2.1.

Non-Contact Diffuse Reflectance Spectroscopy

The Non-Contact DRS system that we use is detailed in a previous study²¹ and its design preserves the same geometrical dimensions of the initial Contact DRS system described. Measurements performed with both systems inspect the same sample volume to allow their direct comparison: Contact DRS is considered as the reference measurement system to validate our Non-Contact DRS results.

The illumination is provided by a Tungsten Halogen (T–H) Lamp HL2000 Ocean Optics and the detection is ensured by a QE65000 Ocean Optics spectrophotometer, cooled to $-15°\mathrm{C}$ to reduce dark noise. The spectral working range of the entire instrument lies between 470 and 880 nm. The probe features a central illumination fiber ($Ø=500\mu \mathrm{m}$) and six detection rings [each composed of seven concentric fibers ($Ø=100\mu \mathrm{m}$)] at six different distances, ranging from 300 to $2488\mu \mathrm{m}$ (center to center). The seven fibers of each detection ring are clustered in a specific bundle referred to as F6 to F1, with F6 and F1 measuring the closest and furthest ring to the source, respectively. F7 is a specific fiber that enables a direct measurement of the illumination spectrum. A step-wise measurement of each $F_i$ signal by the spectrometer provides the diffuse reflectance at all source-detector distances individually and sequentially. The noncontact modality of the measurement is ensured by placing an achromatic doublet pair between the probe and the sample as shown in Fig. 1(a) (right). The specific treatment of these Non-Contact DRS measurements has been precisely described²¹ and will not be detailed here.

2.2.

Multispectral Imaging

The MSI instrumental setup is shown in Fig. 1(a) (left). The system makes use of a 12-bit monochrome VGA PixelFly CCD detector, matrix of $640\times 480\text{pixels}$ with a pixel size of $9.9\mu \mathrm{m}$, which allows good spatial resolution. A Xenoplan 2.0/28 objective lens ensures a FOV of $67\times 50{\mathrm{mm}}^2$ (corresponding to a magnification $G=0.095$) at the focal plane, where the sample is placed. The lamp is a KL2500 LCD Schott source set for a broad spectral distribution (3000 K of temperature). The system sequentially acquires data with a limited number of spectral bands leading to a good tradeoff between spectral sampling and acquisition time. Namely, the source is coupled with a filter wheel that defines four specific illumination wavelengths: 500, 550, 600, and 700 nm having a 10-nm FWHM. The illumination of the sample is achieved through the use of a six-LED illumination ring that is inclined to avoid specular reflection of imaged flat samples.

The multispectral images, noted $S$, are preprocessed according to a background ($S_{\mathrm{offset}}$) subtraction and corrections of integration time ($t$) and flat field (FF). The flat-field image ($S_{\mathrm{flat}}$) is acquired with a Spectralon^® standard SR99%. Background $S_{\mathrm{offset}}$ and signal $S$ images are acquired with the same integration time $t$. The preprocessed or corrected image is referred to as $S_c$ and calculated according to

Depending on the required signal-to-noise ratio, spatial resolution, and calculation time, the $S_c$ image is averaged on neighboring pixels according to Eq. (2), with $T_x=T_y=1$ leading to the primary image spatial resolution

Finally, the resulting averaged image $S_a$ is normalized by an indirect measurement of the illumination intensity, namely the intensity of a fixed plastic corner in the FOV (see Fig. 1, left). This is done to correct the fluctuations of the illumination intensity from one measurement to another and thereby ensure repeatability and accuracy of the quantification analysis. The resulting normalized image is noted as $S_N$.

2.3.

Phantom set and Dual-Step Acquisition Protocol

3.1.

Scattering Coefficient Estimation by Non-Contact Diffuse Reflectance Spectroscopy

Both absorption and scattering coefficients estimation through Non-Contact DRS are performed according to a Monte Carlo (MC) look-up table (LUT)-based approach and the adaptive calibration algorithm ACA-Pro, detailed in a previous study.²¹

In this work, we define the depth of field (DOF) as the focus range achieving an acceptable spatial resolution that allows accurate estimation of optical properties with Non-Contact DRS. As the scattering property is estimated with the closest fibers to the source, its estimation is more sensitive to the focus range than that of the absorption property, estimated with further fibers.²¹ Therefore, the Non-Contact DRS scattering estimation error is used to determine the DOF of the Dual-Step system. This is done according to the average error calculated between the reference ${\mu }_s^{\prime }$ (${\mu }_{s,\mathrm{ref}}^{\prime }$) of a phantom fixed at the focal plane; and ${\mu }_s^{\prime }$ estimations ($\widehat{{\mu }_s^{\prime }}$) of the same phantom placed at different object planes.

An acceptable error of 4.1% on $\widehat{{\mu }_s^{\prime }}$ is achieved for a DOF of 1.2 mm, as shown in Fig. 2.

Fig. 2

Estimated averaged $\widehat{{\mu }_s^{\prime }}$ (blue line) using a fixed focused reference phantom (black dotted line) and measurements at object plane range of 1.2 mm. The average $\widehat{{\mu }_s^{\prime }}$ error, $\mathrm{Er}=4.1\%$, considers the entire working spectrum.

