# A Review of Image Reconstruction Algorithms for Diffuse Optical Tomography
## 4 Optimization Process
### Other Important Topics: Regularization Parameter, Artifacts, Local Minima, and AI
In this method employing SA, which is a type of a Metropolis-Hastings Markov chain Monte Carlo method, \(\delta\mu_{a}\) was expressed as a discrete value as \(\delta\mu_{a}=\delta\mu_{a}{}^{\max}\left\{(S_{i}/M)+(1/2)\right\}\) by employing the spin variable \(S_{i}=\pm 1\), \(\pm 2\),..., \(\pm M/2\) at the \(i\)th position. The cost function \(f\left(S_{i}\right)\) with linearization by Rytov approximation and a type of Tikhonov regularization was used in the Boltzmann distribution with temperature \(T\), formulated as \(p(S_{i})\propto\exp(-f\left(S_{i}\right)/T)\), as the probability density function to sample the spin variables. \(S_{i}\) randomly generated in accordance with \(p(S_{i})\) was adopted when the probability increased (i.e.,\(f\) decreased). Starting from a high \(T\), which provided a highly random selection of \(S_{i}\) and cooling by decreasing \(T\), the spins resulting in the reconstructed image were narrowed from a wide range of candidates. The numerical simulations demonstrated that the SA avoided the local minimum arising in a single-spin case [155] and that the DOT image was reconstructed in the multiple-spin case. AI, as represented by deep learning, can provide a different approach to reconstructing DOT images. Deep learning comprises an artificial neural network (ANN) with several hidden layers. Various ANN architectures for deep learning, such as convolutional neural networks (CNN), generative adversarial networks, and recurrent neural networks (RNN), have been employed. The neural network outputs the data processed by artificial neuron units, the connections of which are adjusted by a learning process using training data that are given input and output pairs. Deep learning and its relationship with the DOT have been detailed in the literature [16; 17]. [PERSON] et al. proposed a deep-learning approach [15] that employed an ANN to solve the following Lippman-Schwinger equation for \(\delta\mu\):  
\[\delta\Phi(\mathbf{r})=-\int_{\Omega}G_{0}(\mathbf{r},\mathbf{\upxi})\delta \mu(\mathbf{\upxi})\Phi(\mathbf{\upxi})d\mathbf{\upxi}, \tag{39}\]  
where \(\delta\Phi=\Phi-\Phi_{0}\), and \(\Phi_{0}\) and \(G_{0}\) are the fluence rate and Green's function of the CW version of the PDE with the squared diffusive wave number of the background substituted for \(\mu_{a}\). Equation (39) can be inverted using a technique related to Hankel matrix decomposition, which is achieved using a fully data-driven neural network with an encoder-decoder structure. The training data were generated by numerical simulations using NIRFAST. This approach output the image of \(\delta\mu\) better than the \(\ell_{1}\)-sparse regularization method and the iterative updating method based on Rytov approximation in the numerical, phantom, and in vivo mouse experiments. [PERSON] et al. proposed an ANN output of the updated optical properties for the Gauss-Newton method used in a nonlinear optimization process [33]. Image reconstruction was formulated using the FD PDE. The Bayesian approach was employed to incorporate prior information on the distribution of the optical properties. In the process to update the optical properties, CNN was employed. The update of the optical properties is expressed as  
\[\mathbf{\upmu}_{t+1}=G_{\theta,t}(\mathbf{\upmu}_{t},\delta\mathbf{\upmu}_{ t}), \tag{40}\]  
where \(\mathbf{\upmu}_{t}\) is the distribution of \(\mu_{a}\) and \({\mu_{s}}^{\prime}\) in the \(t\)th iteration, denoted by the subscript, \(G_{\theta,t}\) is the updating function achieved by the CNN, and \(\delta\mathbf{\upmu}_{t}\) is the gradient of the cost function. \(\mathbf{\upmu}_{t}\) and \(\delta\mathbf{\upmu}_{t}\) are the inputs to the CNN. The selection of the step size, which must be determined in the updating process of the nonlinear optimization, was reduced. Numerical experiments demonstrated that the proposed method worked better than the conventional Gauss-Newton method under modeling errors.