## Abstract

Spatial frequency domain imaging is a non-contact, wide-field, fast-diffusion optical imaging technique, which in principle uses steady-state spatially modulated light to irradiate biological tissue, reconstruct two-dimensional or three-dimensional tissue optical characteristic map through optical transmission model, and further quantify the spatial distribution of tissue physiological parameters by multispectral imaging technique. The selection of light source wavelength and light field spatial modulation frequency is directly related to the accuracy of tissue optical properties and tissue physiological parameters extraction. For improvement of the measurement accuracy of optical properties and physiological parameters in the two-layer tissue, a multispectral spatial frequency domain imaging system is built based on liquid crystal tunable filter, and a data mapping table of spatially resolved diffuse reflectance and optical properties of two-layer tissue is established based on scaling Monte Carlo method. Combined with the dispersion effect and window effect of light-tissue interaction, the study applies numerical simulation to optimize the wavelength in the 650-850 nm range with spectral resolution of 10 nm. In order to minimize the uncertainty of the optical properties, Cramér-Rao bound is used to optimize the optical field spatial modulation frequency by transmitting the uncertainty of optical properties. The results showed that in order to realize the detection of melanin, oxyhemoglobin, deoxyhemoglobin, water and other physiological parameters in two-layer tissue, the best wavelength combination was determined as 720, 730, 760 and 810 nm according to the condition number. The findings of the Cramér-Rao bound analysis reveal that the uncertainty of optical characteristics for the frequency combinations [0, 0.3] mm^{-1}, [0, 0.2] mm^{-1}, and [0, 0.1] mm^{-1} increases successively. Under the optimal combination of wavelength and frequency, the diffuse reflectance of the gradient gray-scale plate measured by the multi-spectral spatial frequency domain imaging system is linearly correlated with the calibration value. The error between the measured liquid phantom absorption coefficient and the collimation projection system based on colorimetric dish is less than 2%. The experimental results of human brachial artery occlusion indicate that under the optimal wavelength combination, the change of the second layer absorption coefficient captured by the three frequency combinations decreases in turn, so as the change of oxygen saturation.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

Spatial frequency domain imaging (SFDI) has been widely used in Alzheimer's disease [1], burn and its severity [2,3], color spots [4], breast cancer [5–7], ovarian cancer [8,9] and so on, with the advantages of non-contact, wide field and high speed. In this technique, biological tissues are irradiated by steady-state spatially modulated light fields with two different spatial modulation frequencies and three different initial phases, and six diffuse reflection images are captured. In combination with the three-phase demodulation method, the inverse model based on diffusion theory or the inverse look-up table method based on Monte Carlo simulation, the optical properties of tissue are reconstructed pixel by pixel to form a two-dimensional optical characteristic topology map [7,10–12].

Combined with multispectral imaging technology, Three-dimensional data cube of tissue optical properties reconstructed at multi-wavelengths by spatial frequency domain imaging, and the topology map of biological tissue physiological parameters can be retrieved according to Lambert Beer's law and non-negative least square algorithm or multiple linear regression. In general, the more the wavelengths, the smaller the inversion error of tissue physiological parameters, but at the same time, the acquisition process will lead to a linear increase in time cost. In order to meet the exact conditions, the theoretical number of the lowest wavelength should not be less than the number of absorbing components in biological tissue [13,14]. In visible and near infrared bands, biological tissue absorption mainly comes from melanin, oxyhemoglobin deoxyhemoglobin and water, so as to determine a reasonable combination of illumination wavelengths to avoid approximate collinearity of extinction coefficients of absorbed substances in tissues. The way can effectively improve the inversion accuracy of physiological parameters. Yafi et al. measured the oxygen saturation of normal skin using four wavelengths of 655, 730, 850 and 970 nm [15]. Nguyen et al. measured the concentration of oxyhemoglobin and deoxyhemoglobin in human breasts using 670, 730, 760, 808, 860 and 980 nm wavelengths [16]. XinlinChen et al. measured the concentrations of melanin, oxyhemoglobin and deoxyhemoglobin in human skin tissue using 460, 540 and 623 nm wavelengths [17]. Omri et al. measured the oxygen saturation of normal skin using four wavelengths of 690, 880, 920 and 970 nm [18]. When the above research teams quantified chromophore concentrations, the selection for wavelength combination was lack of objective theoretical basis. Mazhar et al. based on the extinction coefficients of oxyhemoglobin and deoxyhemoglobin, carried out wavelength optimization by condition number during the range of 650-980 nm with the resolution of 10 nm [19]. However, there is a lack of wavelength optimization research on the effects of melanin on other physiological substances, and a large amount of melanin in the upper tissue will absorb part of the incident light.

