## Parameter Estimation Techniques: A Tutorial with Application to Conic Fitting (1995)

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### BibTeX

@MISC{Zhang95parameterestimation,

author = {Zhengyou Zhang},

title = {Parameter Estimation Techniques: A Tutorial with Application to Conic Fitting},

year = {1995}

}

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### Abstract

Almost all problems in computer vision are related in one form or another to the problem of estimating parameters from noisy data. In this tutorial, we present what is probably the most commonly used techniques for parameter estimation. These include linear least-squares (pseudo-inverse and eigen analysis); orthogonal least-squares; gradient-weighted least-squares; bias-corrected renormalization; Kalman filtering; and robust techniques (clustering, regression diagnostics, M-estimators, least median of squares). Particular attention has been devoted to discussions about the choice of appropriate minimization criteria and the robustness of the different techniques. Their application to conic fitting is described.

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Citation Context ...us derivatives of rst three orders, and yields a unique solution. The 95% asymptotic eOEciency on the standard normal distribution is obtained with the tuning constan c = 1:3998. ffl Huber's function =-=[6]-=- is a parabola in the vicinity of zero, and increases linearly at a given level jxj ? k. The 95% asymptotic eOEciency on the standard normal distribution is obtained with the tuning constant k = 1:345... |

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Citation Context ... a priori information, if any. The linearization of a nonlinear model leads to small errors in the estimates, which in general can be neglected, especially if the relative accuracy is better than 10% =-=[15, 4]-=-. However, as pointed by Maybank [12], the extended Kalman lter seriously underestimates covariance. Furthermore, if the current estimatess iji\Gamma1 is very dioeerent from the true one, the rst-orde... |

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Citation Context ...mic system from noisy data taken at discrete real-time intervals. Without entering into the theoretical justication of the Kalman lter, for which the reader is referred to many existing books such as =-=[7, 13]-=-, we insist here on the point that the Kalman lter yields at t i an optimal estimate of s i , optimal in the sense that the spread of the estimateerror probability density is minimized. In other words... |

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Citation Context ...amma s i )] ; where M is an arbitrary, positive semidenite matrix. The optimal estimatess i of the state vector s i is easily understood to be a least-squares estimate of s i with the properties that =-=[3]-=-: 1. the transformation that yieldsss i from [x T 0 \Delta \Delta \Delta x T i ] T is linear, 2.ss i is unbiased in the sense that E[ s i ] = E[s i ], 3. it yields a minimum variance estimate with the... |

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Citation Context ...y f i . Hence determine the change in the measure of t, i.e. \Deltaf i = f \Gamma f i , when datum i is deleted. 4. Delete datum i for which \Deltaf i is the largest, and goto step 2. It can be shown =-=[22]-=- that the above two techniques agrees with each other at the rst order approximation, i.e. the datum with the largest residual is also that datum inducing maximum change in the measure of t at a rst o... |

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Citation Context ...s. Another criterion of optimality, namely the minimum variance of estimation, is not addressed in this method. 8 Kalman Filtering Technique Nota: The rst four subsections are extracted from the book =-=[24]-=-: 3D Dynamic Scene Analysis: A Stereo Based Approach, by Z. Zhang & O. Faugeras (Springer Berlin 1992). Kalman ltering, as pointed out by Lowe [11], is likely to have applications throughout Computer ... |

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Citation Context ...ates may be completely perturbed. During the last three decades, many robust techniques have been proposed, which are not very sensitive to departure from the assumptions on which they depend. Hampel =-=[5]-=- gives some justications to the use of robustness (quoted in [17]): What are the reasons for using robust procedures? There are mainly two observations which combined give an answer. Often in statisti... |

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Citation Context ... B i p and recompute w i using the new p. 4. Return p if the update has converged; go back to step 2 otherwise. Remark 1: This implementation is dioeerent from that described in the paper of Kanatani =-=[8]-=-. This is because in his implementation, he uses the N-vectors to represent the 2-D points. In the derivation of the bias, he assumes that the perturbation in each N-vector, i.e., \Deltam ff in his no... |

