## Abstract

This paper reports on the optimization and assessment of a hyperspectral imaging-based spatially-resolved system for determination of the optical properties of biological materials over the wavelengths of 500-1,000 nm. Twelve model samples covering a wide range of absorption and reduced scattering coefficients were created to validate the hyperspectral imaging system, and their true values of absorption and reduced scattering coefficients were determined and then cross-validated using three commonly used methods (i.e., transmittance, integrating sphere, and empirical equation). Light beam and source-detector distance were optimized through Monte Carlo simulations and experiments for the model samples. The optimal light beam should be of Gaussian type with the diameter of less than 1 mm, and the optimal minimum and maximum source-detector distance should be 1.5 mm and 10-20 mean free paths, respectively. The optimized hyperspectral imaging-based spatially-resolved system achieved good estimation of the optical parameters.

©2010 Optical Society of America

## 1. Introduction

The study of light propagation in turbid biological materials has attracted considerable attention in the biomedical field because of its potential application in disease diagnosis [1, 2], photodynamic therapy [3], oximetry [4], and glucose monitoring [5]. Likewise, there has been increased interest recently in optical characterization of agricultural and food products for nondestructive quality evaluation [6–8]. The optical properties of biological materials like fruits and vegetables are characterized by the absorption coefficient (${\mu}_{a}$) and the reduced scattering coefficient (${\mu}_{s}\text{'}$) [9]. Quantification of these optical properties can provide rich information about the chemical composition and physical structures of biological materials, which can lead to the development of effective sensing techniques for evaluating the physiological state and quality attributes of agricultural and food products.

Measurement of the optical properties of biological materials can be achieved using either direct or indirect methods. Direct methods (e.g., total transmittance and reflectance) are easy to carry out, but they are destructive or invasive. In contrast, indirect methods can be performed on intact samples nondestructively but need sophisticated instrumentation and complex mathematical models. Therefore, recent research has been focused on indirect methods, which include time-resolved (TR) [10], frequency-domain (FD) [11], and spatially-resolved (SR) [12]. Based on radiation transfer theory, these indirect methods allow noninvasive or nondestructive measurement of the optical properties from intact biological samples. While TR and FD techniques have been extensively researched in the biomedical area, they may not be suitable for food and agriculture because of expensive instrumentation, slow speed in measurement, and the requirement of good contact between the sample and detector. In comparison, spatially-resolved technique is less expensive in instrumentation, and it is potentially easier to use and faster in measurement. The technique is, thus, more viable for food and agriculture applications. Spatially-resolved technique is generally implemented in two sensing configurations; one is based on multiple fiber arrays connected to spectrometers and the other uses non-contact reflectance imagery [13, 14]. The former needs good contact between the detecting probe and the measured medium, while the second sensing configuration can achieve noncontact measurement, which is advantageous for food and agricultural products because of the safety and sanitation requirements. However, most research on non-contact reflectance imagery mode can only provide optical property information at single or several wavelengths.

Recently, our laboratory developed a hyperspectral imaging-based spatially-resolved technique to measure the optical properties of biological materials over a broad spectral range (i.e., 500-1,000 nm) simultaneously [15, 16]. While the technique is promising for optical characterization of food and agricultural products, several critical issues need to be resolved. First, a reference method is needed to provide true values of the optical properties of samples, against which the hyperspectral imaging system can be evaluated and validated. This would require model samples with known optical properties for a specific range of values. Several reference methods including transmittance, integrating sphere and empirical equation have been reported [17, 18]; however, they have not been accepted as ‘standard’. Therefore, it is necessary to test and cross-validate these methods in order to provide accurate and reliable references for the optical system. Second, the hyperspectral imaging system needs to be such designed that it fully meets the requirements of diffusion theory that governs light transfer in turbid biological materials. The hyperspectral imaging-based spatially-resolved system uses a continuous-wave point light beam to illuminate the sample. The shape and size of the beam can directly affect measurement accuracies. It is therefore important to examine and optimize the light beam. Third, an appropriate source-detector distance is critical for accurate estimate of the optical properties. Additionally, optimization of the inverse algorithm for the diffusion theory model is equally important in extracting the optical parameters from the reflectance profiles, which has been described in our previous paper [19].

This research was therefore aimed at addressing key technical issues in the development of hyperspectral imaging-based spatially-resolved technique, so that it can provide accurate and reliable measurement of the optical properties of food and agricultural products. The specific objectives were to:

- 1) Evaluate and cross-validate three reference methods (i.e., transmittance, integrating sphere, and empirical equation) for determining the true values of optical properties of model samples, so as to establish a standard procedure of evaluating the hyperspectral imaging system;
- 2) Optimize the design of the light beam and source-detector distance in the hyperspectral imaging system, using Monte Carlo simulation and experiments for model samples, to improve the measurement of the absorption and reduced scattering coefficients;
- 3) Assess the performance of the hyperspectral imaging system in terms of accuracy, precision/reproducibility, and sensitivity.

