# Computational Analysis of PDE-Based Shape Analysis Models by Exploring the Damped Wave Equation
## 6 Experiments on Shapes
### Study of the Damping Parameter  
First, we wanted to study the impact of the damping parameter. For this experiment, we used 30 different, logarithmic equidistant distributed, damping parameters \(k\) between \(10^{-3}\) and \(1\). Then, we computed the hit rate depending on \(k\) within a shape class. The classes we chose are cat, centaur, dog, horse and wolf. Additionally, we showed the mean of all classes. To compare the results with our previous works [6; 7; 9; 10], we added \(k=0\) (dashed lines) to simulate the non-damping scenario. The results can be viewed in Figure 9.  
Across all classes, we see a similar behaviour of the hit rates. For the first-order method, the hit rate falls with rising \(k\). In contrast, the second-order methods increase the hit rate until \(k\approx 0.1\). This point differs from shape to shape. For \(k>0.1\), hit rates are the same and fall similar to each other.  
Let us take a closer look at the plot from the centaur (cf. Figure 10). In addition, we resized the hit rate axes, so that variations become more evident. Here we notice that the second-order methods have a dent at \(k\in(0.01,0.03)\). While the CN method continues to rise, the SO\(l_{0}\) method decreases in this interval. This inconstant growth in the hit rate can  
Figure 8: (**First row**) We see the Gaussian distribution \(\exp(-x^{2}/0.01)\) as the initial condition. (**Columns**) The solutions of different numerical schemes for the (left to right) IE, CN and SO\(l_{0}\) schemes are presented. (**Rows 2-4**) We see the solution after different iterations. In the second row, we calculated 10 iterations. The third row shows us the solution after 20 iterations, and in the fourth row, the 50 th iteration is presented. The time increment was set to \(\tau=0.01\).  
be observed in all other classes in Figure 9, as well. It is even noticeable in the plot with the mean hit rate.  
In conclusion of these experiments, we see that certain damping is a beneficial mechanism (when about \(k\approx 0.1\)). Since the second-order methods are supposed to give an accurate account of the underlying analytical model, this means that a reasonable analytical model appears to be the damped wave equation (with \(k\approx 0.1\)). Let us note in this context that one cannot expect that the results of the IE method and an analytical model resolved at second-order accuracy coincide exactly, so one may interpret in many cases such as in Figure 10 the peak situation (there again at \(k\approx 0.1\)) and the IE method results as numerically identical.  
Figure 10: Hit rate from the centaur shapes over different damping parameters \(k\). Solid lines indicate changing damping parameters and dashed lines denote a fixed damping parameter (\(k=0\)). The colours distinguish between the different numerical methods: red for IE, blue for CN and green for SO\(l_{0}\).  
Figure 9: We plotted the hit rate at a geodesic error of \(0.25\) over the damping parameter \(k\). With the three different colours, we identify the numerical schemes that were used to solve the damped wave equation on the different shapes. Red indicate the IE scheme, blue denotes the CN method, and green stands for the SO\(l_{0}\) scheme. Solid lines denote a changing damping parameter \(k\), and the dashed lines indicate a fixed \(k=0\).