# A Review of Image Reconstruction Algorithms for Diffuse Optical Tomography
## 4 Optimization Process  
### Use of Regularization Minimizing Norms  
Regularization methods were employed in DOT image reconstruction to solve the inverse problem and alleviate instability, which means that reconstruction is strongly affected by noise in the measurement and error in the forward process. The reconstructed image was disturbed by noise and errors. As shown in Equation (9), the classical Tikhonov regularization that minimizes the squared 2-norm (\(\ell_{2}\)-norm or \(L_{2}\)-norm) of the reconstructed image can be readily applied to image reconstruction. Tikhonov regularization with an appropriate \(\gamma\) is useful for obtaining a smooth image by reducing the effect of measurement noise. However, smoothness is often accompanied by a decline in spatial resolution. In such a case, the changes in the optical properties of the measured object are blurred and are reconstructed in a larger volume than the true one. As a result, the change is reconstructed as a smaller value than true one, which means that the concentration of the photon absorber such as hemoglobin is underestimated by image reconstruction with the Tikhonov reconstruction. Additionally, the effect of diffusive light propagation also causes a low spatial resolution, generally in the image reconstruction of optical imaging, as reported in the literature [102]. To improve the quality of the reconstructed image, the regularization method for minimizing the \(p\)-norm (\(0\leq p\leq 1\)) of the reconstructed image has been used in recent years. The \(p\)-norm is defined as \(\left\|\boldsymbol{\mu}\right\|_{p}=\left(\sum_{k=1}^{K}\left\|\mu_{k}\right|^ {p}\right)^{1/p}\), for the vector \(\boldsymbol{\mu}\) comprising \(K\) entries. Figure 2 illustrates the idea of the regularization minimizing \(p\)-norm [103]. The dashed line represents the set of optical properties \(\boldsymbol{\mu}=(\mu_{1},\,\mu_{2})\) at two positions which provide an identical and single measurement. The image reconstruction of two optical properties from a single measurement is underdetermined. By employing a certain value of the norm of the reconstructed image (solid line), the number of candidates of the solution can be reduced. The candidates exist on intersecting points of the dashed line and circle, which represent the set of \(\boldsymbol{\mu}\) with a certain value norm in the 2-norm case. By choosing smaller \(p\)it is observed that the intersection points approach one or the other of the axes. Thus, one of entries of \(\upmu\) at a position takes a large value, and the other takes a value close to zero. This indicates that the solution becomes sparse. Generally, the DOT image reconstruction has a large number of unknown changes in \(\mu_{a}\) and \(\mu_{s}\), which are localized in a smaller number of voxels (smaller region) with smaller value of \(p\) as well as in the case of Figure 2. Then, the blurriness of the image can be reduced. In the case of Figure 2, by minimizing the \(p\)-norm, the solution can be specified on the tangent point. A difficulty encountered while implementing the regularization minimizing \(p\)-norm is computing the gradient for the optimization because the cost function becomes nonconvex. There are several techniques to implement the regularization minimizing \(p\)-norm. [PERSON] et al. proposed a compressed sensing method for DOT [104]. In this study, the DOT images were reconstructed using measurements acquired via random sampling. Image reconstructions were performed using the regularization method, minimizing the \(L_{1}\) norm (1-norm) of the image. The optimization process was based on the literature [105], in which the \(L_{1}\)-norm, i.e., the absolute value, was approximated as \(|\mu|\approx\sqrt{\mu\cdot\mu+u}\), with parameter \(u\) smoothing the function, and the gradient was approximately computed as \(d|\mu|/d\mu\approx\mu/\sqrt{\mu\cdot\mu+u}\).