1. Introduction

In recent years, considerable efforts have been made on measuring the optical absorption (μ_a) and reduced scattering (μ_s^′) coefficients of biological tissues, including food and agricultural products (e.g., apples, mangos and kiwifruits), in order to better understand light interactions with tissues, properly interpret measurement data, gain better knowledge of the chemical and structural properties of tissues, and enhance food quality and safety assessment [1–4]. With current noninvasive techniques, tissue optical properties are usually determined by using an analytical solution to the diffusion approximation equation (DAE) coupled with an inverse algorithm [5,6]. Much research on optical property measurement in the past has been focused on biological tissues that can be simplified as homogeneous media. However, many biological tissues are composed of distinct layers with different optical properties. Hence it is inadequate to use an analytical solution or model derived for homogeneous media to estimate optical properties of layered media, as it can result in large errors in estimation and also loss of critical physicochemical information for individual layers. It is thus desired that appropriate multilayer models be used in estimating optical properties of each layer.

Several analytical solutions to the DAE for layered turbid media have been derived under the illumination of a point light source (i.e., continuous-wave or steady-state, time-resolved (pulsed), or frequency modulated) [7–11]. Optical techniques (i.e., spatially resolved, time-resolved, and frequency domain) based on these analytical solutions have been developed for measuring the μ_a and μ_s^′ of turbid media in the traverse direction (i.e., along the surface of a turbid medium). As a relatively new optical technique, spatial-frequency domain imaging (SFDI) has capability for quantitative spatial mapping (i.e., in the axial and transverse directions) of μ_a and μ_s^′ over a wide field of view [12,13]. In this technique, diffusely back-scattered images are captured from a turbid sample subjected to the illumination of sinusoidal pattern with different spatial frequencies (f_x). The tissue optical properties are then determined by fitting the demodulated reflectance on a pixel-by-pixel basis using an analytical solution of the diffusion model derived by Cuccia et al. [14,15]. Hereinafter, the spatial frequency is defined as the number of fringes per unit path (e.g., meter or millimeter). Several studies have been reported for estimating the optical absorption and reduced scattering coefficients of layered tissues using SFDI technique. Weber et al. (2009) [16] employed SFDI for measuring the μ_a and μ_s^′ of two-layer custom-constructed optical phantoms and in vivo volar forearms at 650 nm, and they reported that the approach could provide noncontact mapping and quantification of layered tissue optical properties. Saager et al. (2011) [17] presented a method for depth-resolved optical property quantitation using spatially-modulated quantitative spectroscopy, and they reported that the method was capable of determining top-layer thickness and chromophore concentration in highly structured skin. Yudovsky et al. (2011; 2012) [18,19] successfully decoupled the effect of melanin absorption in the epidermis from blood absorption in the dermis, and independently measured the optical thickness of the epidermis and μ_a and μ_s^′ of the dermis, using spatial-frequency domain spectroscopy for two-layer media, coupled with an artificial neural network. However, there still exist great challenges both mathematically and experimentally for estimating the optical properties of two-layer turbid media. A two-layer model is much more complicated in parameter estimation since it has five optical parameters (i.e., μ_a and μ_s^′ of each layer, plus the unknown thickness of the first layer), which may result in larger, unacceptable estimation errors, especially for μ_a and μ_s^′ of the second layer. For instance, Cen et al. (2009) [20] and Weber et al. (2009) [16] estimated the optical properties (μ_a1, μ_s1^′, μ_a2 and μ_s2^′) of two-layer phantoms simultaneously by using spatially resolved and SFDI techniques, respectively, and reported that it was difficult to accurately estimate optical parameter(s) for the second layer. Hereinafter we refer to the method for estimating μ_a1, μ_s1^′, μ_a2 and μ_s2^′ simultaneously as ‘one-step method’, in comparison to the ‘stepwise method’ proposed in this paper. In order to accurately estimate the optical properties of two-layer media, it is important and also necessary to understand the intrinsic properties of the two-layer model (e.g., sensitivity coefficients of optical properties) prior to implementing an inverse algorithm for optical property estimation. Furthermore, a stepwise method, also called sequential method, which first estimates μ_a and μ_s^′ of one (often the top) layer, followed by estimating μ_a and μ_s^′ of the other layer, was reported to have significantly improved the parameter estimation accuracy of two-layer media, when using spatially resolved technique [21,22]. While the same approach may be applied for optical property estimation of two-layer turbid media by SFDI technique, the efficacy and accuracy of the stepwise method should be determined since the two techniques are substantially different in their mathematical models and measurement configurations.

