Robust statistical analysis: LMS and LTS method

4.2 Robust statistical analysis: LMS and LTS method

As described in the introduction the Authors selected two methods for carrying out the analysis; RANSAC Hartley and Zisserman, 2004 and the Forward Search Atkinson and Riani, 2004 and Atkinson et al, 2010 using a “secure clean dataset”. For this purpose two different techniques obtained from the Robust Statistic Rousseeuw and Leroy, 1987 were used: the LMS Least Median Square and the LTS Least Trimmed Square methods Draper and Smith, 1998. In the LMS method, by combining the points used in the analysis, it is possible to minimize the median of the residuals of the totality of points 18 with respect to the five used in the specific combination. The combination of selected points is then used for computing the solution. The LTS method is quite similar although it differs thanks to the minimization of the sum of the square of the residuals of the totality of points with respect to the five used in the specific combination. Also in this case the combination of selected points is used for computing the solution. For each method the value of standardized residual Eq. 1 was compared to an established threshold obtained from the bibliography Draper and Smith, 1998: ∑ ∙ ∑ ↔ 1 where υ stn = standardized residuals υ = unknown residuals of observations S = scale factor p = weight of the observations n = number of unknown parameters 5 m = number of observation in use 18 threshold = adopted according to bibliography = 2.2 The first value of S is shown in the Equation 2: ∙ 1.4826 ∙ .1 0 1 23 45 2 The iterations continue as long as there are no conditions that are capable of stopping the difference between the values of S in two successive cycle has to be negligible and in each iteration the observations are re-weighed as one if ,7 8 meaning that the observation is an inlier, otherwise as they are re-weighed as zero meaning that the observation is an outlier. At the end of the iterations these two robust methods enable us to obtain a first skimming of the data with an initial detection of outliers. The comparison of the results obtained with each method showed that the LMS method enabled us to obtain a more realistic clear subset relative to the LTS method. Figure 7 shows the tie points identified by the “software extractor”, yet it is evident that they are not all inliers: in fact some outliers were found using LMS in green in Figure 8 while no outliers were observed using LTS Figure 10. Figure 7. In green the tie point outliers identified by the software extractor Figure 8. Example of LMS result: outliers in green and inliers in blue the outlier number “5” is shown in red in the zoom of Figure 9 Figure 9. An example of one outlier 5 identified by LMS In Figure 9 we can see that the outliers identified by the LMS method are in actual fact points which are badly associated by the extractor, therefore in this case LTS is not suitable since it does not identify these outliers Figure 10. Figure 10. Example of the LTS result: inliers in blue there are no outliers

4.3 RANSAC and Forward Search