# Improving Localization of Deep Inclusions in Time-Resolved Diffuse Optical Tomography
## 2 Methods
### A Novel Method to Compute Temporal Windows  
Let us define \(u(t)\) as the DTOF simulated at a detector and \(w(t)\) as an arbitrary shaped temporal window. A datatype \(\Gamma\) is defined as  
\[\int_{0}^{\infty}u(t)w(t)\;\mathrm{d}t=\Gamma, \tag{6}\]  
since \(u(t)=0\) for \(t<0\). If \(w(t)=t^{n}e^{-pt}\) then \(\Gamma\) can be calculated directly without computing \(u(t)\) explicitly [30]. However, until now, no method has been published that uses a different window form without computing explicitly \(u(t)\) for each time step. The approach described in this paper proposes to compute the datatypes in the frequency domain; this can be done by using the Plancherel theorem as follows:  
\[\Gamma=\int_{-\infty}^{\infty}u(t)\overline{w(t)}\;\mathrm{d}t=\int_{-\infty} ^{\infty}U(f)\overline{W(f)}\;\mathrm{d}f, \tag{7}\]  
where uppercase character denotes the Fourier transform defined as \(U(f)=\int_{-\infty}^{\infty}u(t)e^{-2\pi ift}\,\mathrm{d}t\) (e.g., \(W(f)\) is the Fourier transform of \(w(t)\)) and \(\overline{W(f)}\) denotes the conjugate of \(W(f)\). If \(\overline{W(f)}\) or \(U(f)\) are non-zero at a small interval of the frequency domain, the integral of Equation (7) can be approximated numerically by using few frequencies, see Figure 2. This new approach allows fast computation of a larger set of temporal windows. To approximate \(\Gamma\) using few frequencies, it is important to ensure that window spectra decay rapidly. Dispersion of the windows after applying Fourier transform can be controlled by using the _uncertainty principle_. This principle states that the narrower a function \(g(t)\) is, the more spread its Fourier transform \(G(f)\) will be. That is, if it is wanted to have a sharp window in time domain then the integral in frequency domain will be more spread around the zero frequency. Defining the dispersion of a function in time and frequency domain as  
\[D_{0}(g)=\frac{\int_{-\infty}^{\infty}f^{2}|g(t)|^{2}\,\mathrm{d}t}{\int_{- \infty}^{\infty}|g(t)|^{2}\,\mathrm{d}t},\qquad D_{0}(G)=\frac{\int_{-\infty}^ {\infty}f^{2}|G(f)|^{2}\,\mathrm{d}f}{\int_{-\infty}^{\infty}|G(f)|^{2}\, \mathrm{d}f} \tag{8}\]  
the _uncertainty principle_ states that if \(g(t)\) is absolutely continuous and the functions \(tg(t)\) and \(g^{\prime}(t)\) are square integrable (i.e., they are in \(L^{2}\) space) then [31]  
\[D_{0}(g)D_{0}(G)\geq\frac{1}{16\pi^{2}}, \tag{9}\]where \(G\) is the Fourier transform of \(g(t)\) and the equality holds only for the Gaussian density function centered at \(t=0\). The reported technique has to be much faster to compute than the resolution of time-dependant Diffusion Approximation equation in order to be useful in practice. In the following section, some important computational aspects related to the time complexity will be shown. First, to compute \(\phi(\mathbf{r},t_{n})\) in Equation (1), it is needed to know the previous values of \(\phi(\mathbf{r},t_{n-i})\), where \(t_{n}\) is the \(n\)-th discrete time bin and \(t_{n-i}<t_{n}\). This implies that to compute \(\phi(\mathbf{r},t_{n})\) previously \(n-1\) linear systems have to be solved. However, in the frequency resolved version of Equation (1), there is no such restriction, since the fluence rate at any frequency can be computed independently. This is the main reason why the integral of Equation (7) is faster to compute in the frequency domain. Therefore, the described technique is easily parallelizable and could be implemented in Graphics Processing Units (GPU) hardware; some labs have already implemented state-of-the-art photon propagation models using GPU technology [32; 33]. Another important property is that for real functions, Fourier transform coefficients will have symmetric real part and anti-symmetric imaginary part.