# Improving Localization of Deep Inclusions in Time-Resolved Diffuse Optical Tomography
## 3 Results and Discussion
### Comparing State-of-the-Art Windows with Tukey and Gaussian Windows  
The purpose of the first simulations is to check whether Tukey or Gaussian windows improve the reconstruction in depth compared to \(w(t)=t^{n}e^{-pt}\) state-of-the-art windows. For the Mellin-Laplace reconstruction \(p=3\) and the first 35 moments were used (Figure 11(a)). For Gaussian windows, the value of \(\sigma\) was set to 0.3 and their centers were located from 0.3 to 9.6 ns in steps of 0.3 ns (32 windows in total, Figure 11(b)). For the Tukey windows, the same centers were used and the parameters were set to \(\alpha=0.25\) and \(t^{*}=0.3\) ns (Figure 11(c)). The given parameters were selected so that in all cases the windows totally covered the simulated signal and the number of windows were similar (35 for Mellin-Laplace and 32 for Tukey and Gaussian). The computed frequencies ranged from 0 to 2 GHz in steps of 200 MHz. The Figure 13 shows reconstruction using Melin-Laplace, Tukey and Gaussian temporal windows for inclusions that ranged from 1 to 2.5 cm deep (in steps of 0.5 cm). In Figure 14, the reconstructions for inclusions from 3 cm to 4 cm deep are given. In Figure 15, the evaluation metrics results are shown for each method. Significant differences are seen for inclusions shallower than 2.5 cm not just in terms of localization accuracy but also regarding absorption quantification. Localization error for Mellin-Laplace windows is around 0.4 cm for the inclusion located at 2 cm depth. However, for the novel windows the maximum localization error is 0.2 cm. Moreover, the absorption quantification is constantly underestimated by Mellin-Laplace windows, see how average contrast metric decreases exponentially and the contrast of Tukey and Gaussian windows is always greater than Mellin-Laplace windows in the range of 1 to 2.5 cm depth. For inclusions deeper than 3 cm Mellin-Laplace windows reconstruction is significantly worse in terms of absorption quantification. Also note that the localization error for Mellin-Laplace windows is much larger for the inclusion 4 cm deep; these results are in agreement with [44] which shows the same problem beyond 2 cm depth. Moreover, the relative reconstructed volume also increases due to the fact that Mellin-Laplace loses the inclusion localization and absorption quantification, and therefore, it tries to compensate it by reconstructing higher volume inclusions, see for example the inclusion at depth 2.5 cm. This phenomenon occurs due to the diffusion of photons  
Figure 12: Black curve represents a simulated DTOF. (**a**) Mellin–Laplace windows (\(p=3\), 35 moments), note that they are highly overlapped. (**b**) Gaussian windows (\(\sigma=0.3\)). (**c**) Tukey windows (\(\alpha=0.25\), \(t^{*}=0.3\) ns). inside the medium. In contrast, the results also show that Tukey and Gaussian windows give very similar reconstructions and their localization error is always less than 0.4 cm. Figure 14: Absorption, \(\mu_{a}\), reconstructions using Mellin–Laplace, Tukey and Gaussian windows for inclusions from 3 cm to 4 cm deep. The red circle indicates the correct location of the inclusion. Figure 13: Absorption, \(\mu_{a}\), reconstructions using Mellin–Laplace, Tukey and Gaussian windows for inclusions from 1 cm to 2.5 cm deep. The red circle indicates the correct location of the inclusion. The fact that proposed windows outperform Mellin-Laplace state-of-the-art windows for deep inclusions can be explained by the theoretical analysis shown in the previous sections. First, Mellin-Laplace windows have a poor late-arrival photon selectivity and windows are highly correlated. This implies that each Mellin-Laplace window has very little new information by itself. As the order of the Mellin-Laplace window increases the new information that it carries decreases. That implies that a huge amount of window should be used to include all the available information in the DTOFs.