# Diffuse Optical Tomography Using Bayesian Filtering in the Human Brain
## 2 Material and Methods
### Forward Model and Image Reconstruction  
In functional neuroimaging, the goal is to model the changes in the light attenuation \(y\) measured from the boundary volume \(\partial\Omega\), generated by changes in the optical properties \(\mu\) within a volume \(\Omega\):  
\[y=J\mu \tag{9}\]  
where \(y\) depicts measurements onto the boundary volume, based on \(\mu\) optical properties (absorption \(\mu_{a}\) and scattering \(\mu_{s}\) coefficients) in a position \(r\) within \(\Omega\). \(J\) is the Jacobian operator, which relates changes in the measured light intensity \(y\) on the boundary volume \(\partial\Omega\) with changes in internal optical properties \(\mu\), and it can be written as \(J=(\partial y)/(\partial\mu)\). This matrix is constructed from a model that is known as the forward model, which is derived from the Radiative Transport Equation (RTE),the mathematic model that is used in DOT and describes light propagation. From simplifying the RTE, we arrive at the diffusion approximation (DA):  
\[\frac{1}{c}\frac{\partial\Phi(r,t)}{\partial t}-D\
abla\Phi(r,t)+\mu_{a}\Phi(r,t )=q(r,t) \tag{10}\]  
where \(\Phi(r,t)\) is the photon density in time \(t\) and position \(r\), \(q(r,t)\) is an isotropic source, \(\mu_{a}\) is the absorption coefficient, \(c\) is the speed of the light in the medium, and \(D\) is the diffusion coefficient.  
The DA equation was described in a finite matrix that can be solved using the finite element model (FEM) [17]. FEM discretizes the continuous domain into a finite number of elements based on basis functions. An important aspect during the light propagation modeling is the anatomy model, which adds more accuracy to the spatial distribution of the optical properties [18]. Therefore, the light propagation modeling in the anatomy model impacts the reconstructed image quality. Anatomy images from MRI were used for light propagation modeling at the layer level. These anatomy images had been segmented into five layers aforehand: scalp, skull, cerebrospinal fluid, and gray and white matter. A mesh was then created from each layer, improving the anatomical details to a submillimeter scale.  
During image reconstruction from a turbid medium, the photon density (or light intensity) \(\Phi\) and optical property \(\mu\) relationship is nonlinear. In DOT measurements, changes in the light intensity \(\Delta\Phi\) that are generated by changes in the inner optical property \(\Delta\mu\) can be assumed as relatively differential changes; therefore, the problem can be considered to be linear [19]. A perturbation method [20] allows for us to construct the changes in the optical properties according to the measured light intensities on the surface. During the image reconstruction, the \(J\) matrix is directly inverted, as it is ill-conditioned and ill-posed. To invert \(J\), the singular value decomposition (SVD) algorithm was used. The diagonal matrix from SVD decomposition contains singular values \(\sigma_{m}\) providing information on the propagation errors from \(\Delta\Phi\) to \(\Delta\mu\). \(\sigma_{m}\) was used as regularization parameter in order to reduce the inverse matrix's dimensionality while using a minimum description length (MDL) index [21].