# Improving Localization of Deep Inclusions in Time-Resolved Diffuse Optical Tomography
## 1 Introduction  
Time-resolved diffuse optical technology is an emerging photonic technique to continuously quantify the concentrations of several physiological chromophores such as hemoglobin, lipid or collagen. Successful measurements have been done at different human body locations such as brain [1; 2], breast [3] or thyroid [4]. An interesting extension of this technology is to perform diffuse optical tomography [5; 6] by computing three-dimensional maps of oxy- and deoxy-hemoglobin; in this approach, photon propagation is modeled in a computer and results are compared with experimental measurements. Then, optical parameters in the model are updated by solving an ill-posed inverse problem until the difference between the model and the data is negligible. In order to have accurate results, it is important to have a realistic model for photon propagation in tissues. The most accurate approach is to use the integro-differential Radiative Transfer Equation (RTE) [7]. Although RTE has some analytical solutions [8; 9; 10] these just hold for simple geometries and cannot be applied to more complex environments, such as an human head or breast models, without making strong assumptions. Apart from classical Monte-Carlo simulations [11; 12], new numerical methods have been proposed, some of which are the one-way RTE [13] or hybrid RTE [14]; nevertheless, they are still highly time-consuming for real-time applications. Due to the previous reasons, usually the first-order time-dependant Diffusion Approximation is used,  
\[\frac{1}{c}\frac{\partial\phi(\mathbf{r},t)}{\partial t}-\
abla\cdot(D(\mathbf{ r})\
abla\phi(\mathbf{r},t))+\mu_{a}(\mathbf{r})\phi(\mathbf{r},t)=S(\mathbf{r},t), \tag{1}\]where \(c\) is the speed of light in the medium, \(\phi\) is the fluence rate, \(D=\left(3\mu_{s}^{\prime}\right)^{-1}\)[15,16], \(\mu_{a}\) and \(\mu_{s}^{\prime}\) are the diffusion, absorption and reduced scattering coefficients respectively, and \(S\) is the source function. This approximation, based on spherical harmonics [7], is valid assuming that (1) the reduced scattering coefficient is much greater than absorption (\(\mu_{s}^{\prime}\gg\mu_{a}\)) and that (2) detectors are sufficiently far from light sources (>10/\(\mu_{s}^{\prime}\)). These assumptions hold in most practical cases and although higher order approximations have also been suggested [17; 18; 19] the extra computational cost usually does not compensate the small gain in accuracy. The time-dependant Equation (1) can be solved using Finite Element Method [20] but it is still a slow process since time has to be discretized in short steps in order to guarantee stability and convergence. For this reason, several authors have proposed to use temporal windows datatypes since they are faster to compute than the fluence rate itself [21; 22]. Nevertheless, as indicated in [23], the temporal windows datatypes are faster to compute than the fluence rate if they have the form \(w(t)=t^{n}e^{-pt}\) where \(n\in\mathbb{N}\) is in the set of natural numbers and \(p\in\mathbb{C}\) is in the set of complex numbers. This formula incorporates the Fourier transform (when \(n=0\) and \(p\) is an imaginary number), Laplace transform (when \(n=0\) and \(p\) is a complex number), standard moments (when \(n\) is an integer and \(p=0\)) and Mellin-Laplace moments (when \(n\) is an integer and \(p\) is a real number). Some authors prefer to use Fourier Transform (FT) datatypes. In [2] the magnitude and phase of 100 MHz frequency from a time-resolved signal was used for the reconstruction. A similar approach was published in [24] to retrieve the fluorescence lifetime distribution. A related idea is to directly use frequency-domain technology [25; 26; 27] to retrieve optical properties or fluorescence lifetime [28; 29]. These reconstruction approaches based on solving the inverse problem in frequency domain are a good alternative to temporal windows but are not equivalent since only MHz order frequencies are used.