Therefore, this research was aimed at employing a stepwise method for improved estimation of optical properties for two-layer turbid media from the spatial-frequency domain reflectance. The specific objectives were to: 1) assess the stepwise method by comparing the reflectance generated by the diffusion model and Monte Carlo (MC) simulation for one-layer and two-layer turbid media; 2) evaluate the efficacy and accuracy of the stepwise method for optical property estimations in comparison with the one-step method; and 3) determine the constraining conditions, beyond which the optical property estimation accuracy is not acceptable, by analyzing and comparing the absolute error contour maps.

2. Materials and methods

2.1 Theory and model

The DAE, which is a simplified form to the radiative transfer equation assuming that the light distribution is almost isotropic in a scattering dominated medium (i.e., μ_s^′ >> μ_a), is adequate to model light propagation. Consider a medium normally illuminated at its surface by a steady-state, planar sinusoidal light pattern, the DAE can be expressed as

2.1.1 One-layer model

For a homogeneous, scattering dominant medium of semi-infinite geometry, the incident optical intensity S₀ in Eq. (2) decays exponentially with the depth [14]. The DAE is then reduced to a one-dimensional ordinary differential equation. By solving this ordinary differential equation with an appropriate boundary condition, one can obtain the fluence rate $\varphi \left(x,z\right)$, from which the reflectance at the surface can be calculated as [13]

2.1.2 Two-layer model

For a two-layer turbid medium with the continuous or discontinuous interface, the incident optical intensity S₀ in Eq. (2) is different in each layer subject to the first-layer thickness (the second layer is assumed to be infinitely thick) and can be expressed approximately in the following forms

2.2 Sensitivity coefficient analyses

Sensitivity coefficients are used to describe how the dependent variables respond to the changes of independent variables, which can be calculated as the first derivative of the dependent variable with respect to the independent variable [20]. Essentially, sensitivity coefficients reflect how the variables in an equation or function are related to the calculated result, and they are helpful to determine if unique solutions for the optical property estimations exist and whether a parameter of interest can be estimated properly. In this study, the scaled sensitivity coefficients of optical parameters (i.e., μ_a and μ_s^′ of each layer) in the two-layer model for SFDI were calculated by

2.3 Simulation samples

In order to evaluate the stepwise method for optical property estimations and determine constraining conditions for the first-layer thickness, a publicly available MC simulation program for multilayered media was used [13,18,24]. The MC program has been proven effective and accurate in previous studies [20,25]. In the MC simulation, a package of 5 × 10⁶ photons were tracked. The maximum radial distance of the medium was set to 50 mm, which is large enough to be treated as semi-infinite. The spatial resolution for both radial distance and depth was set to 0.1 mm. The refractive indexes of the two-layer media were both chosen to be 1.35, which is typical for many biological materials [26,27].

