=-�so we limited ourselves to those that are the most representative. All experiments were conducted using the Matlab system [10]. Eigensystems are solved using the underlying EISPACK routines. We shall... |

4 | Segmenting curves into lines and arcs - Rosin, West - 1990 |

4 | A buyer's guide to conic tting - Fitzgibbon, Fisher - 1995 |

4 | Fitting ellipses and predicting con dence envelopes using a bias cor rected kalman lter - Porrill - 1990 |

4 | A note on the least square tting of ellipses - Rosin - 1993 |

3 | Least-squares tting of circles and ellipses - Gander, Golub, et al. |

3 | Statistical bias of conic tting and renormalization - Kanatani - 1994 |

2 | Conic sections in automatic chromosome analysis - Paton - 1969 |

2 | Fitting ellipses and predicting comfidence envelopes using a bias corrected Kalman filter - Porrill - 1990 |

2 | Part-based Grouping and Recogntion: AModel-Guided Approach - Pilu - 1996 |

2 | Ellipse tting by accumulating ve-point ts - Rosin - 1993 |

1 |
Shape using volumetric primitives
- Yuen, Illingworth, et al.
- 1989
(Show Context)
Citation Context ...lude by presenting some possible extensions. 2 PREVIOUS METHODS AND THEIR LIMITATIONS The literature on ellipse fitting divides into two broad techniques: clustering (such as Hough-based methods [9], =-=[19]-=-) and leastsquares fitting. Least-squares techniques center on finding the set of parameters that minimize some distance measure between the data points and the ellipse. In this section, we briefly pr... |

1 | ARANSAC-basedapproach to model fitting and its application to finding cylinders in range data - Bolles, Fischler - 1981 |

1 | The algebraic eigenvalueproblem - Wilkinson - 1965 |

1 | Fitting ellipses and predicting com dence envelopes using a bias corrected Kalman lter - Porrill - 1990 |