Page - 161 - in Joint Austrian Computer Vision and Robotics Workshop 2020

Figure 2: Examples of area curves resulting from ground truth (blue) and algorithmic (orange) segmentations. Scan 0 is an example of correct segmentation, the remaining three cases (scans 1, 2, 5) correspond to incorrect segmentations shown infigure1. 3.1.1 Curve Embedding To grasp the convex-vs-concave nature of the area curves and to embed them in a lower dimensional space we chose to approximate them by second or- der polynomials a(x;w)≈w0+w1x+w2x2 and to represent themby the three regressioncoefficients wi. Following [3] the regression coefficients w = [w0,w1,w2] > for each area curve are calculated by means of regularized least-squares, i.e, by solving w=(λI+Φ>Φ)−1Φ>a,whereλ is the regulariza- tion term,I the3×3 identity matrix,Φ the200×3 design matrix with rows [1,x,x2], andx indexes the slicesx∈{0..199}. The optimal regularization coefficient was deter- mined close to zeroλ≈ 0, which can be explained by the fact thatfittinga low-gradepolynomial to200 values does not suffer from overfitting. This reduces the curve fitting to ordinary least squares, i.e., mul- tiplication of the area curve vector by the psuedoin- verse of the design matrix: w= (Φ>Φ)−1Φ>a= Φ†a. 3.1.2 RegressionCoefficients Regression coefficients corresponding to all 100 ground truth (blue) as well as algorithmic (orange) segmentations are scatter-plotted in the first row of figure 3. Its second row shows the three correspond- ing kerneldensityestimation (KDE)plots. Thetwow0KDEplots indicateverysimilardistri- butionsandtherefore thew0 coefficientsdonotseem tobediscriminative. The too-thick segmented layers are mapped to concave area curves. Therefore, the distribution of w2 coefficients is of special interest, as they are re- sponsible for the positive/negative curvature of the polynomials. Looking at the KDE plot ofw2 coef- ficients, there is a high peek from the ground truth coefficients between 0 and 0.5, showing that there arealmostnonegativew2 coefficients. Therefore the assumption that ground truth curves tend to exhibit convexity (positive curvature) holds. In contrast, the orangeKDEresulting fromalgorithmsegmentations is more flat in the GT area and also exhibits a minor peek around -0.5. This indicates the presence of a cluster of negativew2 values, which corresponds to concave area curves. This distribution can be con- firmed looking at scatter plots including w2. For example in the w1–w2 plot there is a (blue) clus- ter formed by ground truth coefficients while several negativew2 algorithm coefficients are scattered out- sideof it. Interestingly the w1 coefficients exhibit a very similar behaviour to thew2 coefficients: almost no positive w1 GT coefficients and a tendency to bi- modal distribution of the algorithm ones forming a smallpeekaroundvalueof100. The highly correlated coefficientsw1 andw2 en- courageforfurtherdimensionalityreduction. Indeed, 161