# A Review of Image Reconstruction Algorithms for Diffuse Optical Tomography
## 4 Optimization Process
### Other Important Topics: Regularization Parameter, Artifacts, Local Minima, and AI
examined methods to select \(\gamma\) automatically, such as the L-curve, GCV methods, MRM, projection error method (PEM), and model function method in nonlinear image reconstruction using Tikhonov regularization from the logarithm of light intensities in CW measurements [143]. The PEM calculates \(\gamma\) at the \(t\)th iteration, as follows:  
\[\gamma_{t}=\max\left({{I_{t}}^{T}}{{I_{t}}}\right)/\left[2+\exp\left(-\| \mathbf{M}-\mathbf{F}_{t}\|^{2}\right)\right]. \tag{37}\]  
The absolute bias error (ABE), which is the mean value of the error between the reconstructed and true values; CNR, which compares the values in the region of interest and background; and FWHM were evaluated through numerical simulations. PEM provided better CNR and ABE than the other methods, especially for the image reconstruction of a single target. MRM improved the spatial resolution more than the other methods. [PERSON] et al. proposed \(\gamma\) as a spatial variant [144]. Because of the diffusive nature of light propagation, the spatial resolution of the DOT image degrades as the distance from the target to the sources and detectors increases. A spatially constant \(\gamma\) can excessively smooth the image of a deeper target. For image reconstruction of a circular object with Tikhonov regularization, the spatially variant \(\gamma\) is formulated as follows:  
\[\gamma\left(r_{j}^{n}\right)=\gamma_{e}\exp\left(r_{j}^{n}/R\right)+\gamma_{c}, \tag{38}\]  
where \(r_{j}^{n}\) is the radial position of the \(j\)th pixel and R is the radius of the image. \(\gamma_{e}\) and \(\gamma_{c}\) are empirically determined parameters. In the numerical simulation of the FD measurements with three targets, the spatially variant \(\gamma\) reconstructs the target close to the center of the circular object more clearly than the spatially constant \(\gamma\). The spatially variant \(\gamma\) provided a better image than the spatially constant one by evaluating the peak value of the reconstructed target, the FWHM, and the variance in the homogeneous region. Phantom experiments were also conducted. The artifacts appearing in the high-sensitivity region close to the sources and detectors were suppressed using this regularization method. Another way to suppress the artifacts and influence of measurement noise on the reconstructed image is to reduce the measurement noise. [PERSON] et al. proposed a noise reduction method for TD measurement [145]. The number of photons measured using the time-correlated single-photon counting method is a Poisson-distributed random variable with a mean equal to the noise-free photon count. Based on the probability density function of the DTOF measurements formulated using the Poisson distribution function, a noise-free DTOF that maximizes the posterior log-likelihood was estimated. Using the estimated noise-free DTOF, the artifacts and variations in the homogeneous region were reduced in the numerical and phantom experiments. Errors in the measurement conditions cause a mismatch between the forward process and the actual measurement. Consequently, artifacts may appear in the image to compensate for this mismatch. Therefore, calibration of the sources and detectors is important for accurately measuring light intensities. [PERSON] et al. demonstrated the simultaneous reconstruction of \(\delta\mu\) and calibration factors [146]. In this study, the intensity of the \(i\)th light source at position \(\mathbf{r}_{s,j}\) has a constant calibration factor \(s_{i}\) and is formulated as \(s_{i}\cdot q_{0}(\mathbf{r}_{s,j})\). Then, the measurement by the \(j\)th detector with calibration factor \(d_{j}\) was expressed as \(s_{i}\cdot d_{j}\cdot M_{ij}\). The image reconstruction of the calibration factors and \(\delta\mu\) was performed by linearization using the Rytov approximation. The matrix \(J\) in Equation (7) can be rewritten to include the effects of \(s_{i}\) and \(d_{j}\): The matrix was formulated using [PERSON]'s function for slab geometry.