For the spatially modulated light field, the biological tissue is equivalent to a low-pass filter. With the increase of spatial modulation frequency, the penetration depth of light in biological tissue is lower, and the tissue diffuse reflectance signal decreases [10,20]. The selection of spatial modulation frequency is crucial to the accuracy of optical characteristic reconstruction [21,22]. When different research teams measure the optical characteristic parameters of tissue, the frequency range used is different, and there is a lack of objective spatial frequency evaluation method. Anderson et al. measured the optical property parameters of normal and damaged apple tissue using nine equally spaced spatial frequencies in the 0.0149-0.1344mm^{-1} range [23]. Mazhar et al. used five frequencies between 0-0.2mm^{-1} when detecting thermal burns in pig tissue [3], and Cuccia et al. used up to 30 frequencies when measuring the optical property parameters of liquid optical bionic samples, and the spatial frequency was in the 0-0.13mm^{-1} range [10]. For this reason, Pera et al. proposed to use the Cramér-Rao bound to calculate the uncertainty of tissue optical properties in the spatial frequency domain to evaluate the quality of spatial frequency selection [24], but the research was limited to the one-layer tissue model of most biological tissues. As for measuring the optical parameters of apple, Hu Dong et al. proposed to optimize the starting point, end point and interval of spatial frequency sequentially [25], which is used to reconstruct the optical properties of two-layer tissue. However, the optimization effect is limited by the accuracy of the tissue optical model and the optimal range of the starting point, end point and interval of the selected frequency.

This paper mainly has two novel contributions. Fist, during the wavelength optimization process, some research teams consider the crosstalk of water to oxyhemoglobin or deoxyhemoglobin, while this paper considers the melanin is likely to be the crosstalk to oxyhemoglobin or deoxyhemoglobin when exits a lot of melanin in the skin epidermis, and minimize the interference degree through analog simulation method, then optimize wavelength combination; Second, it is difficult to quantify the uncertainty of optical parameters retrieved by reverse lookup table method. In this paper, the method of evaluating the uncertainty of optical parameters of double layer tissues by Cramér-Rao bound is derived to guide the selection of frequencies.

## 2. Materials and methods

#### 2.1 Multispectral spatial frequency domain imaging system

The structure of the multispectral spatial frequency domain imaging system built in this study is shown in Fig. 1. The system uses broadband halogen lamp (OSL2IR, 150W3200 K, Thorlabs) as the illuminating light source. After passing through the optical fiber bundle (OSL2RFB, Thorlabs), collimating lens L1 and linear polarizer P1 in turn, the output light is irradiated to the digital micromirror device (DMD) (Vmur7001 ViALUX). The effective structured light area is about 60mm×40 mm. The DMD modulation pattern is projected onto the biological tissue sample through the achromatic lens L2 and the plane mirror M. The modulated diffuse reflection image of the interaction between light and biological tissue is collected by CMOS camera C (BFS-U3-123S6M-C) after linear polarizer P2, liquid crystal adjustable filter LCTF (VariSpec) and imaging lens L3. The CMOS camera has an imaging range of 250 mm, the camera resolution of 4096×3000, and the pixel size of 3.45um which represents approximately 810um^{2} dimensions of space. The polarizer P1 and P2 are orthogonal in the transmission direction to eliminate the specular reflection of the sample.

