# Computational Analysis of PDE-Based Shape Analysis Models by Exploring the Damped Wave Equation
## 6 Experiments on Shapes
### Noisy Shape Experiments  
The shapes from the TOSCA dataset [2] are artificially created. Therefore, they have no noise in the coordinates of the points. We want to address this fact and add noise to the points. Again, we use the centaur as the shape class for our considerations. We applied a percentage normally distributed noise of 6% and 12% to each shape.  
With these noisy shapes, we repeat the experiments with the damping parameter from Section 6.1. This means we solve the damping wave equation with the \(\delta\)-peak initial condition and repeat this with different damping parameters \(k\) between \(10^{-3}\) and \(1\).  
The results are plotted in Figure 16. The relations of the colours to the numerical solvers remain the same. Dashed lines still refer to the non-damping scenario (\(k=0\)) and the solid lines indicate the change of \(k\). However, the symbols on the solid lines refer to the percentage of noise now. The triangle denotes a noise of 6%, and the square denotes a 12% noise.  
Figure 15: The comparison of the feature descriptors from the centaur shape. We used the damped wave equation (\(k=0.1\)) with the Gaussian (\(\sigma=10\)) initial condition to generate the feature descriptor. Shape 0, Shape 2 and the points are equal to the points, and shapes shown in Figure 1. (**Rows**) Different solvers. (**Columns**) Different points.  
The first thing we observe is that with higher noise, the hit rate is decreasing, which appears quite natural. The other thing we notice is that, in this scenario, the optimal value for the damping parameter is \(k=0.05\), which is half of the value from the non-noise scenario. What is also somewhat interesting is that we can increase the hit rate for the IE method with certain damping parameters compared to the non-damping case. With a noise of 12% and around \(k=0.02\), we are slightly better compared to the non-damping (\(k=0\)) case. The second-order methods show similar behaviour as in Figure 10.  
Figure 16: We solved the damped wave equation using different numerical schemes and the \(\delta\)-peaked initial condition over different \(k\). Then, we computed the hit rate at a geodesic error of 0.25 and plotted the hit rate over different damping parameters \(k\). From (**Top**) to (**Bottom**), we see the hit rate produced with the IE method (red), the CN scheme (blue) and the SO\(l_{0}\) method (green). With the dashed line, we indicate the results from the \(k=0\) scenario. The triangle and square denote a noise of 6% and a noise of 12%, respectively.  
Next, we wanted to recreate the experiments from Section 6.3. This time, we fixed the damping parameter at \(k=0.05\) and solved the damped wave equation with the Gaussian distributed initial condition. We repeated this with different values for \(\sigma\), the width of the Gaussian distribution. As before, we change \(\sigma\) to be between 1 and 20.  
The results are illustrated in Figure 17. The meaning of the colours and the symbols did not change. We obtained similar results for both noise levels. In the beginning, we see an increase in the hit rate. When entering the interval \(\sigma\approx 7\) to \(\sigma\approx 12\), we notice a plateau in the hit rate. In this interval, a rising \(\sigma\) did not affect the hit rate. For \(\sigma\approx 12\), the hit rate is decreasing again.  
Overall, our results from the noisy shapes are similar to the results from the artificial shapes. This indicates that our approach is robust and could be transferred to real-world shape-matching scenarios.