## 2. Materials and methods

#### 2.1 Diffusion model

Light propagation in biological materials is governed by the light transport equation. Exact solutions to the equation are found only under a few very restricted conditions [20]. For most biological materials in which scattering is dominant (${\mu}_{s}\text{'}\gg {\mu}_{a}$), diffusion approximation is valid [21]. For a homogeneous semi-infinite turbid medium illuminated by an infinitely small continuous-wave point light source, the diffuse reflectance $R(r)$ at its surface can be expressed as the sum of isotropic fluence rate and the flux [22], and it is a function of the source-detector distance *r* and the two optical parameters ${\mu}_{a}$ and ${\mu}_{s}\text{'}$, which is given below

*θ*relative to the normal to the boundary, and

*Ω*is the solid angle. For $n=1.33$, ${C}_{1}$ and ${C}_{2}$ are equal to 0.1320 and 0.3359, respectively. Once ${R}_{eff}$ is determined, the shape of the reflectance profile is uniquely determined by ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ based on Eq. (1). Conversely, if the reflectance profile over the surface of the investigated material from a point light source is known, ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ could be determined through Eq. (1) by an inverse algorithm.

The diffusion model in Eq. (1) is the theoretical basis of this research. A nonlinear least squares curve fitting inverse algorithm was developed to extract the estimates of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ from the spatially-resolved reflectance profiles [19]. A simple wavelength-dependent function (${\mu}_{s}\text{'}=a{\lambda}^{-b}$, where *a* and *b* are parameters for the power series model) was used during the curve fitting to obtain a smooth spectrum of ${\mu}_{s}\text{'}$ [16].

#### 2.2 Hyperspectral imaging-based spatially-resolved technique

In spatially-resolved method, a small continuous-wave light beam perpendicularly illuminates the sample’s surface, and the remitted light is measured at different distances from the light source to obtain a spatially-resolved reflectance profile as shown in Fig. 1(a) .

A hyperspectral imaging system in line scan mode used for acquiring spatially-resolved diffuse reflectance profiles from a sample is schematically shown in Fig. 1(b). The system mainly consisted of a high performance 512 × 512 pixel camera with a back-thinned illuminated CCD detector (Model C4880-21-24A, Hamamatsu Photonics Systems, Bridgewater, NJ, USA) covering the wavelengths of 200-1,100 nm, an imaging spectrograph (ImSpector V10, Spectral Imaging Ltd., Oulu, Finland) covering the wavelength region of 400-1,000 nm, a zoom lens with the focal length of 11-110 (10 × ) mm (Zoom 7000, Navitar Inc., Rochester, NY, USA), a quartz tungsten halogen light source (Oriel Instruments, Stratford, CT, USA) with a feedback controller, an optical fiber coupled with a focusing lens for delivering a small beam to the sample, and a computer installed with a frame grabber board for controlling the camera and acquiring images. Hyperspectral imaging technique, combining imaging and spectroscopy methods, acquires both spectral and spatial information simultaneously, and it is, therefore, ideally suitable for measuring spatially-resolved diffuse reflectance profiles for a broad wavelength region. To improve the repeatability of measurements, 10 line scans were taken from each sample for every 1 mm horizontal displacement over a range of 9 mm as the sample was moving at the predetermined velocity during the image acquisition [Fig. 1(c)]. The sample movement and positioning were controlled by the two motorized stages, and the distance between the sample and the lens is 243 mm. Spectral and spatial calibrations and nonuniform instrument response corrections for the hyperspectral imaging system were performed by following the procedure described in literature [16, 24].

Figure 2(a) shows a typical hyperspectral reflectance image for a liquid model sample made up of Intralipid fat emulsion as scatterers and blue dye as absorbers (see more details in Section 2.3). Each horizontal line taken from the image represents one spatially-resolved reflectance profile at a specific wavelength, as shown in Fig. 2(b). Hence the reflectance image which had a calculated spectral resolution of 4.55 nm by the spectral calibration, in effect, consisted of more than 100 spatially-resolved reflectance profiles for the wavelengths of 500-1,000 nm. Since the spatially-resolved reflectance profiles were symmetric to the light incident point, the two sides were averaged in the extraction of optical properties. Each spatially-resolved reflectance profile was then fitted by the diffusion model with the inverse algorithm, from which the spectra of absorption and reduced scattering coefficients at 500-1,000 nm were obtained.

#### 2.3 Model samples preparation and reference methods

### 2.3.1 Model samples

To optimize and evaluate the hyperspectral imaging system, 12 liquid model samples were created with two different dyes (Direct Blue 71 and Naphthol Green B, Sigma-Aldrich Inc., St. Louis, MO, USA) and their mixture as absorbers and fat emulsion as scatters (Intralipid-20%, Sigma-Aldrich Inc., St. Louis, MO, USA) for different concentration levels. The three dye absorbers were used in the model samples to allow fine adjustment of the absorption coefficient and comprehensive investigation of the accuracy for estimating ${\mu}_{a}$ over the entire spectral region of 500-1,000 nm because each dye had high absorption for a specific wavelength range. For each absorber, aqueous absorbing stock solution with the concentration of 1 mg/mL was first prepared, and then it was added to 250 mL distilled water followed by adding different volumes of Intralipid (10-50 mL) into the absorbing solutions to form the model samples with specified values of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$. The ranges of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ values for these model samples were 0.0-0.934 ${\text{cm}}^{\text{-1}}$ and 7.0-39.9 cm^{−1}, respectively, at the wavelength range of 500-1,000 nm.

