# A Review of Image Reconstruction Algorithms for Diffuse Optical Tomography
## 4 Optimization Process
### Other Important Topics: Regularization Parameter, Artifacts, Local Minima, and AI  
In DOT image reconstruction, the selection of the regularization parameter \(\gamma\) is important, as is the case in other inverse problems. The reconstructed image cannot sufficiently approximate the true optical properties when \(\gamma\) is excessively large because the squared error term in Equation (2) is not sufficiently small. However, the measurement noise, which causes artifacts in the reconstructed image, disturbs the reconstructed image when \(\gamma\) is extremely small. Selecting an appropriate \(\gamma\) is one of the important aspects in image reconstruction. The L-curve and generalized cross-validation (GCV) methods are often used to solve inverse problems. The L-curve method finds \(\gamma\) that minimizes the squared error term and regularization term in good balance by plotting the log of the squared residual error term versus the log of the regularization term with various \(\gamma\), which often draws an L-shaped curve [111, 114, 115, 138]. \(\gamma\) allows the corner of the L-shaped curve to efficiently minimize both terms. The GCV method in the case of Equation (9) with \(W=L=I\) selects \(\gamma\) that minimizes the GCV function,  
\[GCV(\gamma)=\|\delta\mathbf{M}-J\delta\mathbf{\mu}(\gamma)\|^{2}/\bigg{[} \text{trace}\bigg{\{}I-J\Big{(}J^{T}J+\gamma I\Big{)}^{-1}J^{T}\bigg{\}}\bigg{]} ^{2}, \tag{34}\]  
where \(\delta\mathbf{\mu}(\gamma)\) is the Tikhonov regularized image reconstructed using \(\gamma\)[139, 140]. The trace in Equation (34) is the sum of the diagonal elements in the matrix, which can be regarded as a measure of the degrees of freedom in the regularized image \(\delta\mathbf{\mu}\)[116]. Various methods have been proposed to select \(\gamma\). [PERSON] et al. examined 10 methods to select \(g\) including heuristic, L-curve, GCV, unbiased predictive risk estimator, discrepancy principle, normalized cumulative periodogram (NCP), F-slope, quasi-optimality criterion, full width half maximum, contrast-to-noise ratio (CNR), and CNR\(\cdot\Psi^{-1}\) methods [141]. Linearized 3D image reconstruction was performedusing a tissue-mimicking phantom made of epoxy resin with the imaging target. The optimal method was selected based on the relative error in the reconstructed image, objectivity, and the requirement of no prior knowledge. Although it is difficult to find the corner of the L-shaped curve owing to its high ill-posedness, the L-curve method was selected as the optimal method under the conditions of the study. [PERSON] and [PERSON] compared the minimal residual method (MRM) with GCV [139]. MRM updates \(\delta\mu\) in a manner similar to that of the steepest gradient method to solve Equation (8) with \(W=I\) using Tikhonov regularization with \(L=I\). MRM changes \(\gamma\) in each iteration to update \(\delta\mu\). The updating rule at the \(t\)th iteration is  
\[\delta\mu_{t+1}=\delta\mu_{t}-s_{t}\cdot\
abla f_{t}, \tag{35}\]  
\[\
abla f_{t}=J^{T}(J\delta\mu_{t}-\delta\mathbf{M})-\gamma_{t}\cdot\delta\mu _{t}, \tag{36}\]  
where the step size of \(s_{t}=\|\
abla f_{t}\|^{2}/\left(\|J\
abla f_{t}\|^{2}+\gamma_{t}\|\
abla f_{t }\|^{2}\right)\). \(\gamma_{t}\), which minimizes the squared residual error, was searched using a simplex-type method. The image reconstructions based on the Rytov approximation with CW measurements in the numerical simulations and the gelatin phantom experiment demonstrated that the MRM provided a higher spatial resolution and peaks closer to the true values than the GCV method. [PERSON] and [PERSON] proposed the conjugate gradient-type least-square QR (LSQR) method to select \(\gamma\)[142]. The LSQR method is typically used to solve linear equations efficiently and is applied to image reconstruction. The LSQR, MRM, L-curve, and GCV methods were compared using numerical simulations and gelatin phantom experiments. The LSQR and MRM methods improved the spatial resolution and quantitativeness of the reconstructed images. [PERSON] et al.