## Linear combination representation for outlier detection in motion tracking (2005)

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Venue: | In Proc. IEEE Conf. on Computer Vision and Pattern Recognition |

Citations: | 5 - 2 self |

### BibTeX

@INPROCEEDINGS{Guo05linearcombination,

author = {Guodong Guo and Charles R. Dyer},

title = {Linear combination representation for outlier detection in motion tracking},

booktitle = {In Proc. IEEE Conf. on Computer Vision and Pattern Recognition},

year = {2005},

pages = {274--281}

}

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### Abstract

In this paper we show that Ullman and Basri’s linear combination (LC) representation, which was originally proposed for alignment-based object recognition, can be used for outlier detection in motion tracking with an affine camera. For this task LC can be realized either on image frames or feature trajectories, and therefore two methods are developed which we call linear combination of frames and linear combination of trajectories. For robust estimation of the linear combination coefficients, the support vector regression (SVR) algorithm is used and compared with the RANSAC method. SVR based on quadratic programming optimization can efficiently deal with more than 50 percent outliers and delivers more consistent results than RANSAC in our experiments. The linear combination representation can use SVR in a straightforward manner while previous factorization-based or subspace separation methods cannot. Experimental results are presented using real video sequences to demonstrate the effectiveness of our LC + SVR approaches, including a quantitative comparison of SVR and RANSAC. 1.

### Citations

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Citation Context ...restrictive limitations [9] [16]. Furthermore, only 4 parameters are involved for each regression. The linear combination representation can directly take advantage of support vector regression (SVR) =-=[20]-=-, which can deal withsa high percentage of outliers using slack variables. Major contributions of this paper are: (1) showing that linear combination representations [19] [21] can be used effectively ... |

2452 |
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Citation Context ...SVR are not yet known. As part of our evaluation of SVR for outlier detection we will compare its performance to one of the standard methods for robust regression, RANSAC, for RANdom SAmple Consensus =-=[4]-=-, which uses a small number of randomly selected points to estimate the underlying model and uses the remaining points for verification. The process iterates many times, and the model with the largest... |

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Citation Context ...hods for outlier detection on several video sequences. Because of space limits, only two sequences are reported here. For each sequence, feature points were detected and tracked using the KLT tracker =-=[15]-=-. The FLC and TLC algorithms were each applied to the measurement matrix W . The first sequence contains a hotel and has frames of size 480 × 512. We used the first 48 frames, though any length can be... |

1441 |
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Citation Context ...tion is executed for each ɛ. In our linear combination representation, i.e., Eqs. (2)- (5), there is only one output variable for each regression, so standard SVR code can be used. We used SV M light =-=[11]-=- in our experiments. 3.3. Automatic Threshold Determination After using SVR, we measure the regression residual for each input data value and sort them in descending order. Typically, we get a curve o... |

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Citation Context ...itative comparison of SVR and RANSAC. 1. Introduction Robust tracking of feature points in image sequences is of great importance for tasks such as video sequence alignment [2], structure from motion =-=[17]-=-, and motion segmentation [18]. In order to obtain good results in motion-based vision tasks, feature trajectory outliers have to be detected and removed. For tracked features there are typically two ... |

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Citation Context ...mentation [18]. In order to obtain good results in motion-based vision tasks, feature trajectory outliers have to be detected and removed. For tracked features there are typically two types of errors =-=[22]-=-: (1) Location errors, where the location of a 2D feature is distorted and usually assumed to exhibit Gaussian behavior. (2) False matches, where there is a mismatch of 2D features, i.e., two correspo... |

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Citation Context ... constitute a group until all frames are used. Assuming each section has s frames, we get s four-frame groups and therefore s FLC computations. For each four-frame group, SVR or RANSAC (see p. 103 in =-=[7]-=- for algorithm details) is used to estimate the linear combination coefficients [ax bx cx dx] T and [ay by cy dy] T , and classify the corresponding trajectories as inliers or outliers. Dealing with t... |

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Citation Context ...ure trajectories, a different formulation can be employed that results in several improvements over existing methods. The linear combination (LC) of images representation proposed by Ullman and Basri =-=[19]-=- for alignment-based object recognition will be used to detect outliers. The LC relation also holds using 4 motion trajectories [21]. Based on these ideas, we develop two methods, one is frame-based a... |

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Citation Context ...tion [20] resulting in the final solution given by l� w = (9) where αi,α ∗ i i=1 (αi − α ∗ i )xi are Lagrange multipliers. The value of b in Eq. (6) can be determined by plugging Eq. (9) into Eq. (6) =-=[5]-=-. See [20] for more details. 3.2. A Toy Example To illustrate the SVR idea, we use a toy example that contains 30 points in 2D with 10 in a line and the remaining 20 being outliers distributed on both... |

