## Robust Regression for Data with Multiple Structures (2001)

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Venue: | In 2001 IEEE Conference on Computer Vision and Pattern Recognition, volume I |

Citations: | 20 - 3 self |

### BibTeX

@INPROCEEDINGS{Chen01robustregression,

author = {Haifeng Chen and Peter Meer and David E. Tyler},

title = {Robust Regression for Data with Multiple Structures},

booktitle = {In 2001 IEEE Conference on Computer Vision and Pattern Recognition, volume I},

year = {2001},

pages = {1069--1075}

}

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### Abstract

In many vision problems (e.g., stereo, motion) multiple structures can occur in the data, in which case several instances of the same model need to be recovered from a single data set. However, once the measurement noise becomes significantly large relative to the separation between the structures, the robust statistical methods commonly used in the vision community tend to fail. In this paper, we show that all these techniques are special cases of the general class of M-estimators with auxiliary scale, and explain their failure in the presence of noisy multiple structures. To be able to cope with data containing multiple structures the techniques innate to vision (Hough and RANSAC) should be combined with the robust methods customary in statistics. The implications of our analysis are illustrated by introducing a simple procedure for 2D multistructured data problematic for all known current techniques. 1.

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Citation Context ... M-estimators and least median of squares (LMedS), were successfully applied to vision problems. However, some of the “most” robust techniques used in the vision community, Hough transform and RAN=-=SAC [2]-=-, are innate and were developed independently to meet the specific needs of processing visual data. In this paper we show that the robust estimation techniques frequently used in vision applications (... |

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Citation Context ...n � from the corresponding hyperplane. Equivalently, RANSAC is the minimization ����� � � � problem (6) with and a preset scale. RANSAC was also successfully generalized to use a smo=-=oth loss function [16]-=-. In many vision applications an educated guess of the error tolerance can be made, however, once the range of the noisy inliers is not well defined, determining an adequate scale value becomes diffic... |

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Citation Context ...������������� ��� � ��� � � � � �s¡ �s¡�� � � (2) ¡��¢� subject to (1), the solution is the total (orthogonal) least squares (TLS)=-= estimate. See [5] for a discussion of TLS type methods in vision and [19] for an -=-in-depth treatment of TLS estimation. It is well known that in solving (2) the true valuessare replaced by the orthogonal projections¡ of ¡�� onto the hyperplane defined by � . The orthogonal ... |

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Citation Context ...cale parameter � and � � � ��� � � � � �s¡ ��� (6) � a loss function ��� satisfying the following properties: ��� � � � � nonnegative with ; � ��=-=� � � � even symmetric, ; and nondecreasing � with . See [6] for-=- an introduction to M-estimators and [3] for a more in-depth discussion. In the sequel we will consider � � only redescending M-estimators which are characterized by bounded loss functions and w.l... |

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Citation Context ...e is recognized, a diagnosis tool incorporating the advantages of both approaches can be designed and the well documented limitation of the current robust techniques for data with multiple structures =-=[13] c-=-an be relaxed. Figure 2 gives some examples of such data. To define the model employed for the inliers without loss of generality we will assume for the moment that all the available measurementss¢¡... |

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Citation Context ...d, the solution of ��� � ��� � � � � ¡��¢� � � � � � � � � ¡ � � ��� �s¡ ��� ���s� � ��� � (10) � ��� � wh=-=ere is the weight function (9). An example of a vision application employing this approach is [1]. -=-For the errors-in-variables model only the redescending M-estimators (the ones considered in this paper) may have nonzero breakdown point [20] which is bounded upward by , where� is the dimension of... |

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Citation Context ...tures in the input. This fact is well known in the vision literature and to improve the quality of the accumulator � ��� ��� � the replacement of with a smooth loss function was often =-=proposed, e.g., [4, 10]. In-=- the “enhanced” Hough transforms the scale parameter is set based on a guess of the noise corrupting the edge pixels [4], or in relation to the size of the accumulator bin [10]. The chosen value o... |

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Citation Context ...tures in the input. This fact is well known in the vision literature and to improve the quality of the accumulator � ��� ��� � the replacement of with a smooth loss function was often =-=proposed, e.g., [4, 10]. In-=- the “enhanced” Hough transforms the scale parameter is set based on a guess of the noise corrupting the edge pixels [4], or in relation to the size of the accumulator bin [10]. The chosen value o... |

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Citation Context ...tion will prefer a least squares type fit since it yields a smaller scale value. (Recall the S-estimator/LMedS connection discussed at the end of Section 2.) This was first observed experimentally in =-=[9]-=-. The deficiency of robust estimators is not specific to the step-edge data considered above. In [13] two more configurations are discussed and their analysis with the tools introduced in this paper y... |

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Citation Context ...r-structure (Figure 2a). The presence of structured outliers is the most damaging for the performance of robust estimators and almost always the breakdown condition is due to such data configurations =-=[18]-=-. When the outlierstructure contains slightly less than half the data points and both model instances are corrupted by significant noise, none of the four classes of robust methods popular in computer... |

3 |
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Citation Context ...ample of a vision application employing this approach is [1]. For the errors-in-variables model only the redescending M-estimators (the ones considered in this paper) may have nonzero breakdown point =-=[20] which is bo-=-unded upward by , where� is the dimension of the model. The breakdown ����� point, however, reflects a worst case situation since it just specifies that in the presence of severe contamina... |

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Citation Context ...o different approaches to obtain the scale: – it is set (known) prior to the parameter estimation; – it is estimated from the data. In either context (6) is called an M-estimator with auxiliary sc=-=ale [7, 15]-=-. The main distinction between the robust techniques developed in the vision community, Hough transform and RANSAC, and the robust estimators imported from statistics is in the way the scale is determ... |

1 |
On the uniqueness of S and constrained M-functionals under non-elliptical distributions
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Citation Context ...o different approaches to obtain the scale: – it is set (known) prior to the parameter estimation; – it is estimated from the data. In either context (6) is called an M-estimator with auxiliary sc=-=ale [7, 15]-=-. The main distinction between the robust techniques developed in the vision community, Hough transform and RANSAC, and the robust estimators imported from statistics is in the way the scale is determ... |