The optical parameter calibration of the SFDI system was carried out, and the diffuse reflectance of gray-scale plates was measured by SFDI system and compared with plates’ s reference value; the absorption coefficient measured by collimating transmission system was compared with that measured by SFDI system, in this process, the spatial frequencies are [0, 0.2] mm^{-1}, because the phantom used was liquid, the absorption coefficient of the first layer and the second layer were set to be the same during inversion, the liquid phantom was placed in a 5 ml glass beaker, and black sandpaper was placed around and at the bottom of the beaker to prevent the reflection of the glass mirror, and placed the humoral surface in the imaging plane, since there was reflection at the edge of the beaker, the average value was calculated from the middle area. The system calibration flowchart is shown in Fig. 2.

#### 2.2 Spatial frequency domain imaging method

According to the biological tissue structure and optical properties, the biological tissue is regarded as a two-layer tissue optical model. The first layer is the epidermis, which is mainly absorbed by melanin, while the lower layer is mainly absorbed by oxyhemoglobin, deoxyhemoglobin and water. Considering that the thickness of the first layer is thin, the light passing through the first layer is mostly parallel, ignoring the scattering difference between the two layers [17]. The principle of the spatial frequency domain imaging method is shown in the Fig. 3. The digital micromirror successively projects two steady spatially modulated light fields with two different spatial modulation frequencies and three different initial phases to irradiate biological tissue, and the camera captures six diffuse reflection images of biological tissue. Formula (1) and formula (2) are used to demodulate photon density amplitude and spatially resolved diffuse reflectance images of biological tissue. Then, based on the optical characteristics and spatial frequency domain diffuse reflectance data mapping table, the diffuse reflectance image is looked up pixel by pixel, and the two-dimensional topological map of biological tissue optical properties is reconstructed. Finally, according to Lambert Beer's law, the constrained least square is used to solve the physiological parameters.

Among them, ${I_1}({\lambda _i},{f_x})$, ${I_2}({\lambda _i},{f_x})$ and ${I_3}({\lambda _i},{f_x})$ are the original images collected at different wavelengths and spatial frequencies, $M({\lambda _i},{f_x})$ and ${R_d}({f_x})$ are the photon density amplitude and spatially resolved diffuse reflectance of the tissue to be measured, ${M_{\textrm{ref}}}({\lambda _i},{f_x})$ and ${R_{d,ref}}({\lambda _i},{f_x})$ the photon density amplitude and diffuse reflectance of the reference whiteboard, respectively, which ${R_{d,ref}}({\lambda _i},{f_x})$ is equal to 0.99. ${u_{a2}}$ is the absorption coefficient of the second layer of skin tissue, ${\varepsilon _i}$ is the extinction coefficient of oxyhemoglobin, deoxyhemoglobin and water, and ${c_i}$ is the concentration of oxyhemoglobin, deoxyhemoglobin and water, respectively.

#### 2.3 Scaled Monte Carlo simulation

In conventional spatial frequency domain imaging, the multi-layer Monte Carlo method proposed by Lihong Wang et al. [26,27] is used to simulate the vertical irradiation of biological tissues with different optical properties by collimating point light source, and the spatial point spread function of biological tissue is extracted and converted to spatial frequency domain by Fourier transform to establish the data mapping table of optical properties and spatial frequency domain diffuse reflectance. Due to the radial symmetry of spatial point spread function, two-dimensional Fourier transform can be realized by one-dimensional Hankel transform to obtain the diffuse reflectance in spatial frequency domain [10,26,27], which is later called general Monte Carlo simulation (gMC). However, for multi-layer biological tissues, the time cost of Monte Carlo simulation increases significantly with the increase of optical characteristic dimension. Furthermore, this method is greatly affected by sampling density since Hankel transform is very sensitive to discrete error.