The model samples were first measured using the hyperspectral imaging system [Fig. 1(b)]. Thereafter, confocal laser scanning microscopy (CLSM) images of these model samples were acquired for gaining an insight of the distribution and interaction of the dye and Intralipid particles in the solution. This is because when the dye and Intralipid were mixed, the two components could interact with each other, which might in turn affect the stability of its mixture and thus of the optical properties [25]. Finally, the true optical properties of these model samples were measured using the reference methods discussed below.

### 2.3.2 Reference methods

The true values of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ of the model samples are needed to validate the hyperspectral imaging system. However, in a real situation, it is impossible to obtain true ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ of the model samples. Therefore, nominal true values (or reference values) were derived as reliable estimates of the true values by using three commonly used methods, and evaluation and cross-validation of these methods were performed.

A collimated transmittance method was first employed for the standard absorption measurement using a miniature fiber optic spectrometer (USB4000-VIS-NIR, Ocean Optics, Dunedin, FL, USA) with a 10-mm pathlength quartz cuvette sitting in a cuvette holder (Fig. 3 ). The true values of ${\mu}_{a}$ at the wavelengths of 500-1,000 nm were calculated by the Beer-Lambert law

where${I}_{s}$ is the sample intensity,${I}_{d}$ is the dark intensity, and ${I}_{r}$is the reference intensity (empty cuvette). The accuracy of the transmittance measurement was experimentally quantified by testing In-Spec® certified UV and visible standards (Standard 0.10, 0.5, 0.8, and Background solution, GFS Chemicals, Inc., Columbus, OH, USA) with known ${\mu}_{a}$ at selected wavelengths provided by the manufacturer.For each absorber, eight concentration levels of pure absorbing solutions with the absorption coefficient varied from 0.0 to 1.800 ${\text{cm}}^{\text{-1}}$ for the wavelengths of 500-1,000 nm were made for transmittance measurements, which covered the range of ${\mu}_{a}$ for the model samples. The linear relationship between the absorption coefficient and the concentration of each absorber was established at each wavelength, and the absorption coefficient was then derived based on the concentration of the absorber in the model samples.

An integrating sphere for collecting the total transmittance and reflectance was used to obtain true values of the absorption and reduced scattering coefficients of the model samples (Fig. 4
). A 152.4 mm diameter integrating sphere (RT-060-SF, Labsphere, Inc., North Sutton, NH, USA) with a 12.7 mm diameter detector port and a 25.4 mm sample port was used. The sphere was coated with Spectraflect^{®} diffuse reflectance materials with 98% reflectance at 500-1,000 nm. A quartz tungsten halogen light source was used and the light received from the sample port of the integrating sphere by a 400 µm diameter fiber was delivered to the miniature fiber optic spectrometer, the same spectrometer used in the collimated transmittance measurement (Fig. 3). Each model sample held in a 2 mm pathlength cuvette was placed in the sample holder. For measuring total diffuse reflectance [Fig. 4(a)], the sample was placed at the sample port and the light beam was arranged at the beam entrance port. For total transmittance measurements [Fig. 4(b)], the sample was placed at the sample port with other ports covered by the standard masks, and all light transmitted through the sample was collected by the detector. The same beam size was used for both reflectance and transmittance measurements to ensure measurement accuracy.

After completion of the total transmittance and reflectance measurements, an inverse adding-doubling algorithm developed by Prahl et al. [17] was used to calculate ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ values of the model samples based on the total reflectance and transmittance. Prahl et al. [17] reported that the accuracy of the integrating sphere with the adding-doubling algorithm for determining the optical properties (${\mu}_{a}$ & ${\mu}_{s}\text{'}$) was 2-3% for most reflectance and transmittance values. However, a 1% variation in the reflectance and transmittance values would result in errors of 0.4% for ${\mu}_{s}\text{'}$ and 17% for ${\mu}_{a}$. This suggests that the integrating sphere method could be problematic in estimating ${\mu}_{a}$.

For estimating the ${\mu}_{s}\text{'}$ of Intralipid scatterers, an empirical equation derived by van Staveren et al. [18] is widely used, which has been experimentally proved to have accuracies better than 6% in the wavelength range of 400-1,100 nm. Therefore, ${\mu}_{s}\text{'}$ obtained from the integrating sphere were also compared with those from the empirical equation given below

where ${\mu}_{s}\text{'}$ is in ${\text{cm}}^{\text{-1}}$,*λ*is the wavelength in $\text{\mu m}$, and $C\%$ is the percent Intralipid concentration. For easy comparison of the results obtained from the hyperspectral imaging system and the three reference methods, the absorption and reduced scattering coefficients are denoted as ${\mu}_{a}{}^{\Delta}$ & ${\mu}_{s}\text{'}\text{\hspace{0.05em}}{\text{\hspace{0.05em}}}^{\Delta}$ for the hyperspectral imaging system, ${\mu}_{a}{}^{\ast}$ & ${\mu}_{s}{\text{'}}^{\text{\hspace{0.05em} \hspace{0.05em}}\ast}$ for the integrating sphere, ${\mu}_{a}{}^{\circ}$ for the transmittance measurement, and ${\mu}_{s}\text{'}\text{\hspace{0.05em}}{\text{\hspace{0.05em}}}^{\circ}$ for the empirical equation.

