# Improving Localization of Deep Inclusions in Time-Resolved Diffuse Optical Tomography
## 2 Methods
### Numerical Simulations  
To analyze the reconstruction improvements that could be obtained with the decorrelated windows, some numerical simulations were performed. The used computational phantom was a three-dimensional volume with a spherical inclusion at different depths. This phantom can be consider as an approximation to functional near-infrared spectroscopy experiments where the inclusion represents the activation in the cortex. The computational phantom had a size of \(9\times 9\times 5\) cm\({}^{3}\). Numerical simulations were performed in a reflectance geometry by solving the time-domain Diffusion Approximation up to 10 ns; the boundary conditions were implemented as described in [42]. To generate the synthetic measurements the domain was discretized using around 32 and 172 thousand nodes and elements respectively, see Figure 9(right). For the reconstruction, a coarser mesh was used (9 and 48 thousand nodes and elements respectively) in order to avoid inverse crime phenomenon. Each simulated curve contained up to \(2\times 10^{5}\) photons, were convoluted with the instrumental response function of a single photon avalanche diode detector (SPAD, FWHM \(\approx\) 160 ps) and corrupted by Poisson noise, see Figure 9(left).  
In this work, only absorption inhomogeneities were considered, although it can be easily extended to scattering inhomogeneities. Background absorption was set to \(\mu_{a}=0.018\) cm\({}^{-1}\) and the reduced scattering was \(\mu_{s}^{\prime}=14.7\) cm\({}^{-1}\) over all the domain. An inclusion of 0.5 cm radius was included whose optical properties were \(\mu_{a}=0.337\) cm\({}^{-1}\) and \(\mu_{s}^{\prime}=14.7\) cm\({}^{-1}\) (these values were taken from  
Figure 8: Correlation matrices of standard moments (first 50 orders), Mellin–Laplace (first 50 orders, \(p=7\)), Gaussian (\(t_{0}=0.16\) ns shifts, \(\sigma=0.05\)), and Tukey windows (\(t_{0}=0.16\) ns shifts, \(\alpha=0.25\) and \(t^{*}=0.2\) ns).  
the experiment performed at [43] using 550 nm wavelength light and are of similar order as found at several types of human tissue).  
The setup of source and detectors was moved along the top boundary to scan the phantom at multiple positions. The distance between the source and detectors was 3 cm at orthogonal directions, see Figure 10. The scan of the computational phantom with the inclusion was done at 30 different source positions (6 shifts in \(x\)-direction separated by 0.75 cm and 5 shifts in \(y\)-direction separated by 0.75 cm). The inclusions were set from 1 to 4 cm depth.  
During the reconstruction, to avoid large sensitivities in the surface nodes the sensitivity of each node was normalized by a diagonal weight matrix with values proportional to nodes depth. A flowchart of the used algorithm is shown at Figure 11. Simulations and reconstructions were done using a laptop with an Intel Core i7 processor and 4 GB RAM.  
Figure 10: (**Left**) Scheme of the numerical phantom. Simulated signals were convolved with the IRF of a single photon avalanche diode detector SPAD and corrupted by Poisson noise. (**Right**) Source positions; red crosses. Detector positions; blue dots. Note that for each source only the signal at two detectors were considered; 3 cm right and 3 cm down. For example, for source at position \((-2.62,2.25)\)cm only the signal at detectors \(\mathbf{x}_{1}=(0.38,2.25)\) cm and \(\mathbf{x}_{2}=(-2.62,-0.75)\) cm was used.  
Figure 9: (**Left**) Instrumental response function of a single photon avalanche diode detector (SPAD) detector (blue line). Normalized simulated signal without IRF or noise (red line). Normalized simulated signal convoluted with the IRF (yellow line). Normalized simulated signal convoluted with IRF and corrupted with Poisson noise (black line). (**Right**) Tetrahedral mesh used to generate the synthetic data; it had a size of \(9\times 9\times 5\) cm\({}^{3}\) and was discretized using around 32 and 172 thousand nodes and elements respectively.