In this study, we uses the scaling Monte Carlo simulation (sMC) proposed by Gardner et al. to avoid the above limitations [12]. For a specific scattering coefficient, the biological tissue absorption coefficient is initialized randomly, and a general Monte Carlo simulation is carried out to record the total path of all photons moving in each layer and the distance between the photons escaping from the tissue surface and the incident point. For the biological tissues with the same scattering coefficient and different absorption coefficient, the spatial frequency domain diffuse reflectance can be directly scaled by formula (4). Repeat the above process for all scattering coefficients, that is, complete the establishment of optical properties and spatial frequency domain diffuse reflectance data mapping table.

*N*is the total number of emitted photons, ${W_n}$ is the rest weight of each photon, ${s_{1,n}}$ and ${s_{2,n}}$ are the path length of the photon in the first layer and the path length of the photon in the second layer respectively, ${J_0}$ is the zero-order Bessel function, ${f_\rho }$ is the spatial frequency, and ${x_n}$ is the distance between the emission point and the emission point of the photon. Figure 4 is a flow diagram of the gMC and sMC algorithms.

#### 2.4 Wavelength optimization method

In this article, 21 wavelength points of 650-850 nm range and 10 nm interval are used to optimize the wavelength combination. In Lambert Beer's law, the condition number is used to measure the sensitivity of physiological parameters to the error of optical parameters, and the larger the condition number, the worse the sensitivity. Lambert Beer's law can be written in the form of formula (5). The condition number is defined as the ratio of the maximum and minimum singular values of the extinction coefficient matrix [28]. The wavelength combination is selected by the minimum condition number and used to quantify the more accurate physiological parameters. In formula (6), *k* is the condition number and *SVD* is the singular value.

After optimizing the wavelength by condition number, we uses the simulated phantom methods. The absorption coefficient of the simulated phantom is obtained by multiplying the assumed physiological parameters by the corresponding extinction coefficient, and 5% random Gaussian noise was added into absorption coefficient to reconstruct the physiological parameters for different wavelengths.

#### 2.5 Frequency optimization method

For the spatially modulated light field, the biological tissue is equivalent to a low-pass filter. With the increase of spatial modulation frequency, the penetration depth of light in biological tissue is lower, and the tissue diffuse reflectance signal is smaller. The selection of spatial modulation frequency is crucial to the accuracy of optical characteristic reconstruction. The SFDI system was used to measure the diffuse reflectance at 10 different frequencies, and these diffuse reflectance are calculated as variation coefficient *c*. Variation coefficient *c* and ${R_d}$ were calculated to obtain the uncertainty of diffuse reflectance *C*. The uncertainty of optical parameters of biological tissue can be transmitted by Fisher information matrix [29].

The Fisher information matrix is given by Eq. (7), and $\theta \textrm{ = (}{\mu _{a1}}\textrm{,}{\mu _{a2}}\textrm{,}{\mu _s}\textrm{)}$, the measurements of different spatial frequencies are statistically independent, so the data obey multivariate normal distribution, *r* is the spatially resolved diffuse reflectance and ${C_{Rd}}$ is the diagonal covariance matrix.

*r*is diffuse reflectance, obtained by scaling Monte Carlo simulation. ${C_{Rd}}$ is the error model of diffuse reflectance at different frequencies and wavelengths. Taking the inverse of Fisher matrix, the uncertainty of optical parameters is obtained as follows:

#### 2.6 Brachial artery occlusion method

The frequency selection is guided by the computational simulation of the Cramér-Rao bound. In addition, arterial occlusion experiments have been performed on healthy male skin tissue to verify the effect of different frequency combination inversion parameters. During this imaging procedure, the dorsal right hand was imaged while a full arterial occlusion was applied to the brachial artery using a pneumatic cuff capable of near-instantaneous inflation to 210 mm Hg. The duration of imaging consisted of 60s of baseline (no occlusion), followed by 140s of arterial occlusion, and then 120s of postrelease. Two-dimensional (2-D) maps of oxyhemoglobin and deoxyhemoglobin were reconstructed for each time point.