#### 2.4 Procedures for optimization and assessment of the hyperspectral imaging system

### 2.4.1 Optimization of the light beam and source-detector distance

To achieve accurate measurement of the optical properties, optimization of the hyperspectral imaging system was performed for two key factors, i.e., light beam and source-detector distance. In the experimental setup, a finite size beam was used to illuminate the sample, which deviates from the solution of the diffusion model [Eq. (1)] that was derived for an infinitely small beam. Therefore, the effect of the incident beam on the determination of the optical properties was investigated based on Monte Carlo (MC) simulation, which offers a flexible and accurate approach for quantifying the optical features of light transport that are difficult to measure experimentally.

In the simulation experiment, a semi-infinite medium was described by its refractive index (*n*), absorption coefficient (${\mu}_{a}$) and reduced scattering coefficient (${\mu}_{s}\text{'}$). The spatially-resolved reflectance generated by MC simulation for an infinitely small photon beam was first convolved for the finite size beam using the programs ‘MCML’ and ‘CONV’ developed by Wang et al. [26, 27]. Two specific beams, i.e., Gaussian and circularly flat, were investigated. Meanwhile, the error of estimating ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ caused by the finite size Gaussian beam with the diameter of $0.1\le d\le 2\phantom{\rule{.2em}{0ex}}\text{mm}$ was quantified. Six sets of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ with the ranges of $0.060\le {\mu}_{a}\le 2.000\text{\hspace{0.05em} \hspace{0.05em} \hspace{0.05em}}{\text{cm}}^{\text{-1}}$and $4.0\le {\mu}_{s}\text{'}\le 40.0\text{\hspace{0.05em} \hspace{0.05em} \hspace{0.05em}}{\text{cm}}^{\text{-1}}$, and the ratios of ${\mu}_{s}\text{'}/{\mu}_{a}=20$ and ${\mu}_{s}\text{'}/{\mu}_{a}=70$ were investigated.

In conjunction with the MC simulations, actual beam profiles for the hyperspectral imaging system were measured and characterized. This was accomplished by moving and imaging the beam for every 0.12 mm distance (determined by the width of scan line) in the direction perpendicular to the scan line. Twelve 512 × 512 pixel images were acquired with the resolution of 0.1 mm along the scan line direction, and the 3-D beam profiles were then reconstructed from these images at each wavelength, from which the beam profile type (i.e., Gaussian and flat) and beam circularity were determined. The circularity of the beam was calculated based on the roundness ($Rd$), which is given by

where*A*and

*P*is the beam area and perimeter, respectively, and $Rd=\text{1}$ for circle.

Accurate information on the source-detector distance, including minimum source-detector distance (${r}_{\mathrm{min}}$), maximum source-detector distance (${r}_{\mathrm{max}}$) and spatial resolution, is required for determining the range of the spatially-resolved reflectance profile as shown in Fig. 1(a). MC simulation experiments were first conducted to investigate the effects of the minimum and maximum source-detector distances on estimating 29 sets of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ with $0.040\le {\mu}_{a}\le 3.000{\text{cm}}^{\text{-1}}$ and $4.0\le {\mu}_{s}\text{'}\le 40.0{\text{cm}}^{\text{-1}}$. In experimental measurements, signal-to-noise ratio (SNR) should be considered when determining the optimal maximum source-detector distance since SNR varies at different radiation locations and decreases with the decreasing signal. Ten images from a model sample were acquired at the same location. The mean profile divided by the standard deviation of the CCD count at each wavelength was considered as an estimate of SNR.

Sufficient data points for the spatially-resolved reflectance profile are needed to obtain good estimates of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$. Hence the effect of spatial resolution on calculating the optical coefficients was also examined. The reflectance profile with the resolution of 0.01 mm for ${\mu}_{a}=1.000{\text{cm}}^{\text{-1}}$ and ${\mu}_{s}\text{'}=20.0{\text{cm}}^{\text{-1}}$ was firstly obtained from the diffusion model. Seven new reflectance profiles with the spatial resolution ranging from 0.07 mm to 0.25 mm were then derived from the original profile by integration of the light intensity over the area covered between the two adjacent data points divided by the corresponding spatial resolution. These seven profiles were fitted by the diffusion model to obtain the estimated ${\mu}_{a}$ and ${\mu}_{s}\text{'}$, and they were then compared with the true values of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ obtained from the reflectance profile with the resolution of 0.01 mm.

