# Improving Localization of Deep Inclusions in Time-Resolved Diffuse Optical Tomography
## 2 Methods
### Time- and Datatypes-Based Reconstruction  
Time-resolved diffuse optical tomography can be posed in terms of time bins or datatypes; in both cases time-resolved measurements from the detectors are used. Absorption of the medium \(\Omega\) can be recovered by using iteratively the linearized Born approximation,  
\[\delta\phi_{sd}(t)=\int_{\Omega}\left[\phi_{s}^{(k)}*G_{d}^{(k)}\right]\left( \mathbf{r},t\right)\,\delta\mu_{a}^{(k)}(\mathbf{r})\,\mathrm{d}\mathbf{r}, \tag{2}\]  
where \(\delta\phi_{sd}=\phi_{sd}-\phi_{sd}^{(k)}\) is the fluence rate difference at detector \(d\) when source at position \(s\) was activated; in \(\phi_{sd}\) the experimental factors are included. \(G_{d}^{(k)}(\mathbf{r},t)\) indicates the value of the Green's function at every point in the domain assuming the source is located at position \(d\). \(\phi_{s}^{(k)}(\mathbf{r},t)\) is the fluence rate at every point in the domain for a source located at \(s\). \(\delta\mu_{a}^{(k)}(\mathbf{r})\) is the absorption update at each iteration. The superscript indicates that absorption values obtained at iteration \(k\) were used. The reconstructed absorption will be the result of adding all \(\delta\mu_{a}\) to the initial absorption estimation,that is, \(\mu_{a}=\delta\mu_{a}^{(1)}+\delta\mu_{a}^{(2)}+\dots\) A simplified sketch of the reconstruction procedure is shown at Figure 1. Equation (2) can also be extended to include changes in scattering. Equation (2) can be discretized into a full time-resolved linear system,  
\[ \begin{bmatrix}L_{sd}(t_{1})\\ L_{sd}(t_{2})\\ \vdots\\ L_{sd}(t_{N})\end{bmatrix} = \begin{bmatrix}J_{sd}(t_{1},r_{1})&J_{sd}(t_{1},r_{ 2})&\cdots&J_{sd}(t_{1},r_{M})\\ J_{sd}(t_{2},r_{1})&J_{sd}(t_{2},r_{2})&\cdots&J_{sd}(t_{2},r_{M})\\ \vdots\\ J_{sd}(t_{N},r_{1})&J_{sd}(t_{N},r_{2})&\cdots&J_{sd}(t_{N},r_{M})\end{bmatrix} \begin{bmatrix}\delta\mu_{a}(r_{1})\\ \delta\mu_{a}(r_{2})\\ \vdots\\ \delta\mu_{a}(r_{M})\end{bmatrix} , \tag{3}\]  
where \(J_{sd}(t,r)\) and \(L_{sd}(t)\) are column vectors with one entry for each source-detector pair. \(J\) matrix is known as sensitivity matrix and \(L\) is the discretization of \(\delta\phi_{sd}(t)\). In this case the system matrix size is \((N\cdot S\cdot D)\times M\) where \(N\) is the number of time bins considered, \(S\) is the number of sources, \(D\) is the number of detectors and \(M\) is the number of mesh nodes. Expressing the reconstruction in terms of time has two main drawbacks: (1) the number of time bins is high (usually of a few thousands order) which makes the system very large and (2) it is necessary to simulate the distribution of photon time of flight (DTOF) for each source and detector, which is very time-consuming. Due to previous reasons, it is better to solve the problem of Equation (2) in terms of datatypes. A datatype is a filter that is applied to the DTOF to reduce the size of the problem. For example, when a temporal window filter is applied to the DTOF, the problem is reduced to  
\[ \begin{bmatrix}L_{sd}(w_{1})\\ L_{sd}(w_{2})\\ \vdots\\ L_{sd}(w_{W})\end{bmatrix} = \begin{bmatrix}J_{sd}(w_{1},r_{1})&J_{sd}(w_{1},r_ {2})&\cdots&J_{sd}(w_{1},r_{M})\\ J_{sd}(w_{2},r_{1})&J_{sd}(w_{2},r_{2})&\cdots&J_{sd}(w_{2},r_{M})\\ \vdots\\ J_{sd}(w_{W},r_{1})&J_{sd}(w_{W},r_{2})&\cdots&J_{sd}(w_{W},r_{M})\end{bmatrix} \begin{bmatrix}\delta\mu_{a}(r_{1})\\ \delta\mu_{a}(r_{2})\\ \vdots\\ \delta\mu_{a}(r_{M})\end{bmatrix} , \tag{4}\]  
whose matrix size is \(W\cdot S\cdot D\times M\), where \(W\) is the number of windows filters used. In practice, this system will be much smaller than the system of Equation (3) since only a few dozens of windows are usually needed. Figure 1: Absorption reconstruction algorithm based on linearized Born equation (Equation (2)). The Fourier transform is also a datatype that converts the reconstruction problem to,  
\[ \begin{bmatrix}L_{sd}(f_{1})\\ L_{sd}(f_{2})\\ \vdots\\ L_{sd}(f_{F})\end{bmatrix} =