In order to verify the accuracy of wavelength and frequency optimization, based on the prior knowledge of extinction coefficient of physiological parameters, combined with the “condition number” wavelength optimization method, the wavelengths of human skin tissue experiments are set as 720, 730, 760 and 810 nm. [0, 0.3] mm^{-1}, [0, 0.2] mm^{-1}, and [0, 0.1] mm^{-1} frequency pairs were used to carry out the experiment of arm artery occlusion in healthy men. The stripe patterns of 0mm^{-1}, 0.1mm^{-1}, 0.2mm^{-1} and 0.3mm^{-1} uploaded in DMM were collected at the same time during arterial occlusion, and then divided into [0, 0.3] mm^{-1}, [0, 0.2] mm^{-1}, and [0, 0.1] mm^{-1} frequency combinations for post-processing.

In the process of arterial occlusion experiment, the back of the hand is placed on the desktop in the imaging plane. The first three times were unpressurized ground state, seven times were collected after compression, and six times were collected after release, which had a total of 16 times, with an interval of 20 seconds.

## 3. Results

#### 3.1 Scale Monte Carlo simulation results

gMC is used to simulate ${\mu _{\textrm{a}1}} = 0.05m{m^{ - 1}}$ and ${\mu _{\textrm{a2}}} = 0.05m{m^{ - 1}}$, ${\mu _s}$ in the 10mm^{-1}-20mm^{-1} range with the interval 0.1mm^{-1}, the number of photons emitted *N *= 10^{6}, the thickness of the first layer is set to 0.11 mm [17], the refractive index (n) is set to 1.4, and the anisotropic scattering factor (g) is 0.9. The range of optical parameters for scaling is set as follows: ${\mu _{\textrm{a}1}}$ and ${\mu _{\textrm{a2}}}$ are both in the range of 0-0.095mm^{-1}, with an interval of 0.005mm^{-1}. ${\mu _s}$ is between 10mm^{-1} and 20mm^{-1}, and an interval of 0.1mm^{-1}, with a total of 40400 points. The simulation results are used in the inverse look-up table of optical parameters and the data source of frequency optimization. In order to compare the results of scaled Monte Carlo (sMC) and general Monte Carlo simulation (gMC), four groups of optical parameters of human skin tissue were selected in the frequency range of 0-0.3mm^{-1}. The results of ua1, ua2 and us were 0.08mm^{-1}, 0.03mm^{-1} and 18.9mm^{-1}; 0.08mm^{-1}, 0.03mm^{-1} and 10mm^{-1}; 0.04mm^{-1}, 0.02mm^{-1} and 18.9mm^{-1}; 0.04mm^{-1}, 0.02mm^{-1} and 10mm^{-1}; respectively. Figure 5 shows the comparison between sMC and gMC simulations. The results indicate that sMC can quickly build tables instead of gMC.

#### 3.2 Diffuse reflectance calibration and optical parameter calibration of the system

In order to verify the linearity of the multispectral spatial frequency domain imaging system in diffuse reflectance measurement, gray-scale plates with different diffuse reflectivity (2%, 5%, 10%, 20%, 40%, 60%, 80%, 99%) have been photographed, and 21 wavelengths of zero frequency diffuse reflectance are obtained and linearly fitted with the diffuse reflectance provided by the manufacturer, as shown in Fig. 6 (a). Figure 6 (b) shows the diffuse reflectance histogram of the system at 21 wavelengths, and B-U is the 21 wavelengths of 650-850 nm. The results indicate that the system has good linearity (R^{2} > 0.99) and the fitting slope is not 1, but the diffuse reflectance is calibrated by the whiteboard with no effect on the measurement of optical parameters.

Using 10% long-chain Intralipid as scatterer and Hemoglobin (HGB) and Melanin (CAS 8005-03-6) as absorber respectively, liquid phantoms with different optical parameters are made, and the distribution ratio is shown in Table 1. The absorption coefficients of phantom 1 and 2 are measured by multispectral spatial frequency domain imaging system, and the absorption coefficients of phantom 3 and 4 are measured by collimation transmission system. The absorption coefficient of tissue phantom is calculated by the following formula (11), where *l* is the optical path, ${I_0}$ is the transmitted light intensity without adding phantom, and $I$ is the transmitted light intensity of with adding phantom.