### 2.4.2 Assessment for accuracy, precision/reproducibility, and sensitivity

Assessment of the performance of the system is an important step to determine its validity, reliability, and robustness. The hyperspectral imaging system was evaluated for accuracy, precision/reproducibility, and sensitivity. The accuracy of a measurement is defined as the closeness of agreement between the measured values and the nominal true values of parameter α (${\alpha}_{meas}$&${\alpha}_{true}$) such as ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ obtained from the reference methods under optimal experimental conditions. Accuracy can be quantified by the relative error of the measurement, given in Eq. (5)

Accuracy is important for the absolute measurement, i.e., for simulating light transport and deriving the physiological information about the biological tissue.

Precision/reproducibility is a term frequently used to characterize the random error and describe the system consistency of measurement over a long time period. It quantifies how the system is self-consistent over different times, and it is particularly important for a long time experiment like fruit quality monitoring during postharvest. Repetition of the measurement on the same model samples under the same experiment conditions over a period of four days was performed. The precision of the system was evaluated by calculating the coefficient of variation (CV) with respect to the average values calculated over the entire experiment.

Sensitivity is described by the minimum detectable value under an acceptable noise level, and it determines the detection limit. Clearly, the noise level is affected by the signal intensities and detection efficiencies, and thus the sensitivity of the system depends on the amount of signal acquired at each measurement point. Sensitivity analysis is especially important in measuring biological materials with small values of absorption coefficient. The analysis was performed on a blue dye model sample which had minimal levels of the absorption coefficient for the wavelengths of 500-1,000 nm.

## 3. Results and discussion

#### 3.1 Accuracy of the reference methods for measuring optical properties

Figure 5
shows the measured absorption spectra of the three standard solutions by transmittance as well as the true absorption values provided by the manufacturer. The average error of ${\mu}_{a}$ by the transmittance measurement was 3% at 500-850 nm, and less than 5% at 850-900 nm with the minimum detectable value of 0.050 cm^{−1}. The performance of the method for measuring ${\mu}_{a}$ above 900 nm was not evaluated because the true absorption values of these standard solutions were not available. However, it was found that the water absorption at 970 nm measured by the transmittance method was 0.414 cm^{−1}, which is within 1.4% of the reported data [28]. Hence, it was concluded that the transmittance method gave accurate measurement of the absorption coefficient with errors less than 5%, and it was therefore appropriate for determining ${\mu}_{a}$ of the model samples.

Integrating sphere measurements of ${\mu}_{s}\text{'}$ for the 12 model samples were compared with the calculated values from the empirical equation. It was found that the empirical equation, on average, overestimated ${\mu}_{s}\text{'}$ measured with the integrating sphere method by 9% for the wavelengths of 500-900 nm when the Intralipid concentration ranged between 0.8% and 3.2%. However, at low Intralipid concentrations (i.e., <1.2%), the measured values of ${\mu}_{s}\text{'}$ from the integrating sphere matched well the calculated values from the empirical equation with the differences being less than 4.5% (Fig. 6 ). As the concentration of the Intralipid increased, the measured value deviated more from the calculated value. This may be because the relationship between the reduced scattering coefficient and the particle concentration at high density Intralipid solutions was no longer linear, as reported by Giusto et al. [29] and Zaccanti et al. [30]. The empirical equation was derived based on the linear relationship between ${\mu}_{s}\text{'}$ and the particle concentration. In addition, variation in batches of Intralipid might have also existed, which, in turn, could have introduced errors in calculating ${\mu}_{s}\text{'}$ values. In view of these results, it was concluded that the integrating sphere method would give more accurate measurement of ${\mu}_{s}\text{'}$ than the empirical equation for a wider range of ${\mu}_{s}\text{'}$, and it was therefore used to determine the true ${\mu}_{s}\text{'}$ of the model samples.

#### 3.2 Optimization of the system

### 3.2.1 Beam characteristics

Since Gaussian beam, like most optical beams, does not have sharp physical edges, the beam size is usually determined between the two points that contain a selected percentage of the ‘useful’ energy. In this study, the size of Gaussian beam was defined by the 1/e^{2} diameter, where the beam’s power is at 13.5% of the maximum height. Figure 7
shows the spatially-resolved reflectance profiles obtained for an infinitely small beam and flat and Gaussian beams of different sizes through Monte Carlo simulations. The Gaussian beam and the flat beam nearly gave the same results for the same beam diameter of $d=1\text{mm}$. Finite circular beams with different sizes generated reflectance similar to that produced by the infinitely small beam when the source-detector distance was larger than the beam size ($r>d$). The average errors of estimating six sets of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ from the reflectance generated by the Gaussian beam of different sizes are presented in Fig. 8(a)
. The error produced by the finite beams relative to the infinitely small beam was less than 1% for the beam size of less than 0.5 mm; it increased linearly with the beam size larger than 0.5 mm. Generally, a 1-mm light beam introduces around 5% error of estimating ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ compared to the infinitely small beam. These results suggest that the beam in the system should be less than 1 mm in size in order to control the error to within 5%. The error introduced by a finite beam was also influenced by the value of the optical properties, as shown in Fig. 8(b) and (c). The error curves are relatively flat for small values of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$, which indicates that beam size had less effect on the smaller ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ than on large ${\mu}_{a}$ and ${\mu}_{s}\text{'}$.

