# A Review of Image Reconstruction Algorithms for Diffuse Optical Tomography
## 4 Optimization Process
### Use of Spectral Prior Information  
While DOT can image the hemoglobin concentration and oxygen saturation of tissues by using the reconstructed \(\mu_{a}\) and \(\mu_{s}{{}^{\prime}}\), DOT image reconstruction faces the cross-talk problem caused by nonuniqueness, which means that a reconstructed image with a combination of \(\mu_{a}\) and \(\mu_{s}{{}^{\prime}}\) different from the true values is obtained from the measurements because the different combinations of \(\mu_{a}\) and \(\mu_{s}{{}^{\prime}}\) cause the measurements to be identical or very close to each other. [PERSON] and [PERSON] [132] summarized the conditions for combinations of \((\mu_{a1},D_{1})\) and \((\mu_{a2},D_{2})\) that cause the same CW measurements, as follows:  
\[\eta_{0}(\mu_{a1},D_{1})=\eta_{0}(\mu_{a2},D_{2})\text{ in }\Omega_{0}, \tag{26}\]  
with  
\[\eta_{0}(\mu_{a},D)=\Bigl{[}\Bigl{(}\
abla^{2}D^{1/2}\Bigr{)}/D^{1/2}\Bigr{]} +(\mu_{a}/D), \tag{27}\]\[D_{1}=D_{2}\text{ in }\Omega_{1}, \tag{28}\]  
where \(\delta_{0}\) is the measured region surrounded by \(\Omega_{1}\) including isotropic light sources. It was reported that the cross-talk problem was reduced by the image reconstruction method, which directly reconstructed the concentrations of photon absorbers using multi-spectral measurements because the spectral prior information of photon absorbers could function as a constraint to maintain the consistency of the spectra in the reconstructed images [133; 134]. In DOT image reconstruction employing prior spectral information, \(\mu_{a}\) at the \(o\) wavelengths, \(\lambda_{1}\), \(\lambda_{2}\),... \(\lambda_{o}\), was formulated with the concentrations of \(q\) photon absorbers, \(\mathbf{c(r)}=[c_{1},\,c_{2},\,\ldots\,c_{q}]^{T}\), as follows:  
\[ \begin{bmatrix}\mu_{a}(\lambda_{1},\mathbf{r})\\ \vdots\\ \mu_{a}(\lambda_{o},\mathbf{r})\end{bmatrix} = \begin{bmatrix}\varepsilon_{1}( \lambda_{1})&\cdots&\varepsilon_{q}(\lambda_{1})\\ \vdots&\ldots&\vdots\\ \varepsilon_{1}(\lambda_{o})&\cdots&\varepsilon_{q}(\lambda_{o})\end{bmatrix} \begin{bmatrix}c_{1}(\mathbf{r})\\ \vdots\\ c_{q}(\mathbf{r})\end{bmatrix} , \tag{29}\]  
where \(\varepsilon_{1}\), \(\varepsilon_{2}\),... \(\varepsilon_{q}\), are the extinction coefficients of the photon absorbers. In contrast, \(\mu_{s}{{}^{\prime}}\) is defined as \(\mu_{s}{{}^{\prime}}(\lambda,\,\mathbf{r})=a(\mathbf{r})\cdot\lambda^{-b( \mathbf{r})}\), by assuming a simplified Mie-scattering coefficient, where \(a\) and \(b\) are the parameters depending on the size, refraction index, and concentration of the scatterers [133; 135]. By substituting the equations relating \(\mu_{a}\) and \(\mu_{s}{{}^{\prime}}\) to \(a\), \(b\), and \(c\) into Equation (1) or (8) and constructing matrix \(J\), image reconstructions with prior spectral information were performed. [PERSON] et al. proposed an image reconstruction method for multispectral (four wavelengths) CW measurements using nonlinear gradient-based optimization [133]. In this study, \(b\) was assumed constant, and \(a\) and \(\mathbf{c}\) were reconstructed. To reduce the cross-talk problem, the condition of increasing nonuniqueness in the multispectral-CW DOT was derived based on the conditions in Equations (26)-(28). (\(a\), \(\mathbf{c}\)) and (\(a+\delta a\), \(\mathbf{c}+\delta\mathbf{c}\)) yield the same measurements when the following equation holds. \[ \begin{bmatrix}\frac{\varepsilon_{1}(\lambda_{1})}{\lambda_{1}^{A}}&\cdots& \frac{\varepsilon_{q}(\lambda_{1})}{\lambda_{1}^{A}}\\ \vdots&\cdots&\vdots\\ \frac{\varepsilon_{1}(\lambda_{o})}{\lambda_{o}^{A}}&\cdots&\frac{\varepsilon _{q}(\lambda_{o})}{\lambda_{o}^{A}}\end{bmatrix} \left(\frac{\delta a}{a\cdot h (a+\delta a)} \begin{bmatrix}c_{1}\\ \vdots\\ c_{q}\end{bmatrix} +\frac{1}{h(a+\delta a)} \begin{bmatrix}\delta c_{1}\\ \vdots\\ \vdots c_{q}\end{bmatrix} \right)= \begin{bmatrix}1\\ \vdots\\ 1\end{bmatrix}