Figure 7(a) shows the absorption coefficient of the liquid phantom in the range of 650-850 nm, while Fig. 7 (b) shows the correlation between the measured results of the multi-spectral spatial frequency domain imaging system and the collimation transmission system. The dotted line data represents the absorption coefficient of No. 1 and No. 2 measured by the collimation transmission system, and the full line represents the absorption coefficient of No. 3 and No. 4 measured by the multi-spectral spatial frequency domain imaging system. The results indicate that when the absorption composition and concentration are identical, the results of the multispectral spatial frequency domain imaging system are consistent with those of the collimation transmission system, and the maximum deviation is less than 1.85%.

#### 3.3 Reconstructions with optimal and nonoptimal sets of wavelengths

In this article, 21 wavelength points of 650-850 nm range and 10 nm interval are used to optimize the wavelength combination, which are 5985 groups of optional wavelength combinations. The condition number of all wavelength combinations is calculated and sorted. Figure 8 (a) shows the statistical results of the condition number in different wavelength combinations. The wavelength combination corresponding to the minimum condition number (Set1: 720 nm, 730 nm, 760 nm, 810 nm) and the wavelength combination corresponding to the maximum condition number (Set2: 700 nm, 810 nm, 830 nm, 840 nm) are selected. In addition, the two optimized wavelength combinations are plotted on the spectrum of oxygenated, deoxygenated, melanin and water proteins, as shown in Fig. 8 (b), the wavelengths with some distance can better extract chromophore concentration.

A square simulated phantom with pixel of 150 × 150 is established by numerical simulation, which has four circular absorption units with diameter of 30 pixel are embedded. Oxyhemoglobin, deoxyhemoglobin melanin and pure water are added to No. 1 absorption unit, No. 2 absorption unit, No. 3 absorption unit, and No. 4 absorption unit respectively, and the specific concentration is shown in Table 2.

According to Lambert Beer's law, the absorption coefficient of each absorption unit in the simulated phantom under three groups of wavelength combination is calculated, and 5% random Gaussian noise is added. Three groups of wavelength combination are shown in Table 3. Set3 is the conventional wavelength combination used by omri. With the generated optical parameters as the input, the constrained least square is used to quantify the physiological parameters of the four absorption units under three groups of wavelength combination. The inversion results of the four components are shown in the Fig. 9. Melanin in Set2 has serious crosstalk to oxyhemoglobin and deoxyhemoglobin, and the error of melanin reconstruction is larger. Therefore, it can be seen that wavelength combination in Set1 has better separation effect than that of Set2, and melanin in Set3 interferes with deoxyhemoglobin a little more than that of Set1.

#### 3.4 Cramér-Rao bound estimates two-layer optical uncertainty

In the optical range of human skin tissue, four groups of optical parameters have been selected (${\mu _{\textrm{a}1}}$, ${\mu _{\textrm{a2}}}$ and ${\mu _s}$ were set to 0.08mm^{-1}, 0.03mm^{-1} and 18.9mm^{-1}; 0.08mm^{-1}, 0.03mm^{-1} and 10mm^{-1}; 0.04mm^{-1}, 0.02mm^{-1} and 18.9mm^{-1}, 0.04mm^{-1},0.02mm^{-1} and 10mm^{-1}, respectively). The ${R_d}({f_x})$ of four groups of optical parameters at different frequencies is calculated by scaling Monte Carlo, and the uncertainty of the system to each optical parameter is calculated by using the formula (7)–(10). The average values of the four results are taken as the uncertainty of the system estimation of optical parameters. ${f_{x1}}$ and ${f_{x2}}$ are in the range of 0-0.3mm^{-1}, with the same frequency, so the result is diagonally symmetrical.