Simulation results for the light beam showed that it is necessary to understand beam characteristics (i.e., beam profile, shape, and size) in the experimental measurements. Figure 9 gives an example of the measured 3D beam profiles and 2D contours at the wavelengths of 650 nm and 950 nm. The beam at the visible and short near-infrared region had a good Gaussian distribution and its shape was circular with the roundness $Rd=0.986$($\approx \text{1}$). Based on the commonly accepted method for defining the size of Gaussian beam, the beam size in the current setup was 0.83 mm, which would have contributed to less than 4% error in estimating ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ based on the simulation experiments. Although a smaller beam is preferred, other factors such as light intensity and measurement repeatability also need to be considered in the optical design.

### 3.2.2 Source-detector distance

Two examples of the relative errors between the true (values selected for Monte Carlo simulations) and estimated values of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ with ${\mu}_{a}=0.290{\text{cm}}^{\text{-1}}$& ${\mu}_{s}\text{'}=20.0{\text{cm}}^{\text{-1}}$, and ${\mu}_{a}=0.430{\text{cm}}^{\text{-1}}$& ${\mu}_{s}\text{'}=30.0{\text{cm}}^{\text{-1}}$ at different ranges of the source-detector distance are shown in Fig. 10 . For each set, 23 fittings were performed with $0.2\le {r}_{\mathrm{min}}\le 4\phantom{\rule{.2em}{0ex}}\text{mm}$&${r}_{\mathrm{max}}=10\phantom{\rule{.2em}{0ex}}\text{mm}$, and $4\le {r}_{\mathrm{max}}\le 10\phantom{\rule{.2em}{0ex}}\text{mm}$& ${r}_{\mathrm{min}}=1.5\phantom{\rule{.2em}{0ex}}\text{mm}$. The estimated parameters from the reflectance data containing the smallest minimum source-detector distance (${r}_{\mathrm{min}}=0.2\phantom{\rule{.2em}{0ex}}\text{mm}$) resulted in the largest errors for both ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ [Fig. 10(a1) and (a2)]. Values of the absorption coefficients were systematically underestimated, while those of the reduced scattering coefficient were overestimated. These results may indicate the failure of the diffusion model which does not account for the nondiffusive component of the reflectance at these small source-detector distances. As the minimum source-detector distance was increased, the errors decreased, reaching minimum for ${r}_{\mathrm{min}}\approx $0.5 mm (1 mfp', i.e., mean free path) to 2 mm (4 mfp') [$1\text{\hspace{0.05em} \hspace{0.05em} mfp'}={({\mu}_{a}+{\mu}_{s}\text{'})}^{-1}$], depending on the values of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$. For the minimum source-detector distance from 2 to 4$\phantom{\rule{.2em}{0ex}}\text{mm}$, the errors were relative stable. Also, the symmetrical curves of the relative errors of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ indicated that these errors were highly correlated. Patterns similar to Fig. 10 were obtained for a wide range of the optical parameters.

The optimal minimum source-detector distance of ${r}_{\mathrm{min}}\approx $1-4$\text{mfp'}$ found in this study is different from the study of Nichols et al. [31] (0.75-1$\text{mfp'}$) but is consistent with the finding of Farrell et al. [32] that the minimum source-detector distance should be larger than 1$\text{mfp'}$. However, in practice, there is no a *priori* information about the exact values of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ over a specified wavelength range, and thus it is difficult to determine the optimal minimum source-detector distance based on these loose criteria. But considering the average errors of 29 sets of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ that were investigated in the simulation experiments, the smallest average estimated errors of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$ were obtained when ${r}_{\mathrm{min}}=1.5\phantom{\rule{.2em}{0ex}}\text{mm}$. Experimentally, the minimum source-detector distance should also be larger than the incident beam size (${r}_{\mathrm{min}}>0.83\phantom{\rule{.2em}{0ex}}\text{mm}$). Hence, the 1.5 mm minimum source-detector distance selected in the system was considered optimal.

Similar error patterns were found for the maximum source-detector distance, as shown in Fig. 10(b1) and (b2). With ${r}_{\mathrm{max}}$ changing from 4 mm to 10 mm, the error varies between 0.03%-19% for ${\mu}_{a}$, and 0.09%-13% for ${\mu}_{s}\text{'}$. The optimal maximum source-detector distance varied with the values of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$, approximately between 10 and 20 $\text{mfp'}$, which is also in agreement with Nichols et al. [31] and Pham et al. [33]. Hence, the maximum source-detector distance should be optimized to obtain accurate estimates of ${\mu}_{a}$ and ${\mu}_{s}\text{'}$. In practice, the optimal maximum source-detector distance may be largely determined by the SNR. As shown in Fig. 11 , to control the minimum SNR of 20 (corresponding to 5% of the variation coefficient), the signal level should be greater than 150 CCD counts at 650 nm with the maximum source-detector distance around 7.7 mm. A similar threshold of the signal level was obtained at other wavelengths with 150 ± 12 CCD counts, while the maximum source-detector distance varied at different wavelengths for this signal level. Therefore, the threshold of 150 CCD counts (SNR = 20) was used in the curve fitting to determine the maximum source-detector distance at each wavelength.

