# Improving Localization of Deep Inclusions in Time-Resolved Diffuse Optical Tomography
## 2 Methods
### Temporal Windows
#### 2.3.5 Trade-Off between Temporal Selectivity and Computation Complexity  
An optimal temporal window is the one that is easy to compute (it has few oscillations and small dispersion at frequency domain) and at the same time its temporal selectivity is good enough. For some windows, optimality can be described by \(D_{0}^{*}=D_{0}(g)D_{0}(G)\) (see Section 2.2). The \(D_{0}^{*}\) of a window quantifies the trade-off between the temporal selectivity and easiness to compute as expressed by uncertainty principle. The lower the value \(D_{0}^{*}\) is for a given window, the better the window temporal selectivity will be and the less frequencies will be needed to perform the numerical integration. In Table 1, the main optimality results are summarized. The Gaussian is the window that has the lowest \(D_{0}^{*}\) value; this was expected from the uncertainty principle theorem and this is what makes the Gaussian window one of the most promising windows. The exponential window has a twice bigger value compared to the Gaussian window; this means that the numerical integral could be a bit more difficult to compute but it is still a good candidate. Mellin-Laplace windows will increase its \(D_{0}^{*}\) value almost linearly for increasing orders although it is independent of \(p\) value. Expression \(D_{0}^{*}\) is not in \(L^{2}\) space for Tukey windows, nevertheless from Figure 7 it is evident that for some \(\alpha\) values is still feasible to perform numerical integration at frequency domain. Standard moments are neither \(L^{2}\) integrable. Those moments were analyzed in detail in [39; 40] and one of the conclusions that was reached is that standard moments high orders are only profitable when more than \(10^{8}\) photons are detected. Moreover, since by nature they are highly correlated (see Figure 8a) they will not be considered in this work. A point that should also be taken into account is the shift theorem of the Fourier transform. It states that given a function \(g(t)\), if this function is shifted an amount of \(t_{0}\), then its Fourier transform is  
\[G_{t-t_{0}}(f)=G(f)e^{-i2\pi f_{0}}, \tag{20}\]  
where \(G_{t-t_{0}}(f)\) indicates the Fourier transform of shifted function \(g(t-t_{0})=g(t)*\delta(t-t_{0})\) and the complex exponential does not affect the spectral magnitude since \(|e^{-i2\pi f_{0}}|=1\) but it affects the phase. This theorem implies that as temporal windows are shifted to later times, more oscillations will appear. However, in practice this phenomenon did not prevent us to perform the numerical integration accurately. \begin{table}
\begin{tabular}{l c} \hline
**Window** & \(D_{0}^{*}\) \\ \hline Mellin-Laplace & \(\frac{1}{16\pi^{2}}\frac{(2n+2)!}{(2n)!}\left(\frac{2n}{2n-1}-1\right),\quad(n \geq 1)\) \\ Gaussian & \(1/(16\pi^{2})\) \\ Exponential & \(1/(8\pi^{2})\) \\ \hline \end{tabular}
\end{table} 
Table 1: \(D_{0}^{*}\) for each analyzed window. Standard moments and Tukey windows are not \(L^{2}\) integrable. See Appendix A for calculations. The new method to compute temporal windows is also useful for windows with the form \(w(t)=t^{n}e^{-pt}\), such as Mellin-Laplace or standard moments. Both of these moments suffer from an important handicap if traditional computation methods are used: in order to compute the \(n\)-th order window then all the previous orders, \(1,...,n-1\), must be computed (in the case of Mellin-Laplace if \(p\) parameter is modified then all orders must be computed again). These problem can be critical in the case of Mellin-Laplace since as the order of the moments increase, the window temporal selectivity decreases (see Figure 3(left)). If it is wanted to have better sensitivity to later photons (i.e., photons that probabilistically have go deeper into the tissue) then higher \(p\) values must be used.