Three groups of two-layer simulation samples were prepared. For the first group, a total of 26 samples (five samples with different combinations of μ_a or μ_s^′ for each layer, plus six samples with different first-layer thicknesses) covering a large range of biological materials (0.001 mm⁻¹ ≤ μ_a ≤ 0.1 mm⁻¹, 0.5 mm⁻¹ ≤ μ_s^′≤ 4.0 mm⁻¹, and 0.1 mm ≤ d ≤ 4.0 mm) were used for reflectance comparison between the diffusion model and MC simulation and sensitivity coefficient analyses. One optical parameter was varied with five different values, while the other four parameters were held constant. For the second group, 35 samples with varying optical properties (0.01 mm⁻¹ ≤ μ_a1 ≤ 0.05 mm⁻¹, 0.45 mm⁻¹ ≤ μ_s1^′≤ 3.5 mm⁻¹, 0.005 mm⁻¹ ≤ μ_a2 ≤ 0.036 mm⁻¹, and 0.35 mm⁻¹ ≤ μ_s2^′≤ 3.0 mm⁻¹) were used for determining the parameter estimation efficacy and accuracy of the stepwise method. For each combination, five thicknesses for the first layer (i.e., 0.25 mm, 0.5 mm, 1.0 mm, 2.0 mm, and 3.0 mm), were investigated, making up a total of 175 samples. For the last, third group, five samples were used for determining the constraining conditions for the first-layer thickness, as shown in Table 1. For each combination, 16 thicknesses for the first layer, ranging from 0.1 mm to 6.0 mm with increments of 0.1 mm and 0.5 mm for 0.1-0.5 mm and 0.5-6.0 mm, respectively, were investigated, making up a total of 80 samples. These values of optical properties were selected based on the published data for fruit skin and flesh [26,28,29].

Table 1. Five two-layer simulation samples with different combinations of μ_a and μ_s^′ and their corresponding mean free paths of the first layer (mfp₁^′) for determining the constraining conditions for the first-layer thickness

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2.4 Inverse algorithm for estimating optical properties

To extract optical properties from the spatial-frequency domain diffuse reflectance of two-layer turbid media, a suitable inverse algorithm has to be developed. A nonlinear least-squares fitting method was used to minimize the sum-of-squares of the differences between the true reflectance and predicted reflectance with estimated parameters using Eq. (3) or Eq. (12). A subspace trust-region method based on the interior-reflective Newton approach was used to achieve the algorithm optimization [30,31], which is defined by minimizing a quadratic function subject to an ellipsoidal constraint. Moreover, it is globally and locally convergent under reasonable assumptions. The optical property estimation procedure was implemented using the Toolbox function ‘lsqcurvefit’ in Matlab 8.4 (The MathWorks, Inc., Natick, MA, USA).

A flow chart of the stepwise method for optical property estimations from the reflectance generated by MC simulation is shown in Fig. 1. To implement this method, the one-layer model in Eq. (3) was first used for curve fitting to obtain the initial values of μ_a1 and μ_s1^′. The sample-based calibration was conducted to decrease inherent discrepancies of the reflectance between the diffusion model and MC simulation, which would improve accuracy for optical property estimations. In this study, a calibration method, based on calibration samples with different values of μ_s1^′, was proposed by using Eq. (15)

 

Fig. 1 Flow chart of the stepwise method for inversely estimating absorption (μ_a) and reduced scattering (μ_s^′) coefficients of two-layer turbid media from Monte Carlo-generated reflectance, where MC and DM stand for Monte Carlo and diffusion model, respectively.

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Table 2. Selection of calibration samples based on the initial estimation of reduced scattering coefficient (μ_s1^′) for the first layer of two-layer samples for calibrating Monte Carlo-generated or experimentally measured spatial-frequency domain reflectance

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The first-layer thickness, as an important optical parameter, is closely related with the accuracy of optical property estimation. In principle, μ_a1 and μ_s1^′ can be estimated more accurately when the layer is thick, because detected photons would carry more information about the first layer. However, a thick first layer could hinder light interaction with the second-layer tissues, thus resulting in less accurate estimation for μ_a2 and μ_s2^′. It is thus important to determine the constraining conditions of the first-layer thickness for the stepwise method. To address this problem, 16 first-layer thicknesses from 0.1 mm to 6.0 mm, as mentioned above, were investigated. Different frequency ranges (start and end spatial frequencies and frequency resolution) were used for parameter estimation, which are summarized in Table 3. In this study, spatial frequencies for optical property estimation of the first layer were expressed in terms of the reciprocal of mean free path of the first layer (1/mfp₁^′), while those for the second layer were varied with absolute values (mm⁻¹). Here, the unit of mm⁻¹ means the number of fringes per millimeter. The scale of 1/mfp^′ was not used for the second layer because the mfp^′ for the two-layer model has not been defined [22]. The frequencies for the second layer were generally smaller than those for the first layer due to the fact that increasing the frequency leads to shallower light penetration [32]. All these frequency ranges were selected based on a preliminary simulation study. Finally, the constraining conditions for the first-layer thickness were determined based on the absolute error contour maps of μ_a and μ_s^′, beyond which the accuracy of optical property estimations is not acceptable.