Figure 10 (a), Fig. 10 (b) and Fig. 10 (c) are ${\mu _{\textrm{a}1}}$ uncertainty, ${\mu _{\textrm{a2}}}$ uncertainty, ${\mu _s}$ uncertainty respectively, and Fig. 9 (d) is an enlarged view of the shaded part of Fig. 10 (b). The results show that the closer the two frequencies, the greater the optical uncertainty. The lower tissue is mainly absorbed by oxyhemoglobin and deoxyhemoglobin and water, while oxygen saturation can be obtained by ${\mu _{\textrm{a2}}}$.

Based on the above simulation results, the uncertainty of ${\mu _{\textrm{a2}}}$ is simulated for [0, 0.3]mm^{-1}, [0, 0.2]mm^{-1}, [0, 0.1]mm^{-1}, when ${\mu _{\textrm{a}1}}$ keeps unchanged and ${\mu _{\textrm{a2}}}$, ${\mu _s}$ is changed. ${\mu _{\textrm{a}1}}$ is 0.05mm^{-1}, ${\mu _{\textrm{a2}}}$ is 0-0.1mm^{-1}, and ${\mu _s}$ is 10-20mm^{-1}. Figure 11(a) shows the uncertainty of ${\mu _{\textrm{a2}}}$ simulated by using frequency combinations [0, 0.3]mm^{-1}, Fig. 11(b) shows the uncertainty of ${\mu _{\textrm{a2}}}$ simulated by using frequency combinations [0, 0.2]mm^{-1}, and Fig. 11(c) shows the uncertainty of ${\mu _{\textrm{a2}}}$ simulated by using frequency combinations [0, 0.1]mm^{-1}. The results indicate that the optical uncertainty of ${\mu _{\textrm{a2}}}$ increased successively for frequency combinations [0, 0.3]mm^{-1}, [0, 0.2]mm^{-1}, [0, 0.1]mm^{-1}, and the significance for guiding physiological parameters decreases successively.

Simulate the uncertainty of ${\mu _{\textrm{a2}}}$ when ${\mu _s}$ keeps unchanged and ${\mu _{\textrm{a}1}}$, ${\mu _{\textrm{a2}}}$ is changed. The ${\mu _s}$ of (a), (c) and (e) in Fig. 12 is 12.5mm^{-1}, and the ${\mu _s}$ of (b), (d) and (f) in Fig. 12 is 15mm^{-1}, ${\mu _{\textrm{a}1}}$ and ${\mu _{\textrm{a2}}}$ are in the range of 0-0.095mm^{-1}. (a) and (b) in Fig. 11 are ${\mu _{\textrm{a2}}}$ uncertainty of frequency combinations [0, 0.3]mm^{-1}, (c) and (d) in Fig. 11 are ${\mu _{\textrm{a2}}}$ uncertainty of frequency combinations [0, 0.2]mm^{-1}, and (e) and (f) in Fig. 11 are ${\mu _{\textrm{a2}}}$ uncertainty of frequency combinations [0, 0.1]mm^{-1}. The results indicate that the optical uncertainty of ${\mu _{\textrm{a2}}}$ with frequency pairs of 0mm^{-1} and 0.3mm^{-1}, 0mm^{-1} and 0.2mm^{-1}, 0mm^{-1} and 0.1mm^{-1} still increases in turn.

#### 3.5 Human brachial artery occlusion experiment

Figure 13 shows the changes of ${\mu _{\textrm{a2}}}$ captured by 720, 730, 760 and 810 nm at three frequency pairs of [0, 0.3] mm^{-1}, [0,0.2] mm^{-1} and [0, 0.1] mm^{-1} during arterial occlusion. The variation coefficient of diffuse reflectance at different frequencies on the baselines is less than 0.2%, the spacing of the absorption coefficients is 0.05mm^{-1}, and the absorption coefficient can not be reflected by using function “griddata” mapping in MATLAB. The variation coefficient of diffuse reflectance at different frequencies of 810 nm wavelength is less than 0.6%, and the absorption coefficient does not change in the same way, which may be due to the change of chromophores during the occlusion experiment. The results indicate that the capture range of [0, 0.3]mm^{-1} is the largest, the capture range of [0,0.2]mm^{-1} is the second, and the [0, 0.1] mm^{-1} is the smallest, which is consistent with the results of CRB simulation.