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Citation Context ...only 4 parameters to estimate (note that Eqs. (2) and (3) deal with the x and y coordinates independently), which is one benefit of the FLC representation. On the contrary, the affine trifocal tensor =-=[18]-=- uses 3 frames, but there are 16 parameters to estimate together with 9 constraints. It is wellknown that the more parameters there are to estimate, the more susceptible random sampling type methods [... |

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Citation Context ...edian of squares (LMedS) [22] and RANSAC [4] are to instability. In two-view affine geometry, there are 6 parameters to estimate, which is also larger than the 4 in our case. In affine point transfer =-=[14]-=-, correspondence between two images must be established before the corresponding point in a third image can be computed by transfer, which is different from FLC. Finally, FLC is easy to understand and... |

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Citation Context ... for outlier removal and motion analysis based on the affine fundamental matrix and affine trifocal tensor. Recently some other methods were developed that remove outliers in a video sequence [9] [1] =-=[16]-=- based on factorization methods [17] [3]. One of these approaches measured subspace distance based on a rank4 property, but five observations were used instead of four by claiming that location errors... |

17 |
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Citation Context ... methods for outlier removal and motion analysis based on the affine fundamental matrix and affine trifocal tensor. Recently some other methods were developed that remove outliers in a video sequence =-=[9]-=- [1] [16] based on factorization methods [17] [3]. One of these approaches measured subspace distance based on a rank4 property, but five observations were used instead of four by claiming that locati... |

14 |
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(Show Context)
Citation Context ...to a factorization method [3], which only works when the percentage of outliers is low because the M-estimator they used has a low break point. Other recent work dealt with missing features [17] [10] =-=[6]-=- [8]. All these approaches are based on the factorization method or subspace separation, and need to factorize a big matrix [1] or impose restrictive limitations [9] [16]. In this paper we show that f... |

13 |
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Citation Context ...hods for outlier removal and motion analysis based on the affine fundamental matrix and affine trifocal tensor. Recently some other methods were developed that remove outliers in a video sequence [9] =-=[1]-=- [16] based on factorization methods [17] [3]. One of these approaches measured subspace distance based on a rank4 property, but five observations were used instead of four by claiming that location e... |

12 | Multi-body segmentation: Revisiting motion consistency
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(Show Context)
Citation Context ...(5). The coefficients for the x and y coordinates may or may not be the same, but this does not matter for outlier detection. We call this approach linear combination of trajectories. Machline et al. =-=[12]-=- discussed the distinction between the row space and column space of the measurement � � matrix Wx based on subspace analysis. They called the “trajecWy tory matrix” and [Wx|Wy] the “flow-field matrix... |

10 |
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- 2003
(Show Context)
Citation Context ... factorization method [3], which only works when the percentage of outliers is low because the M-estimator they used has a low break point. Other recent work dealt with missing features [17] [10] [6] =-=[8]-=-. All these approaches are based on the factorization method or subspace separation, and need to factorize a big matrix [1] or impose restrictive limitations [9] [16]. In this paper we show that for o... |

6 | Tracking of object with SVM regression
- Zhu, Wang, et al.
- 2001
(Show Context)
Citation Context ...fixed value, ɛ = 0.01, for SVR in all our experiments. We used the knee point of the residuals (see Section 3.3) to automatically choose a threshold to separate outliers from inliers. SVR was used in =-=[23]-=- for two-view affine matching. They iteratively changed ɛ from big to small, and classified the SVs as outliers. As shown in the experiment in Figure 1, this approach not only misclassifies many outli... |

4 |
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Citation Context ...based on the affine fundamental matrix and affine trifocal tensor. Recently some other methods were developed that remove outliers in a video sequence [9] [1] [16] based on factorization methods [17] =-=[3]-=-. One of these approaches measured subspace distance based on a rank4 property, but five observations were used instead of four by claiming that location errors and outliers inflate the rank [9]. Subs... |

1 |
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(Show Context)
Citation Context ...bination (LC) of images representation proposed by Ullman and Basri [19] for alignment-based object recognition will be used to detect outliers. The LC relation also holds using 4 motion trajectories =-=[21]-=-. Based on these ideas, we develop two methods, one is frame-based and the other is trajectory-based. The approach uses linear regression without the need to factorize a large matrix [1] or impose res... |

1 |
Tracking of object with SVM regression,” Proc. Conf. Computer Vision and Pattern Recognition, v. 2, 240-245, 2001. 100 150 200 250 300 350 400 450 500 The initial trajectories tracked by KLT The initial trajectories tracked by KLT (a) (b) (a) (b) The inli
- Zhu, Wang, et al.
(Show Context)
Citation Context ...fixed value, ɛ = 0.01, for SVR in all our experiments. We used the knee point of the residuals (see Section 3.3) to automatically choose a threshold to separate outliers from inliers. SVR was used in =-=[23]-=- for two-view affine matching. They iteratively changed ɛ from big to small, and classified the SVs as outliers. As shown in the experiment in Figure 1, this approach not only misclassifies many outli... |