Table 3. Summary of different frequency ranges for optical property estimations

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3. Results and discussion

3.1 Sensitivity coefficient analyses

Figure 2 shows the reflectance predicted by Eq. (12) and the scaled sensitivity coefficients of μ_a1, μ_s1^′, μ_a2 and μ_s2^′, which were calculated as a function of the spatial frequency. For the two combinations of optical properties, the sensitivity coefficients of μ_s^′ were much larger than those of μ_a due to the fact that the values of μ_s^′ were much larger than that of μ_a. This implied that better estimations of μ_s^′ could be achieved, compared to that for μ_a. Since the shapes of the sensitivity coefficients of μ_a1, μ_s1^′, μ_a2 and μ_s2^′ were different, the four parameters were uncorrelated, which is desirable for estimating the optical properties in real applications. Moreover, in general, the sensitivity coefficients of μ_a1, μ_a2 and μ_s2^′ were closer to the reflectance at smaller spatial frequencies, while the sensitivity coefficients of μ_s1^′ showed almost an opposite pattern. This may be explained by the fact that signals at larger spatial frequencies depended strongly on μ_s1^′, while those at smaller spatial frequencies exhibited large dependence on μ_a1, μ_a2 and μ_s2^′. This finding can be useful for optimizing spatial frequency region for estimating the optical properties of two-layer turbid media.

 

Fig. 2 Sensitivity coefficients as a function of spatial frequency for the two combinations of optical properties: (a) μ_a1/μ_a2 = 0.5 and μ_s1^′/μ_s2^′ = 2.0 (0.01, 2, 0.02, 1, 2); and (b) μ_a1/μ_a2 = 1.5 and μ_s1^′/μ_s2^′ = 0.7 (0.03, 2, 0.02, 3, 2). The values in the brackets are μ_a1, μ_s1^′, μ_a2, μ_s2^′ and d with the unit of mm⁻¹ for optical properties and mm for the first-layer thickness.

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The sensitivity coefficients of μ_a1, μ_s1^′, μ_a2 and μ_s2^′ for the four thicknesses of the first layer with the same optical properties are presented in Fig. 3. It was observed that the magnitudes of the sensitivity coefficients of μ_a1 and μ_s1^′ were almost held constant, while those of μ_a2 and μ_s2^′ decreased, with the increased first-layer thickness, which indicated that a larger thickness of the first layer would cause more difficulties for accurately estimating optical properties of the second layer. That is because a smaller number of photons, which propagate through the second layer, would be reemitted and detected when the first layer is of relatively large thickness. Another observation from Fig. 3 was that μ_s2^′ could be estimated more accurately than μ_a1 when the first-layer thickness was relatively small (i.e., 0.5 mm and 1 mm in Figs. 3(a) and 3(b)). All these findings demonstrated that the first-layer thickness is rather important for parameter estimation of two-layer turbid media, and thus deserves further investigation in this study.

 

Fig. 3 Sensitivity coefficients as a function of spatial frequency for μ_a1 = 0.03 mm⁻¹, μ_s1^′ = 2 mm⁻¹, μ_a2 = 0.02 mm⁻¹, μ_s2^′ = 1 mm⁻¹, and the first-layer thickness: (a) d = 0.5 mm, (b) d = 1.0 mm, (c) d = 2.0 mm, and (d) d = 4.0 mm.