Figure 14 shows the changes of physiological parameters during arterial occlusion, Fig. 14 (a), (b) and (c) show the changes of oxyhemoglobin and deoxyhemoglobin at [0,0.3]mm^{-1}, [0,0.2]mm^{-1} and [0,0.1]mm^{-1}, respectively, and Fig. 14(d) shows the changes of blood oxygen saturation. The results indicate that the frequency pairs of [0,0.3]mm^{-1}, [0,0.2]mm^{-1} and [0,0.1]mm^{-1} decreased sequently in the process of arterial occlusion.

## 4. Discussion

This study builds a multispectral spatial frequency domain imaging system, and studies the wavelength optimization and frequency optimization. The results of wavelength optimization are shown in Fig. 8(b). When the absorption of the four wavelengths to each chromophore is not colinear, the separation effect of the concentration of each chromophore will be better. The simulated phantom experiment results for separation of chromophores with different wavelength combinations are shown in Fig. 9. Set1 has a good separation degree, followed by Set3, and Set2 has serious crosstalk. The results are of benefit to extract more accurate physiological parameters. In this study, we proposes a method for evaluating the optical uncertainty of two-layer tissues with Cramér-Rao bound, with the results shown in Fig. 10. The changes of ${\mu _{\textrm{a2}}}$ of each wavelength during the experiment are shown in Fig. 13. It can be drawn from the results that different frequency combinations are sensitive to different degrees in the whole process of arterial occlusion experiment. Frequency combinations [0, 0.3] mm^{-1} are the most sensitive, with absorption changes captured at the beginning of occlusion, followed by frequency combinations [0, 0.2] mm^{-1}, while the frequency combinations [0, 0.1] mm^{-1} is weakest. The changes of oxygenated hemoglobin, deoxyhemoglobin and blood oxygen saturation in different frequency pairs during the experiment are shown in Fig. 14. The uncertainty of the inverse look-up table method is related to the optical parameters extracted and the optical corresponding table used, which can guide the selection of frequency and extract optical parameters more accurately.

One important aspect of SFDI frequency optimization that we have not explored here is the impact of the first layer thickness. In this study, the thickness of the first layer was set as 0.11 mm, instead of obtaining the first layer thickness through inversion. The difference of skin thickness is crucial for the accurate measurement of optical parameters [25]. Considering the factors of the first layer thickness, the first layer thickness will be evaluated by the Cramér-Rao bound for further study.

Figure 11 and 12 show that under the condition of different optical parameters of two-layer structure, the optical parameter uncertainty of different frequency combinations is different. By studying the influence of the thickness of the first layer on the uncertainty of inversion optical parameters, the variation rule may be found. Different frequencies have different penetration depths. As shown in Fig. 13, the sensitivity of optical parameters retrieved by different frequency combinations in the process of arterial occlusion is different.

## 5. Conclusions

In this study, wavelength optimization and frequency optimization in spatial frequency domain imaging are studied. A large amount of melanin in the upper layer of the skin tissue has strong absorption, and melanin interfered with the extraction of other chromophores. The crosstalk of melanin to other chromophores was studied during wavelength optimization to better separate chromophores. Cramér-Rao bound can effectively quantify the uncertainty of optical parameters inversion by table lookup method and guide the selection of frequency in a simulated way, redu cing a large number of verification experiments with different frequencies, which is of great benefit to the spatial frequency domain imaging technology to extract optical and physiological parameters more accurately.

## Funding

Hefei Institutes of Physical Science, Chinese Academy of Sciences (YZJJZX202009); Anhui Provincial Key Research and Development Plan (202004a07020016); Natural Science Foundation of Anhui Province (1908085QH365).

## Disclosures

The authors declare that they have no conflicts of interest related to this article.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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