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Citation Context ... zero for an ellipse, the arbitrary scale factor in the coeOEcients of the conic equation can be naturally removed by the normalization A + C = 1. This normalization has been used by many researchers =-=[16, 20]-=-. All ellipse can then be described by a 5-vector p = [A; B; D;E;F ] T ; and the system equation Q(x i ; y i ) = 0 becomes: f i 4 = a T i p \Gamma b i = 0 ; (2) where a i = [x 2 i \Gamma y 2 i ; 2x i ... |

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Citation Context ...ements. Due to the requirement of the back propagation of the state estimate, the application of the global IEKF is very limited. Maybe it is interesting only when the state does not evolve over time =-=[1]-=-. In that case, no back propagation is required. In the problem of estimating 3D motion between two frames, the EKF is applied spatially, i.e., it is applied to a number of matches. The 3D motion (the... |

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(Show Context)
Citation Context ...rization of a nonlinear model leads to small errors in the estimates, which in general can be neglected, especially if the relative accuracy is better than 10% [15, 4]. However, as pointed by Maybank =-=[12]-=-, the extended Kalman lter seriously underestimates covariance. Furthermore, if the current estimatess iji\Gamma1 is very dioeerent from the true one, the rst-order approximation, (22 and 23), is not ... |

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Citation Context ... zero for an ellipse, the arbitrary scale factor in the coeOEcients of the conic equation can be naturally removed by the normalization A + C = 1. This normalization has been used by many researchers =-=[16, 20]-=-. All ellipse can then be described by a 5-vector p = [A; B; D;E;F ] T ; and the system equation Q(x i ; y i ) = 0 becomes: f i 4 = a T i p \Gamma b i = 0 ; (2) where a i = [x 2 i \Gamma y 2 i ; 2x i ... |

3 |
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Citation Context ...od cannot t such conics because they require to set F = 0. This might suggest that the other normalizations are superior to the F = 1 normalization with respect to singularities. However, as shown in =-=[19]-=-, the singularity problem can be overcome by shifting the data so that they are centered on the origin, and better results by setting F = 1 has been obtained than by setting A + C = 1. INRIA Parameter... |

3 | A RANSAC-based approach to model tting and its application to nding cylinders in range data - Bolles, Fischler - 1981 |

2 |
Reliability analysis of parameter estimation in linear models with application to mensuration problems in computer vision
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(Show Context)
Citation Context ... a priori information, if any. The linearization of a nonlinear model leads to small errors in the estimates, which in general can be neglected, especially if the relative accuracy is better than 10% =-=[15, 4]-=-. However, as pointed by Maybank [12], the extended Kalman lter seriously underestimates covariance. Furthermore, if the current estimatess iji\Gamma1 is very dioeerent from the true one, the rst-orde... |

1 |
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Citation Context ...ral Parameter estimation is a discipline that provides tools for the eOEcient use of data for aiding in mathematically modelling of phenomena and the estimation of constants appearing in these models =-=[2]-=-. It can thus be visualized as a study of inverse problems. Much of parameter estimation can be related to four optimization problems: ffl criterion: the choice of the best function to optimize (minim... |

1 |
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Citation Context ...The rst four subsections are extracted from the book [24]: 3D Dynamic Scene Analysis: A Stereo Based Approach, by Z. Zhang & O. Faugeras (Springer Berlin 1992). Kalman ltering, as pointed out by Lowe =-=[11]-=-, is likely to have applications throughout Computer Vision as a general method for integrating noisy measurements. The behavior of a dynamic system can be described by the evolution of a set of varia... |

1 | Review of TINA: The She eld AIVRU vision system by J.Porrill et al - Lowe |

1 | Artificial Vision for Mobile Robots: Stereo Vision and Multisensory Perception - PI - 1991 |

1 | Stochastic Processes and Filtering Theory, Aca- demic - Jazwinsky - 1970 |

1 | Robust regres- sion methods for computer vision: A review, Int - Meer, Mintz, et al. - 1991 |