Finding the Maximum Consistent Set

In our algorithms, we follow the basic strategy of diagnostic statistics, outlier detection and then fitting after removal of the outliers using the classical methods (Rousseeuw and Leroy, 2003). After finding outliers in a local neighbourhood, we remove the outliers and fit the plane to the remaining points with PCA, and estimate the local saliencies: normals and curvatures using the best-fit-plane parameters and the estimated eigenvalues. The algorithms serve two purposes: (i) outlier detection, and (ii) robust saliency feature estimation. Outlier detection by using the new methods takes less computation time and are more accurate than the diagnostic methods in Chapter 3. Outlier detection can also be used in point cloud denoising, see Section 4.5.3.1. It is known that off-surface points can appear as noise which may be treated as outliers or gross errors (Sotoodeh,2006). Robust saliency features can also be used for local neighbourhood based point cloud processing e.g. sharp feature preserving (Sections 4.5.3.2) and segmentation (Sections 4.5.3.3).

Our algorithms (an earlier version published in Nurunnabi et al., 2013c) can be classified based on the outlier detection procedures involved. The two proposed robust outlier detection methods use the robust z-score (Rz-score) and robust Mahalanobis Distance (RMD). The algorithms couple the idea of using point to plane Orthogonal Distance (OD) and the λ0 (variation along the surface normal) for identifying outliers. Only the h-subset (a set of points that contains h points) of the majority of good points in a local neighbourhood that are most reliable, homogenous and have minimum sorted ODs are used to fit the plane and to calculate respective λ0 values. The decision based on majority of consistent points is a fundamental idea of robust statistics (Rousseeuw and Leroy, 2003). Moreover, fixing h can remove the problem of choosing an explicit value of the error threshold, which is a major problem in the RANSAC

paradigm (Subbarao and Meer, 2006). In general, we set h = d0.5ke to get the majority of consistent points. In order to get the best h-subset, the algorithm starts with a random h0-subset, where theh0-subset has minimal points (in case of plane fitting, h0 = 3). If the rank of this subset is less than h0, randomly add more points gradually to the subset until the rank is equal to h0. The technique of finding the h-subset by using the h0-subset also reduces the iteration time, because h0 is considerably less thanh. Based on the outlier-free minimal subset (MS) the h-subset can produce a better set of plane parameters. Consequently it gives a better and more accurate normal and the relevant error scale (point to plane orthogonal distance) for the most consistent h-subset, which is used to get the best-fit-plane and robust saliency features. To get an outlier-free h0-minimal subset, one could iterate (randomly sampling) k_C

h0 times (C means

combination), but the number of iterations increases rapidly with the increase in k (the number of points in a local neighbourhood). We employ a Monte Carlo type probabilistic approach (Rousseeuw and Leroy, 2003) to calculate the number of iterations It. If we set Pr for the desired probability that at least one outlier-free h0-subset can be found from the percentage (outlier rate: probability that a point is an outlier) of outlier contaminated data, then Pr = 1−(1−(1−)h0)It, and It can be expressed as:

It = log(1−Pr)

log(1−(1−)h0). (4.1)

Therefore, It = f(pr, , h0), where h0 = 3. We use Pr = 0.9999. Users have the freedom to choose Pr based on their knowledge about the data. Fixing a larger probability increases the number of iterations giving a more accurate, more consistent subset with a high probability of the subset being outlier-free. It is known that the number of iterations is a trade-off between accuracy and efficiency. The outlier rate is generally unknown a priori. A smaller than the real outlier percentage in the data can be influenced by the masking effect. However, an excessively large value of can create swamping. Experience of MLS data reveals that generally, the majority (more than 50%) of points are inliers within a local neighbourhood. To keep the computation safe, we assume= 0.5 for real data. The user can change based on knowledge about the presence of outliers in their data. We find a h-subset for every iteration, based on the minimum Orthogonal Distance (OD) respective to the corresponding fitted plane of the h0-subset, and calculate λ0 values for all theh-subsets from theItiterations. It is reasonable to assume that the plane that is related to the least λ0 value also has

maximum surface consistency (i.e. minimum variation along the normal) among all the h-subsets. Since, the maximum consistency is attained at λ0 ≈ 0, we get maximum surface point consistency from the points that have minimum ODs to the fitted plane. We name the method of getting most consistent h-subset as Maximum Consistency with Minimum Distance (MCMD). The algorithms for outlier detection and robust saliency feature estimation are described in the next two Sections 4.3.3 and 4.3.4, respectively.

Our algorithms are motivated by the concept of robust outlier detection in statistics: detecting the outliers by searching for the model fitted by the majority of the data that have some similarities (Rousseeuw and Leroy, 2003;

Rousseeuw and Hubert, 2011). The subset of majority points used in our algorithms includes the most homogenous and consistent points to each other. The Maximum Consistent Set (MCS) of homogeneous points in a local neighbourhood can be derived by the steps outlined in Algorithm4.1.

Algorithm 4.1: MCS (Maximum Consistent Set)

1. Input: neighbourhood N pi of a pointpi of size k.

2. To get the h-subset for the MCS in a local neighbourhood of a point of interest pi,

we randomly choose h0 points (in our case h0 = 3, the minimum required number

of points for fitting a plane).

3. For the aboveh0-subset, we fit a plane by PCA and calculate ODs for all the points

in the local neighbourhood to the fitted plane and sort them according to their ODs (Figure4.2a) as:

|OD(p1)| ≤, . . . ,≤ |OD(ph)|, . . . ,≤ |OD(pk)|,

where OD(pi) = (pi−p¯)T ·nˆ is the OD for the point pi to the fitted plane, and ¯p

and ˆn are the mean and the unit normal of the fitted plane, respectively.

4. Fit the plane to the above sorted h-subset in Step 3, and calculate theλ0 value for

that plane and store it to the list of previousλ0 values, defined as S(λ0).

5. Iterate Steps 2 to 4 forIttimes given by Eq. (4.1). We get theλ0 values forIttimes

inS(λ0).

6. Find the h-subset of points for which λ0 is minimum in S(λ0). This is the required

MCS (magenta ellipse in Figure4.2c) in the local neighbourhood. 7. Output: The MCS forpi.