## 2 Materials and Methods
### Stage-1 LUT for Quantifying Optical Properties  
Monte Carlo (MC) simulation has been widely used for simulating light transport in turbid media. The simulation results are often used to generate LUTs for quantifying optical properties through inverse modeling [19,20]. Traditionally, MC simulation was implemented on a CPU [21], and is considered computationally intensive and slow [22]. Due to its highly parallel nature of execution, graphic computing unit (GPU)-based MC simulation programs have recently been developed, which significantly improve the simulation efficiency [23]. This study adopted the Monte Carlo eXtreme (MCX) simulation package developed by [PERSON] to simulate reflectance in the spatial frequency domain [24]. We simulated reflectance images under structured illumination with a spatial frequency of 0.125 mm\({}^{-1}\) on large virtual phantoms (9.6 cm \(\times\) 9.6 cm \(\times\) 5 cm) with a grid size of 0.1 mm in the x, y, and z directions. The sample thickness of 5 cm was chosen to satisfy the requirement for infinite geometry in MC simulations [18]. To reflect the general properties of biological tissue, the virtual phantom was assumed to have an anisotropy g = 0.9 and a refractive index \(n\) = 1.37 [25]. A total of 208 pairs of biologically relevant optical properties were simulated with 16 absorption coefficient (\(\mu_{a}\)) values from 0.001 to 1 mm\({}^{-1}\), and 13 reduced scattering coefficient values (\(\mu_{s}\)') from 0.5 to 3 mm\({}^{-1}\). For each sample, diffuse reflectance images were simulated under 3 phase-stepped (0\({}^{\circ}\), 120\({}^{\circ}\), and 240\({}^{\circ}\)) structured illuminations. Thus, a total of 624 MC simulations were performed, and the corresponding reflectance images were saved. The MCX program assumes that one unit of energy (for example, 1 J) is deposited/launched into the medium. The simulation returns the detected optical energy density at the surface. The reflectance image was then calculated by normalizing the detected optical energy density at the surface to the incident optical energy density. Based on our analysis (data not shown), a total of 200 million photons were used in each simulation, which resulted in flat AC and DC reflectance profiles with low levels of noise (example shown in Figure 2C). It is worth noting that MCX simulation is not based on the white Monte Carlo approach; rather, each simulation is based on the actual optical properties of the medium. Following the simulation, a sub-region of size 2.5 cm \(\times\) 2.5 cm (red-dash square in Figure 2A) of each reflectance image was isolated to reduce the signal fall-off toward the edge of the image. The resultant diffuse reflectance image was then averaged along the pattern direction (yellow arrows in Figure 2A) to further improve the signal quality. Three intensity profiles, \(I_{1}\), \(I_{2}\), and \(I_{3}\), were obtained for three phase-stepped illuminations (Figure 2B). The demodulation process shown in Equations (1) and (2) was then performed to extract DC (\(I_{DC}\)) and AC (\(I_{AC}\)) signals (Figure 2C). \[I_{DC}=\sfrac{1}{3}(I_{1}+I_{2}+I_{3}) \tag{1}\]  
\[I_{AC}=\sfrac{\sqrt{2}}{3}\sqrt{(I_{1}-I_{2})^{2}+(I_{2}-I_{3})^{2}+(I_{3}-I_ {1})^{2}} \tag{2}\]  
\(I_{DC}\) and \(I_{AC}\) signals were further converted to reflectance (\(R\)) and modulation (\(M\)) with Equations (3) and (4), respectively. The definitions of reflectance and modulation followed the convention that was initially introduced by [PERSON] et al. [18]. It is worth noting that MC simulation generates normalized reflectance images; thus, no normalization to a spectralon image is needed. \[R(\mu_{a},\mu^{\prime}_{s}) = I_{DC}(\mu_{a},\mu^{\prime}_{s}) \tag{3}\] \[M(\mu_{a},\mu^{\prime}_{s}) = \frac{I_{AC}(\mu_{a},\mu^{\prime}_{s})}{I_{DC}(\mu_{a},\mu^{\prime} _{s})} \tag{4}\]  
Maps of \(R(\mu_{a},\mu^{\prime}_{s})\) and \(M(\mu_{a},\mu^{\prime}_{s})\) were converted to LUTs, \(\mu_{a}(R,M)\) and \(\mu^{\prime}_{s}(R,M)\), using a coordinate mapping technique similar to the one described elsewhere [15].