# 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
This evidence demonstrate the superior performance of Gaussian and Tukey window compared to Mellin-Laplace because of the better photon time arrival selectivity and less noise correlation. Regarding the computation time of the reconstructions, in Figure 16b the number of seconds per iteration is given for each method. To compute windows with form \(t^{n}e^{-pt}\) is much faster than the proposed approach and full-time reconstruction. Specifically, \(t^{n}e^{-pt}\) windows are around six times faster to compute than the proposed approach because \(LU\) factorization can be applied. However, for the proposed method factorization is not efficient since the matrix changes at each frequency. Full-time resolved approach is 9.5 times slower than \(t^{n}e^{-pt}\) windows and around 1.6 slower than the proposed approach. These results shows the trade-off of the proposed method: more information in depth is obtained but computation time is larger than \(t^{n}e^{-pt}\) windows. However, it should also be taken into account that novel method approach is highly parallelizable, that is, the Diffusion Approximation equation can be solve independently for each frequency. Note that this property does not hold for the direct computation of \(t^{n}e^{-pt}\) windows [22] and full-time approach. Therefore, the computation time of novel approach should considerably improve with GPU usage due to its intrinsic parallelizable nature. In Figure 16a, the number of computed iterations until convergence is given. Convergence was reached when relative difference \(\delta\mu_{a}/\mu_{a}^{(k)}<5\times 10^{-3}\). It can be seen that Mellin-Laplace windows converge faster mainly because it underestimates absorption quantification. As explained before, the simulated DTOFs were convolved with an SPAD IRF and corrupted with photon noise. These DTOFs were deconvoluted with a Wiener filter before performing the reconstruction. In some papers, such as [21], a different approach is proposed: DTOFs from a reference medium \(A\) and objective medium \(B\) are used. The optical properties of medium \(A\) are already known because an homogeneous medium is taken as reference and it includes implicitly all the information regarding instrumental factors. The medium \(B\) is the medium to be reconstructed. Absorption of medium \(B\) can be recovered by using iteratively the linearized cross-Born equation,  
\[G_{sd}^{A}(\mathbf{r},t)*M_{sd}^{B}(\mathbf{r},t)-M_{sd}^{A}(\mathbf{r},t)*G_{ sd}^{B(k)}(\mathbf{r},t)=-M_{sd}^{A}(\mathbf{r},t)*\int_{\Omega}\left[G_{s}^{B(k)}*G_{ d}^{B(k)}\right](\mathbf{r},t)\,\delta\mu_{a}\,\,\mathrm{d}\mathbf{r}, \tag{29}\]  
where \(*\) is the convolution operator, \(M_{sd}^{A}\) and \(M_{sd}^{B}\) indicate the measurement obtained at reference and to be reconstructed media by detector \(d\) when source \(s\) was activated, \(k\) indicates the iteration number, \(G_{sd}^{A}\) and \(G_{sd}^{B(k)}\) is the simulated Green's function (free of noise and instrumental factors) from source  
Figure 15: Mellin-Laplace (blue), Tukey (red) and Gaussian (yellow). Quantitative evaluation metric values for each window and inclusion depth. For localization error and relative reconstructed volume, the standard error shadows indicate the uncertainty given by selecting interest regions whose thresholds range from 60% to 80% of maximum absorption. at detector \(d\) by using the photon propagation mathematical model described in Equation (1) and \(G_{s}^{B(k)}\) indicates the Green's function value at every point in the domain given by the propagation model. The cross-Born equation offers the ease of not having to deal with instrumental factors directly (no deconvolution of the measurements is needed). However, it also implies to convolve the data and lose some information in the process. In Figure 17a, reconstructions using Mellin-Laplace and Tukey windows with the cross-Born approach are shown. We chose Tukey windows since they give very similar results to Gaussian windows. In Figure 17b-d, the quantitative metrics for those simulations are given.