Further, the change of spatial resolution from 0.07 mm to 0.25 mm introduced an insignificant error of 0.53% for ${\mu}_{a}$ and 0.36% for ${\mu}_{s}\text{'}$ relative to the optical properties obtained for the resolution of 0.01 mm, as shown in Fig. 12 . In the system setup, the highest spatial resolution that the imaging sensor can achieve was 0.1 mm without binning. To reduce measurement time and also enhance the signal-to-noise ratio, the 0.2 mm spatial resolution with 2 × 2 binning was selected in our measurements since high resolution is not critical in this case.

#### 3.3 Assessment of the system performance

### 3.3.1 Accuracy

Figure 13
shows the measured absorption and the reduced scattering spectra for three model samples over 500-1,000 nm, determined by the hyperspectral imaging system and the reference methods. As can be seen from the plots on the left pane of Fig. 13 (i.e., a1, b1 and c1), the shape of the absorption spectra for the three dyes obtained by transmittance, integrating sphere and hyperspectral imaging system was similar. They all showed a blue dye absorption peak at 585 nm [Fig. 13(a1)], green dye absorption peak at 714 nm [Fig. 13(b1)], mixed dye absorption peak around 600 nm [Fig. 13(c1)], and water absorption at 970 nm in all three plots. However, the ${\mu}_{a}$ values measured by the integrating sphere (${\mu}_{a}{}^{\ast}$) were far off compared with those from transmittance (${\mu}_{a}{}^{\circ}$) and hyperspectral imaging (${\mu}_{a}{}^{\Delta}$). Similar results were reported by Saeys et al. [8] when they compared the integrating sphere measurement with the time-resolved method. It could be due to the fact that light losses at the sample edge are not accounted for in the integrating sphere measurement, and also, ${\mu}_{a}$ is very sensitive to the measured reflectance and transmittance according to Prahl et al. [17] and Chen et al. [34]. Hence, the accuracy of hyperspectral imaging measurements for ${\mu}_{a}$ was evaluated against the transmittance method. The average errors of ${\mu}_{a}$ were calculated at 530-700 nm for the blue dye, 600-850 nm for the green dye, and 530-800 nm for the mixed dye, because the transmittance method was unable to measure ${\mu}_{a}$ at very small levels for some wavelengths. Overall, the average error of estimating ${\mu}_{a}$ for all model samples was 23% at 530-850 nm, and 16%, 26% and 26% for samples with blue, green, and mixed dyes, respectively. For the three model samples with small amount of blue dye, the absorption caused by the blue dye was almost zero at 750 nm, and thus the absorption at 750 nm for these samples was mainly attributed to the water. The reported results show that water absorption at 750 nm is between 0.0247 to 0.0286 cm^{−1} [35], and our result of${\mu}_{a}=0.0275\pm 0.0006\phantom{\rule{.2em}{0ex}}{\text{cm}}^{\text{-1}}$falls in this range and is only 1% different compared with the recent report from Martelli et al. [36] with ${\mu}_{a}=0.0278\phantom{\rule{.2em}{0ex}}{\text{cm}}^{\text{-1}}$.

For most samples, ${\mu}_{a}$ from the hyperspectral imaging measurement, on average, was underestimated. A possible explanation for the discrepancies is that the dye might have interfered with the Intralipid particles. The confocal laser scanning microscopy (CLSM) images of the Intralipid solutions with blue and green dyes in Fig. 14 showed that some dye particles, which were much smaller than Intralipid particles, went into the lipid vesicles in the Intralipid, resulting in a low dye concentration in the solution. It was also observed during the CLSM experiment that the Intralipid particles were relatively stable in the blue dye sample, while they were actively moving in the green dye sample. Since unstable scattering particles in the model samples could have an effect on the measurement, this may explain why better estimations of ${\mu}_{a}$ were obtained from the blue dye samples compared with those from the green and mixed dye model samples. It was also found that the hyperspectral imaging-estimated absorption above 900 nm was consistently lower for all the samples, particularly at the water absorption band of 970 nm. This might have been caused by the beam position change due to the different diffraction of the focusing lens in the near-infrared region, which needs to be confirmed in further research. Therefore, the absorption above 900 nm was not used in calculating the accuracy.

The spectra of ${\mu}_{s}\text{'}$ presented in Fig. 13(a2), (b2) and (c2) show that the measured values (${\mu}_{s}{\text{'}}^{\phantom{\rule{.2em}{0ex}}\Delta}$) matched well with those obtained from the integrating sphere measurement (${\mu}_{s}{\text{'}}^{\phantom{\rule{.2em}{0ex}}\ast}$) at 500-900 nm, and lower than those calculated from the empirical equation (${\mu}_{s}\text{'}\text{\hspace{0.05em}}{\text{\hspace{0.05em}}}^{\circ}$). The average error of estimating ${\mu}_{s}\text{'}$ was 7% at 500-900 nm compared with the integrating sphere measurement. The divergence of ${\mu}_{s}\text{'}$ above 900 nm appeared between the integrating sphere method and the wavelength-dependent exponential decay function of ${\mu}_{s}\text{'}$. It is difficult to know which method is more accurate for determining ${\mu}_{s}\text{'}$ at the wavelength above 900 nm because no published paper reported the ${\mu}_{s}\text{'}$ values of Intralipid in these wavelengths. However, this wavelength-dependent exponential decay function of ${\mu}_{s}\text{'}$ could be violated with the variation of the refractive index at long wavelengths according to the study of Mourant et al. [37].

