# A Review of Image Reconstruction Algorithms for Diffuse Optical Tomography
## 3 Forward Process
### \(Sp_{n}\) Approximation  
[PERSON] and [PERSON] introduced a simplified spherical harmonics (\(SP_{N}\)) approximation, which was originally proposed for neutron transport, for CW light propagation in biological media [62]. In the \(SP_{N}\) approximation, the \(P_{N}\) approximation of 1D light propagation in the planar geometry was directly applied to 3D light propagation by replacing the 1D differential operator, d/dx, by the 3D gradient operator, \(\
abla\). In the 1D \(P_{N}\) approximation, the Legendre moments of radiance \(\phi_{n}(x)=\int P_{n}(\hat{s})I(x,\hat{s})d\Omega\) are defined. A set of four equations for the \(SP_{7}\) approximation was formulated for composite moments \(\varphi_{m}\) (\(m=1\), 2, 3, 4) defined by the sum of the series of \(\varphi_{n}\) (\(n=1\), 2,...,6). The equations for the \(SP_{N}\) approximation (\(N<7\)) were readily derived from those for the \(SP_{7}\) approximation, and the equation for \(SP_{1}\) approximation was equivalent to the PDE described in Section 3.4. The \(SP_{N}\) approximation was calculated using a set of coupled diffusion-like equations with second-order spatial derivative terms. The numerical simulations in [62] indicated that the precision of the approximation improved rapidly from \(N=1\) to \(7\). The computational time of the \(SP_{N}\) approximation is significantly shorter than that for the RTE computation by the discrete ordinate method discretizing \(\mathbf{s}\).  
[PERSON] et al. developed a 3D FEM for \(SP_{N}\) approximation in FD measurements [63]. The computational results were compared with those of MC simulations. The \(SP_{N}\) approximation (\(N>1\)) provides a more accurate phase and amplitude of the modulated light than \(SP_{1}\) approximation. [PERSON] and [PERSON] reported image reconstruction in FD measurements using the \(SP_{7}\) approximation [64]. The 2D image reconstructions of \(\mu_{a}\) and \(\mu_{s}\) from the phase and amplitude obtained by the numerical simulation with the \(SP_{5}\) approximation were performed by employing the Jacobian matrix, \(J\), calculated using the \(SP_{1}\) and \(SP_{5}\) approximations. Image reconstruction using the \(SP_{5}\) approximation provided more accurate quantitative and qualitative images.  
[PERSON] and [PERSON] reported image reconstruction using time-domain parabolic \(SP_{N}\) equations [65,66]. In addition to the aforementioned CW and FD cases, a system of diffusion-like equations for the \(SP_{7}\) approximation was formulated for the TD measurement. Numerical simulations of image reconstruction employing \(N=1\), 3, 5, and 7 were performed. In the simultaneous reconstruction of the heterogeneous medium where the high-absorption inclusion and high-scattering inclusion existed, the reconstructed \(\mu_{a}\) and diffusion coefficient, \(D=\)1/(3\(\mu_{a}+\mu_{s}\)'), were evaluated. The \(SP_{N}\) approximations (\(N>1\)) reconstructed the high \(\mu_{a}\) inclusion and low \(D\) inclusions better than the \(SP_{1}\)approximation, which underestimated \(\mu_{a}\). The error in the reconstructed value with the \(SP_{7}\) approximation was slightly larger than that with the \(SP_{5}\) and \(SP_{5}\) approximations [66].  
Some studies on image reconstruction with \(SP_{N}\) approximations for FDOT and QPAT have been reported [67; 68; 69].