# A Review of Image Reconstruction Algorithms for Diffuse Optical Tomography
## 4 Optimization Process
### Use of Regularization Minimizing Norms
\(L_{1}\)-norm regularization was found to be more robust than Tikhonov (\(L_{2}\)-norm) regularization in terms of reducing the number of measurement samples used in image reconstruction. [PERSON] and [PERSON] proposed image reconstruction with dynamic CW-domain DOT that employed image reconstruction with Rytov approximation for video-rate imaging [106]. They applied the \(\ell_{1}\)-norm (1-norm) minimization implemented with the open-source YELL1 algorithm [107] derived from the alternating direction method (ADM), which minimizes the augmented Lagrangian function equivalent to Equation (8)  
\[L(\delta\upmu,\mathbf{e},\Gamma)=\|\delta\upmu\|_{1}+\frac{1}{2\gamma_{1}}\| \mathbf{e}\|_{2}^{2}-\mathbf{\Gamma}^{T}(\delta\mathbf{M}+\mathbf{e}-J\delta \upmu)+\frac{\gamma_{2}}{2}\|\delta\mathbf{M}+\mathbf{e}-J\delta\upmu\|_{2}^{2}, \tag{19}\]  
by updating sequentially the optical properties \(\delta\upmu\), the residual error norm \(\mathbf{e}\), and the Lagrange multiplier \(\mathbf{\Gamma}\)[108]. The update of \(\delta\upmu\) was performed using the product of the absolute value and the sign of the gradient with a threshold for minimizing the \(\ell_{1}\)-norm. Phantom experiments demonstrated that \(\ell_{1}\)-norm regularization improved the image contrast compared to traditional \(\ell_{2}\)-norm regularization. [PERSON] et al. reported that \(L_{1}\)-norm regularization improved DOT image reconstruction in terms of spatial resolution and depth localization in phantom experiments [109]. Figure 2: Idea of the regularization minimizing \(p\)-norm of the reconstructed image with (**a**) \(p=2\) (Tikhonov regularization), (**b**) \(p=1\), and (**c**) \(0<p<1\). The solid lines represent the points of (\(\mu_{1}\), \(\mu_{2}\)) with a constant value of the \(p\)-norm. The dashed lines represent the group of points providing identical measurements. The cross-section points of the solid and dashed lines are solutions with a certain value of the \(p\)-norm. By minimizing the norms, the tangent points are selected as the optimal reconstructed image. \(L_{1}\)-norm regularization was implemented using the depth-compensation method. The Rytov approximation was employed to linearize image reconstruction. In this study, matrix \(M\) for depth compensation was computed using the singular value decomposition of matrix \(J\). Then, matrix \(J\) in Equation (7) is replaced by \(J\)M. \(L_{1}\)-norm regularization was implemented with a logarithmic barrier penalty function bounding the reconstructed values at a certain interval, which was minimized simultaneously with the cost function. [PERSON] et al. implemented sparsity regularization with \(L_{1}\)-norm by employing an expectation minimization (EM) algorithm [110]. They modeled the measurements (Rytov approximation) using additive noise and optical properties as random Gaussian variables. Subsequently, a Bayesian framework was applied for image reconstruction. The cost function was formulated as the log-likelihood function of the posterior probability to be maximized, which is equivalent to the minimization in Equation (1), where matrix \(W\) is composed of the variance in the Gaussian distribution of the measurement additive noise, and the regularization term is the \(L_{1}\)-norm of the optical properties. The expectation-maximization (EM) algorithm consisted of E- and M-steps. In the E-step, the log-likelihood (cost function) is computed using the measurements and optical properties in each iteration. In the M-step, the log-likelihood is maximized. The soft threshold method was used, in which the gradient of the \(L_{1}\)-norm was computed as \(\text{sgn}(\delta\mu)\,|\,\delta\mu\,|\) and as zero when \(\,|\,\delta\mu\,|\) was smaller than the threshold. [PERSON] et al. used the \(\ell_{p}\)-norm (\(0<p\leq 1\)) of a reconstructed image as a regularization term using the focal underdetermined system solver (FOCUSS) algorithm [111; 112].