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3.2 Reflectance comparison between diffusion model and MC simulation

Figure 4(a) shows reflectance versus spatial frequency, predicted by the two-layer model and MC simulation, on a logarithm scale for four values of mfp₁^′. The parameters μ_a1, μ_a2, μ_s2^′ and d were held constant with the values of 0.03 mm⁻¹, 0.02 mm⁻¹, 1 mm⁻¹ and 2 mm, respectively, while μ_s1^′ was varied from 0.5 mm⁻¹ to 4 mm⁻¹. It was observed that the reflectance decreased with spatial frequency. A larger mfp₁^′ generated smaller reflectance because photons had propagated for much larger distances and thus had a higher possibility of absorption in this case. Comparison of the results between the diffusion model and MC simulation revealed that the diffusion model overestimated the reflectance when the frequency was approximately smaller than 0.5/mfp₁^′ and underestimated the reflectance when the frequency was larger than 0.5/mfp₁^′. Similar findings were also reported by Cuccia et al. (2009) [13] and Hu et al. (2017) [25] when they compared the reflectance generated by the one-layer model and MC simulation. The intrinsic difference between the diffusion model and MC simulation (i.e., an approximation solution to the RTE and a numerical method) and the exponentially decreasing depth-dependent light intensity in the derivation of the two-layer model may both have introduced part of the discrepancy. Smaller discrepancies were observed at lower frequencies. Further analyses showed that the average relative errors of the reflectance generated by the diffusion model were 12.2%, 12.5%, 11.7% and 10.9% for the mfp₁^′ values of 2.0 mm, 1.0 mm, 0.5 mm and 0.25 mm, respectively, compared with that from MC simulation, when the frequencies were smaller than 0.5/mfp₁^′. After the calibration, the reflectance generated by MC simulation was much closer to that of the diffusion model, with the average relative errors of 2.3%, 3.6%, 0.6% and 5.7% for the mfp₁^′ values of 2.0 mm, 1.0 mm, 0.5 mm and 0.25 mm, respectively. In real applications, a similar calibration procedure can be performed to calibrate these discrepancies by replacing$R_{MC\_R}\left(f_x\right)$ with that of reference samples with known optical properties and $R_{MC\_O}\left(f_x\right)$ with the measured reflectance of actual samples [13]. Fewer calibration samples would be needed for the calibration because the μ_s1^′ values for a given type of tissue, like apple, are usually in a relatively narrow range.

 

Fig. 4 Comparison of spatial-frequency domain reflectance obtained from (a) the two-layer model, MC simulation and calibrated MC simulation for different mfp₁^′ values, and (b) the diffusion model and MC simulation for one-layer and two-layer turbid media (μ_a1 = 0.03 mm⁻¹, μ_s1^′ = 3 mm⁻¹, μ_a2 = 0.02 mm⁻¹, μ_s2^′ = 1 mm⁻¹ and d = 2 mm). mfp₁^′ denotes mean free path of the first-layer tissue.

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Figure 4(b) shows reflectance comparisons between the diffusion model and MC simulation for one-layer and two-layer turbid media. It was observed that, for both diffusion model and MC simulation, reflectance for one-layer and two-layer turbid media deviated from each other when the frequency was relatively small and matched better with the increasing frequency. This is due to the fact that light penetration depth is largely related with spatial frequency [32], and shallower tissue contributes more to the reflectance under larger frequency. This phenomenon demonstrates that it is feasible to use the one-layer model for estimating optical properties of the first layer with proper selection of frequencies.

