# A Review of Image Reconstruction Algorithms for Diffuse Optical Tomography
## 4 Optimization Process
### Use of Spectral Prior Information
, \tag{30}\]  
where \(\delta\mathbf{c}=[\delta c_{1},\,\,\delta c_{2},\,\,\ldots\,\,\delta c_{q}]^{T}\) and \(h(a+\delta a)=\{1/3(a+\delta a)\}\cdot[\
abla^{2}(1/3a)^{1/2}/(1/3a)^{1/2}- \
abla^{2}[1/3(a+d)]^{1/2}/\{1/3(a+\delta a)\}^{1/2}]\). Equation (30) was rewritten using the vector matrix formula \(A\mathbf{x}=\mathbf{1}\). The least-squares solution of the equation is \(\mathbf{x}_{0}=(A^{\overline{1}}A)^{-1}A^{\overline{1}}\) when \(o\)\(>q\). The residual error norm, \(E=\|Ax_{0}-\mathbf{1}\|\), was used as the measure of the ability to distinguish \(\mu_{a}\) from \(\mu_{s}{{}^{\prime}}\). When the residual error norm was close to zero, the measurement involved cross-talk problems in image reconstruction. In the numerical simulation, it was demonstrated that the cross-talk between \(\mu_{a}\) and \(\mu_{s}{{}^{\prime}}\) was reduced by a combination of wavelengths that provided an \(E\) that was as large as possible. Simultaneously, the concentrations of oxy- and deoxyhemoglobin were clearly separated by the combination of wavelengths with the small condition number of matrix \(A\) of \(\varepsilon\) in Equation (30). The condition number is the ratio between the maximum and minimum singular values, and indicates the degree of separation ability among \(c_{1}\), \(c_{2}\),... \(c_{q}\). This image reconstruction method was applied to the FD DOT image of oxy-/deoxyhemoglobin, water, and \(\mu_{s}{{}^{\prime}}\) in a real breast tissue at four wavelengths: 690, 750, 786, and 830 nm [136]. [PERSON] et al. used a spectral prior method for linearized image reconstruction [134]. The linear equation, \(\delta\mathbf{M}=J\delta\mathbf{c}\), was solved for the changes in the concentrations of oxy- and deoxyhemoglobin, and \(\delta\mathbf{c}\) was the regularized Moore-Penrose generalized solution, \(\delta\mathbf{c}\)\(=J^{\overline{1}}(J\overline{1}^{T}+\lambda I)^{-1}\delta\mathbf{M}\). To compare the image reconstructions of \(\delta\mathbf{c}\) using a spectral prior (direct method) and calculation from the reconstructed \(\delta\mu_{a}\) (indirect method), a numerical simulation and a blood phantom experiment were conducted. By measuring the contrast to-noise (CNR) ratio (peak value of deoxyhemoglobin divided by the mean standard deviation of the reconstructed values in voxels) and cross-talk (changes in oxyhemoglobin concentration divided by deoxyhemoglobin concentration), we verified that the indirect method could improve the contrast-to-noise ratio and reduce cross-talk. [PERSON] et al. employed the spectral prior method with a formulation of \({\mu_{s}}^{\prime}\) that was different from other methods [137]. Based on the Mie theory and the assumption that the size of the scattering particle is Gaussian distributed, the \(u\)-th scattering particle has a reduced scattering coefficient:  
\[{\mu_{su}}^{\prime}\left(\lambda\right)=\int_{0}^{\infty}\big{\{}3Q_{scat}(v,n,\lambda)\big{[}1-g(v,n,\lambda)\big{]}/2v\big{\}}f(v)dv, \tag{31}\]  
\[f(v)=\bigg{(}1/\sqrt{2\pi\beta^{2}}\bigg{)}\exp\Big{\{}-\big{(}v-\alpha_{u} \big{)}^{2}/(2\beta^{2})\Big{\}}, \tag{32}\]  
where \(v\) is the particle size, \(n\) is the refractive index, \(\alpha_{u}\) is the average particle size, \(b\) is the standard deviation, and \(Q_{scat}\) is the scattering efficiency. Then, \({\mu_{s}}^{\prime}\) is formulated as  
\[{\mu_{s}}^{\prime}\left(\lambda,\mathbf{r}\right)=\sum_{u=1}^{U}\varphi_{u}( \mathbf{r}){\mu_{s_{u}}}^{\prime}(\lambda), \tag{33}\]  
where \(\varphi_{u}\) is the volume fraction of the \(u\)th scattering particle. Equation (33) can be written as a vector matrix formula, similar to Equation (29). \(\varphi_{u}\) was reconstructed with \(c\) simultaneously. This algorithm was examined in numerical simulations with various geometries including particles with sizes of 150, 1000, and 6000 nm in a phantom experiment using blood, intralipid, and agar powder. The algorithm was also tested on the clinical measurements of patients with breast cancer.