When comparing our measurement accuracy with other techniques or other spatially-resolved methods, one should notice the differences of experimental conditions, model samples and wavelength ranges. Bays et al. [38] reported 32% accuracy of ${\mu}_{s}\text{'}$ at 633 nm for polyoxymethlene model samples by using a spatially-resolved reflectometry. Pifferi et al. [39] applied a general protocol for assessing several optical methods for determining optical properties, and they reported large differences from different instruments in measuring the same model samples with the maximum discrepancies of 32% for ${\mu}_{a}$ at 970 nm and 41% for ${\mu}_{s}\text{'}$at 820 nm. Spichtig et al. [40] used frequency-domain technique to measure the optical properties of Intralipid model samples with accuracies being less than 10% for ${\mu}_{a}$ and 31% for ${\mu}_{s}\text{'}$ at eight wavelengths. While covering a wider wavelength range of optical properties, the hyperspectral imaging-based spatially-resolved method has achieved superior results for measuring ${\mu}_{a}$ and ${\mu}_{s}\text{'}$, compared with these reported studies that were only conducted at single or several wavelengths.

### 3.3.2 Precision/reproducibility

The system precision or reproducibility was calculated by repeated measurements of a blue dye model sample at four different days under the same experimental conditions with a 30 min warm-up time and the same level of source power and acquisition time. The coefficient of variation (or CV) in the absorption peak at 585 nm was 2.4% for ${\mu}_{a}$ and 3.8% for ${\mu}_{s}\text{'}$ with the maximum discrepancies of 4.9% and 6.3%, respectively. However, the model sample could have undergone some small changes over the four days, thus inducing some variations in the optical properties of the tested sample. Hence the system has demonstrated good reproducibility and consistency over a prolonged time period.

### 3.3.3 Sensitivity

Ten measurements were performed on a model sample made up with the blue dye as the absorber because its minimum absorption coefficient was almost zero for the wavelengths of 700-900 nm. The result of the sensitivity test on ${\mu}_{a}$ is presented in Fig. 15
, with the CV of ${\mu}_{a}$ for the 10 measurements being plotted as a function of ${\mu}_{a}$ for different wavelengths. The minimum detectable value of ${\mu}_{a}$ was around 0.0082 cm^{−1} for a noise level of 10%. The sensitivity of measuring ${\mu}_{s}\text{'}$, as determined by the CV values, was always less than 4% because ${\mu}_{s}\text{'}$ was much larger than ${\mu}_{a}$ for the investigated range of $7.0\le {\mu}_{s}\text{'}\le 40.0\phantom{\rule{.2em}{0ex}}{\text{cm}}^{\text{-1}}$.

## 4. Conclusions

This research examined critical factors in the development of hyperspectral imaging-based spatially-resolved technique (i.e., methods for providing reference measures of the optical properties, light beam and source-detector distance, etc.) for determining the absorption and reduced scattering coefficients of biological materials over the wavelengths of 500-1,000 nm. Monte Carlo simulations, coupled with experiments for model samples, demonstrated that to achieve the best performance for the hyperspectral imaging system, the light beam should be of circular shape and Gaussian type with the diameter of less than 1 mm, the optimal minimum source-detector distance should be about 1.5 mm, and the optimal maximum source-detector distance should be equivalent to 10-20 $\text{mfp'}$ or determined by the minimum signal-to-noise ratio of 20 (or 150 CCD counts for the system). Under these optimal conditions, the hyperspectral imaging system achieved average accuracies of 23% for the absorption coefficient at 530-850 nm and 7% for the reduced scattering coefficient at 500-900 nm, when it was evaluated using the model samples against the transmittance and integrating sphere methods. These results are better than, or similar to, those obtained with other spatially-resolved sensing configurations, and frequency-domain and time-resolved techniques for single or several wavelengths. The system also had good reproducibility and sensitivity with the minimum detectable ${\mu}_{a}$ value of 0.0082 cm^{−1}. The research provided a systematic guide for optimizing and evaluating the optical system and offered solutions to improve accuracy in measuring the absorption and reduced scattering coefficients, which will be valuable for nondestructive quality evaluation of food and agricultural products.

## Acknowledgements

This research was conducted as a part of the USDA ARS inhouse CRIS project 3635-43000-004-00D “Technologies for Assessing and Grading Cucumbers and Tree Fruits”. The authors thank Dr. Ariana Diwan and Mr. Benjamin Bailey for providing technical support in the system setup.

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