3.3 Optical parameters estimation from calibrated MC-generated reflectance

Table 4 summarizes the mean absolute errors for estimating μ_a1 and μ_s1^′ of all 35 two-layer samples, by using the stepwise and one-step methods from the MC-generated reflectance before and after the calibration. It was observed that the estimations of both μ_a1 and μ_s1^′ using the calibrated reflectance were much better than that using the uncalibrated reflectance for both the stepwise and one-step methods; the relative improvements for μ_s1^′ were more pronounced than for μ_a1. These results clearly indicated that the sample-based calibration method was effective in improving estimation accuracy. For instance, when the first-layer thickness was 2.0 mm, the mean errors for estimating μ_a1 and μ_s1^′ using the calibrated reflectance for the stepwise method and the one-step method were 11.9%, 2.1% and 38.3%, 1.4%, respectively, which represented improvements of 26.5%, 83.3% and 51.0%, 89.4%. Overall, the relative estimation accuracy for both μ_a1 and μ_s1^′ by the stepwise and one-step methods increased with the increasing first layer thickness. Regardless of the data calibration, the stepwise method resulted in consistently better estimations of μ_a1 than the one-step method for all the five groups of samples, while opposite results for μ_s1^′ were obtained. However, it should be mentioned that in the latter case, the relative estimation accuracies for μ_s1^′ by both methods after the data calibration were mostly within the acceptable level (15% or lower). There were two cases for the one-step method, where the relative errors for estimating μ_a1 became larger when using the calibrated reflectance, which might be due to the fact that the one-step method cannot estimate μ_a1 accurately when the first layer is too thin (e.g., 0.25 mm and 0.5 mm). A similar pattern of variation, but with larger errors for estimating μ_a2 and μ_s2^′, was observed for the second layer.

Table 4. Mean absolute errors for estimating μ_a1 and μ_s1^′ of all 35 samples with different first-layer thicknesses by using stepwise and one-step methods from Monte Carlo-generated reflectance before and after the calibration

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To compare the parameter estimation efficacy and accuracy more intuitively between the stepwise method and the one-step method, the mean errors for estimating μ_a1, μ_s1^′, μ_a2 and μ_s2^′ using the calibrated reflectance are presented in Fig. 5, with the first-layer thickness of 2.0 mm. The relative errors for estimating μ_a1 and μ_s1^′ using the stepwise method and the one-step method are 11.9%, 2.1% and 38.3%, 1.4%, respectively, while those for the second layer are 31.9%, 23.5% and 67.4% and 31.8%. Relatively smaller errors were obtained for the optical properties of the first layer, compared with those in the second layer, and the errors for estimating the reduced scattering coefficients are smaller than those for the absorption coefficients, which are consistent with the sensitivity coefficient analyses in Section 3.1. Comparison of mean errors for estimating μ_a1, μ_s1^′, μ_a2 and μ_s2^′ demonstrated that the parameter estimation efficacy and accuracy of the stepwise method are much better than that for the one-step method, which is especially true for estimating the absorption coefficient. This might be because that the stepwise method reduced the number of free variables from four to two at every step, which improved overall estimation accuracy, compared to the one-step method. Moreover, the stepwise method has the advantage that it does not require a priori knowledge of the first-layer thickness for estimating μ_a1 and μ_s1^′. The estimation accuracies were also comparable or superior to the results reported by other researchers. Cen et al. [20] reported that the estimation accuracies for μ_a1 and μ_s1^′ using the spatially resolved technique were 23.0% and 18.4%, respectively, while the technique did not give acceptable estimations for the second layer. Weber et al. [16] utilized the spatial-frequency domain technique for estimating the optical properties of two-layer media, and found that the average accuracies for estimating μ_a1 and μ_s1^′ were around 2% and 17%. Even assuming known values of μ_a1, μ_s1^′ and d, the estimation accuracy for μ_a2 was around 25% and it did not give acceptable results for estimating μ_s2^′. It should be noted that the frequency range for the parameter estimation reported in this section is fixed, and frequency optimization, including start and end frequencies, can improve estimation accuracy, which will be studied in further research.

 

Fig. 5 Relative errors (absolute values) for estimating the optical properties of all 35 two-layer samples with the first-layer thickness of 2.0 mm, by using the stepwise and the one-step methods from the calibrated Monte Carlo-generated reflectance.

