# A Review of Image Reconstruction Algorithms for Diffuse Optical Tomography
## 1 Introduction
### Optimization Process for Nonlinear Image Reconstruction  
Let \(F_{i,j}(\mu_{a},\mu_{s}{}^{\prime})\) be the mathematical operation for calculating the measured light with optical properties \(\mu_{a}\) and \(\mu_{s}{}^{\prime}\) and the given positions of the light sources and detectors to obtain the predicted measurements: \(\mu_{s}{}^{\prime}=(1-g)\mu_{s}\) is defined by the anisotropy (asymmetry) factor, \(g\), which is the average cosine of the scattering angle. Image reconstruction can be formulated as an optimization problem to find \(\mu_{a}\) and \(\mu_{s}\) while minimizing the cost function, which consists of the squared error and additive regularization term, as follows:  
\[\min_{\mu_{a},\mu_{s}{}^{\prime}}\left\{\frac{1}{2}\sum_{i=1}^{l}\sum_{j=1}^{ l}w_{i,j}\big{(}F_{i,j}\big{(}\mu_{a},\mu_{s}{}^{\prime}\big{)}-M_{i,j}\big{)}^{2}+ \gamma\cdot R\big{(}\mu_{a},\mu_{s}{}^{\prime}\big{)}\right\}, \tag{1}\]  
where \(M_{i,j}\) are the measurements with the \(i\)th source and \(j\)th detector, respectively, and \(w_{i,j}\) is the weight adjusting the contribution of \(M_{i,j}\) to the image reconstruction. \(R(\mu_{a},\mu_{s}{}^{\prime})\) is a regularization term. The image-reconstruction process used to solve the inverse problem is generally ill-posed. Image reconstruction often suffers from the nonuniqueness of solutions and instability owing to noise contaminating the measurements. \(R(\mu_{a},\mu_{s}{}^{\prime})\) is a function for evaluating the extent to which the solution (reconstructed image) satisfies the condition expected in the desired solution and for reducing the overfitting of \(F_{i,j}\) to \(M_{i,j}\) contaminated by noise. By minimizing \(R(\mu_{a},\mu_{s}{}^{\prime})\) with a residual error, this ill-posed nature can be alleviated. Various types of \(R(\mu_{a},\mu_{s}{}^{\prime})\), such as the squared \(L_{2}\)-norm of the reconstructed image (Tikhonov regularization), \(L_{p}\)-norm (\(0\leq p\leq 1\)), and total variation, which is the norm of the gradient of the image, can be applied. \(g\) adjusts the regularization effect. The relationship between the optical properties and measurements as a result of light propagation is nonlinear. Therefore, image reconstruction is performed by solving Equation (1) by employing nonlinear optimization methods with iterative updating processes, such as the Newton-Raphson, quasi-Newton, and conjugate gradient methods. For the nonlinear optimization process, the gradient of the cost function \(f\), the sum of the squared error term, with the regularization term in Equation (1) must be computed, as follows:  
\[\frac{\partial f}{\partial\mu_{k}}=\sum_{i=1}^{l}\sum_{j=1}^{l}w_{i,j}\big{(} F_{i,j}(\mu_{k})-M_{i,j}\big{)}\frac{\partial F_{i,j}}{\partial\mu_{k}}+\gamma \cdot\frac{\partial R(\mu_{k})}{\partial\mu_{k}}, \tag{2}\]  
where \(\mu_{k}\) is the optical property of interest at the \(k\)th position \(\mathbf{r}_{k}\), where one of the nodes for the FEM to compute \(F_{i,j}\) exists. The vector matrix formula in Equation (2) is often used. \[\
abla f(\mathbf{\mu})=\frac{\partial f}{\partial\mathbf{\mu}}=WJ^{T}(\mathbf{ F}-\mathbf{M})+\gamma\cdot\frac{\partial R}{\partial\mathbf{\mu}}, \tag{3}\]  
where \(\mathbf{F}\) and \(\mathbf{M}\) are column vectors in which \(F_{i,j}\) and \(M_{i,j}\) are aligned, respectively. \(\mathbf{\mu}\) is a vector of \(\mu_{k}\). \(W\) is the diagonal matrix with \(w_{i,j}\), and \(J\) is the Jacobian matrix with \(\partial F_{i,j}/\partial\mu_{k}\) as its elements, which can be calculated by the perturbation method or the adjoint methodderived in the early important work on the DOT image reconstruction by [PERSON] and [PERSON] [23; 28]. Using the Newton-Raphson method, iterative image reconstruction can be expressed as  
\[\mu_{t+1}=\mu_{t}-H^{-1}(\mu_{t})\
abla f(\mu_{t}), \tag{4}\]  
where \(H=\
abla^{2}f(\mu)\) is a Hessian matrix and the subscript \(t\) is the number of iterations. The updating process in Equation (4) is repeated until the solution converges.