# A Review of Image Reconstruction Algorithms for Diffuse Optical Tomography
## 4 Optimization Process
### Use of Regularization Minimizing Norms
The change in the absorption coefficient \(\delta\mu_{a}\) from the background was formulated to differentiate the \(\ell_{p}\)-norm as \(\delta\mu_{ak}=\text{sgn}(z_{k})\,|\,z_{k}\,|^{2/p}\) at the \(k\)-th FEM node with parameter \(z_{k}\). Then, the optimization problem of the image reconstruction in Equation (1) was reformulated as follows:  
\[\min_{\mathbf{z}}\,\left\{\sum_{i=1}^{l}\sum_{j=1}^{J}\left(F_{i,j}(\mathbf{z} )-M_{i,j}\right)^{2}+\gamma\cdot\sum_{k=1}^{K}|z_{k}|^{2}\right\}, \tag{20}\]  
where \(\mathbf{z}\) is a vector composed of \(z_{k}\) (\(k=1,\,2,\,\ldots,\,K\)). Images were reconstructed using the mean time-of-flight of the time-resolved measurements. Numerical and phantom experiments demonstrated that the reconstructed area with changes in the absorption coefficient decreased as \(p\) approached zero. [PERSON] et al. reported the \(\ell_{p}\)-norm (\(0\leq p\leq 1\)) regularization for CW domain measurement [113]. They implemented the majorize-minimization framework, which replaced the original nonconvex cost function with the \(\ell_{p}\)-norm (\(0<p\leq 1\)) by a sequence of convex cost functions equal to the original cost function at a certain point and larger than the original cost function at the other points. Moreover, the regularization minimizing smooth \(\ell_{0}\)-norm was introduced. The smooth \(\ell_{0}\)-norm was approximated as \(\left\|\delta\mu_{a}\right\|_{0}=K-\sum_{k=1}^{K}\rho(\delta\mu_{ak})\) with \(\rho\) (\(\delta\mu_{ak}\)) = \(\exp(-\delta\mu_{ak}{}^{2}/\sigma^{2})\), which takes values of unity and zero when \(\Delta\mu_{ak}>\sigma\) and \(\Delta\mu_{ak}<\sigma\), respectively. Improvements in image quality were demonstrated in a numerical experiment with irregular geometry and a gelatin phantom experiment with optical properties mimicking typical breast tissue. Various applications of sparsity regularization have been proposed [114; 115]. Although regularization minimizing the norm of change in the optical properties effectively reduces blurriness and can locate the localized cerebral blood flow change caused by neural activity and cancer tissues with angiogenesis in the early stage, it is difficult to reconstruct a certain volume of a relatively large tissue/organ with nearly uniform optical properties. In such cases, the total variation (TV) method may be useful. The TV regularization method minimizes the norm of the image derivative and reconstructs the image using a nearly piecewise constant with jump discontinuities [116]. The TV regularization term is represented as  
\[R_{TV}(\mu)=\int|\
abla\mu(\mathbf{r})|d\mathbf{r}, \tag{21}\]where \(\mu\) is the reconstructed distribution of optical properties; \(|\cdot|\) indicates the Euclidean (\(I_{2}\)) norm. In the actual calculation of the TV norm, a difference approximation of the gradient of \(\mu\) is performed. By minimizing the gradient of \(\mu\) (approximated with differences in \(\mu\) on neighboring discretized nodes for numerical calculation using FEM), the image tends to be flat and has piecewise constant parts. [PERSON] and [PERSON] applied the TV regularization method to the FD-domain DOT to reconstruct \(D\) and \(\mu_{a}\) simultaneously [117]. Image reconstruction was attempted through numerical simulation with additive random noise and in the phantom experiment, where the ratios of the imaging target with a radius of 12.5 mm to the background were 2:1 and 10:1, respectively. Image reconstruction with TV regularization also reconstructed the area and values of \(D\) and \(\mu_{a}\) correctly, whereas image reconstruction without TV regularization was unable to correctly reconstruct the area and values. [PERSON] et al. compared Tikhonov, TV, and Huber regularizations [118]. Although TV regularization reconstructs a volume with piecewise constant optical properties, it is difficult to reconstruct the changes in optical properties for a small volume included in a large volume.