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3.4 Constraining conditions for first-layer thickness on optical property estimation

Figure 6 illustrates the effect of the first-layer thickness on reflectance prediction from the two-layer model. Samples ‘Homogeneous-1’ and ‘Homogeneous-2’ had the same optical properties as that of the first layer and the second layer, respectively. Generally, the reflectance closely matched that of sample ‘Homogeneous-1’ when the first-layer thickness was relatively large and moved closer toward that of sample ‘Homogeneous-2’ with the decreased thickness of the first layer. A thin first layer offers more possibilities of photon interactions with the second-layer tissues. Consequently, μ_a2 and μ_s2^′ can be estimated more accurately; but at the same time, it would become more challenging to estimate μ_a1 and μ_s1^′ because the reflectance carries less information about the first layer. It is thus important to determine the constraining conditions for the first-layer thickness (i.e., the minimum and maximum values) in order to have accurate estimation of the optical properties of two-layer turbid media.

 

Fig. 6 Reflectance predicted by the two-layer model versus spatial frequency for different thicknesses of the first layer, compared with two semi-infinite homogeneous samples, denoted as ‘Homogeneous-1’ and ‘Homogeneous-2’, which had the same optical properties as that of the first layer and the second layer, respectively. μ_a1 = 0.03 mm⁻¹, μ_s1^′ = 2 mm⁻¹, μ_a2 = 0.02 mm⁻¹, μ_s2^′ = 1 mm⁻¹ and the first-layer thickness was varied from 0.1 mm to 4.0 mm.

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Absolute error contour maps are plotted in Figs. 7 and 8 for investigating the constraining conditions for the first-layer thickness on parameter estimations. The stepwise method of first using the one-layer model and then the two-layer model was used for estimating optical properties of the first layer and the second layer, respectively. The acceptable error bound for optical property estimations using spatially resolved technique was reported to be around 10% [8,20]. However, there were still discrepancies in the spatial-frequency domain reflectance generated by the diffusion model and MC simulation, even after the calibration, which would, in turn, result in relatively large estimation errors. Moreover, optical property estimations using the diffusion model for two-layer tissues are more challenging than one-layer tissues. Therefore, the acceptable error bound for optical property estimations was expanded to 20% in this study. Figure 7 shows the absolute error contour maps for estimating μ_a1 and μ_s1^′ of a two-layer sample. The acceptable regions for μ_s1^′ were much larger than those for μ_a1, indicating that it is easier to estimate μ_s1^′ than μ_a1. As the first-layer thickness increased from 0.1 mm to 1.0 mm, the acceptable regions for μ_a1 and μ_s1^′ were expanded and the overall errors of estimation decreased. Moreover, the absolute errors for estimating μ_s1^′ were all acceptable except for the case shown in Fig. 7(a2), while the acceptable region for μ_a1 varied with the start frequency, and it expanded and also shifted to lower frequencies as the thickness of the first layer increased. After comparison and analysis of the absolute error contour maps for all five samples listed in Table 1, it was found that the minimum thickness of the first layer could be as small as 0.1 mm for four samples (1, 2, 3 and 5) and 0.2 mm for sample No. 4. These results indicate that the stepwise method, in principle, can offer acceptable estimations for the μ_a1 and μ_s1^′ of two-layer turbid media, when the first-layer thickness is no smaller than 0.2 mm, which applies to many raw food products such as fruits whose skin is usually greater than 0.2 mm.

 

Fig. 7 Absolute error contour maps for estimating μ_a1 (left panel) and μ_s1^′ (right panel) of a two-layer sample (μ_a1 = 0.04 mm⁻¹, μ_s1^′ = 1.6 mm⁻¹, μ_a2 = 0.015 mm⁻¹ and μ_s2^′ = 1.25 mm⁻¹) when using different start and end spatial frequencies. The first-layer thicknesses are 0.1 mm, 0.2 mm, 0.5 mm and 1.0 mm for (a), (b), (c) and (d), respectively. The one-layer model was used here. Note that the absolute errors of μ_a1 and μ_s1^′ larger than 60% and 30% were clipped to be 60% and 30% for better visual effect.

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Fig. 8 Absolute error contour maps for estimating μ_a2 (left panel) and μ_s2^′ (right panel) of a two-layer sample (μ_a1 = 0.04 mm⁻¹, μ_s1^′ = 1.6 mm⁻¹, μ_a2 = 0.015 mm⁻¹ and μ_s2^′ = 1.25 mm⁻¹) when using different start and end spatial frequencies. The first-layer thicknesses are 1.0 mm, 1.5 mm, 2.0 mm and 2.5 mm for (a), (b), (c) and (d), respectively. The two-layer model was used here assuming that μ_a1, μ_s1^′ and the first-layer thickness were known. Note that the absolute errors of μ_a2 and μ_s2^′ larger than 60% and 30% were clipped to be 60% and 30% for better visual effect.

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Figure 8 shows the absolute error contour maps for estimating μ_a2 and μ_s2^′ of a two-layer sample. When the first-layer thickness was 1.0 mm (Fig. 8(a)), the maximum absolute errors for μ_a2 and μ_s2^′ for the given ranges of start and end frequencies were smaller than 30% and 10%, respectively. As the first-layer thickness increased, the overall absolute error for both μ_a2 and μ_s2^′ also increased, because fewer photons would interact with the second-layer tissues and the measured reflectance carried less useful information about the second-layer tissues. When the first-layer thickness increased to 2.5 mm (Fig. 8(d)), there was no region for which the absolute errors of μ_s2^′ were smaller than 20%, which indicated that the maximum first-layer thickness should be smaller than 2.5 mm. In view of the effect of the optical properties and thickness of the first layer on estimating μ_a2 and μ_s2^′, and after comparison of the absolute error contour maps for all five samples in Table 1, the maximum first- layer thickness was recommended to be no greater than 2.0 mm. These findings suggested that the stepwise method could obtain acceptable estimations for the μ_a2 and μ_s2^′ of two-layer turbid media when the first-layer thickness is no greater than 2.0 mm. It should be noted that these results were carried out using reflectance generated from calibrated MC simulations under ideal situations. However, for real applications, low signal-to-noise ratio and other experimental uncertainties may result in errors larger than the levels that were found in the current study.

4. Conclusion

This research proposed a stepwise method for estimating the absorption and reduced scattering coefficients of two-layer turbid media using spatial-frequency domain imaging technique from MC-generated reflectance. The method was carried out by first using the one-layer model for estimating μ_a1 and μ_s1^′, followed by the two-layer model for μ_a2 and μ_s2^′. Sensitivity coefficient analyses of the two-layer model indicated that the optical properties of the first layer could be estimated more accurately than for the second layer, and estimation accuracy for each layer was dependent on the first-layer thickness. Reflectance for one-layer and two-layer turbid media deviated from each other when the frequency was relatively small and matched better with the increasing frequency, demonstrating that it is feasible to use the one-layer model for estimating μ_a1 and μ_s1^′ with proper selection of frequencies. Discrepancies of the reflectance between the diffusion model and Monte Carlo simulation were partly calibrated by using a sample-based calibration method, and the parameter estimation accuracy was greatly improved after the calibration, which was especially significant for the absorption coefficients of both layers, when compared with the results without the calibration. Results showed that the efficacy and accuracy of the stepwise method were much better than that for the one-step method. Analyses of the absolute error contour maps for μ_a1 and μ_s1^′ showed that when the optical properties of two layers are approximately within the range of 0.005 mm⁻¹ ≤ μ_a ≤ 0.04 mm⁻¹ and 0.69 mm⁻¹ ≤ μ_s^′≤ 2.2 mm⁻¹, the first layer could be accurately estimated, even when its thickness was as small as 0.2 mm. To achieve acceptable estimations for the optical properties of the second layer, the first-layer thickness should be no greater than 2.0 mm.