However, this DOF (1.2 mm) leads to an error on absorption estimation of about 27%, which is unusable in the real context of biological samples. Thus, the Dual-Step technique uses Non-Contact DRS to estimate only the scattering coefficient, noted $\widehat{{\mu }_s^{\prime }}$. All the results in this work are obtained with a scattering coefficient $\widehat{{\mu }_s^{\prime }}$ estimated by Non-Contact DRS.

3.2.

Absorption Coefficient Estimation by Multispectral Imaging

As explained in the introduction, MSI cannot quantify both absolute optical properties. However, quantification of one optical property is possible if the other is known. Consequently, the Dual-Step technique uses the scattering $\widehat{{\mu }_s^{\prime }}$ estimations of Non-Contact DRS to allow absorption estimation with MSI. For the latter, we use an LUT specific to MSI, built according to the diffusion model described in a previous study.¹⁰ This model has been used for various MSI techniques⁴ including the reference SFDI.⁹ The model expresses diffuse reflectance $R$, originated from a uniform illumination, as a function of the wavelength and the albedo $a^{\prime }$, which combines both optical properties ${\mu }_a$ and ${\mu }_s^{\prime }$. We use the equation defined by Cuccia et al.⁹ under the assumption of a planar illumination

Figure 3 (bottom left) shows the modeled diffuse reflectance $R^{\mathrm{LUT}}$ in terms of optical properties (${\mu }_s^{\prime }$, ${\mu }_a$) saved in an LUT.

Fig. 3

Flowchart describing the absorption quantification method of MSI. On the bottom left, $R^{\mathrm{LUT}}$ proper of MSI computed with the diffusion model [Eq. (3)]. A preliminary calibration step (in blue) corrects the treated measure $S_N$, (in pink) from the instrumental response. The comparison of the calibrated signal ($R_{\mathrm{CF},}$) to the entire LUT ($R^{\mathrm{LUT}}$) via a minimization procedure (in black) leads to the identification of the possible optical properties couples ${\mu }_s^{\prime }\text{-}{\mu }_a$ (red line in bottom right plot). With the given $\widehat{{\mu }_s^{\prime }}$ previously estimated with DRS, estimation of absorption $\widetilde{{\mu }_a}$ is possible with MSI (green arrow).

The general method for MSI absorption quantification relies on a calibration process (blue path in Fig. 3) that uses the measurement of a single reference phantom with known optical properties ${\mu }_{s,\mathrm{ref}}^{\prime }$ and ${\mu }_{a,\mathrm{ref}}$. The measured reflectance signal $S_{\mathrm{ref}}$ of a given reference phantom is treated according to Eqs. (1) and (2), respectively. The comparison of $S_{N,\mathrm{ref}}$ with $R_{\mathrm{ref}}^{\mathrm{LUT}}$ (modeled reflectance for the same reference phantom) determines a correction ratio CF. This CF calibrates the treated signal $S_N$ of a measured unknown sample (having unknown optical properties) from the instrumental response. The resulting calibrated signal $R_{\mathrm{CF}}$, is compared with the entire $R^{\mathrm{LUT}}$ to identify the possible couples of ${\mu }_s^{\prime }-{\mu }_a$ optical properties of the unknown phantom. These couples correspond to the minimal difference between $R_{\mathrm{CF}}$ and $R^{\mathrm{LUT}}$ (red line of bottom right plot in Fig. 3). From these possible optical properties ${\mu }_s^{\prime }-{\mu }_a$ couples, and for a given $\widehat{{\mu }_s^{\prime }}$, previously estimated with DRS, the absorption coefficient $\widetilde{{\mu }_a}$ can be determined with MSI (green arrow of Fig. 3).

The improvement in the wide-field absorption estimation accuracy through the use of Non-Contact DRS estimated $\widehat{{\mu }_s^{\prime }}$, compared to theoretically modeled scattering properties,^4–7 has been quantified and validated in a previous study.²³

3.3.

Adaptive Calibration Algorithm Procedure for Absorption Quantification

The model expressed by Eq. (3) makes the approximation that scattering properties dominate over absorption properties. One should keep in mind that this is not always the case for skin, especially for dark skin phototypes and at $\lambda <600\mathrm{nm}$ due to the high absorption of melanin and hemoglobin.²⁴

Randeberg et al.²⁵ approximated the error between the diffusion model and a Monte Carlo simulation for $\lambda =400$ to 850 nm and reduced it through the application of a scaling factor, as a function of the absorption coefficients, on the diffusion model.

In the first stage of this work, we did not consider the influence of optical properties on the calibration factor CF and used a single reference phantom defining a single CF (Sec. 3.2). However, this CF highly depends on the scattering coefficient $({\mathrm{IL}}_{\mathrm{ref}})$ and the absorption coefficient $({\mu }_{a,\mathrm{ref}})$ of the reference phantom (see Fig. 5). Hence, to further improve absorption estimation results we integrate an optical property-dependent correction through the MSI ACA-Pro approach that makes use of several reference phantoms with different known reference optical properties (${\mu }_{s,\mathrm{ref}}^{\prime }$, ${\mu }_{a,\mathrm{ref}}$). These reference phantoms are used to obtain a reference base of various CFs, noted as $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$.

Figure 4 shows the flowchart of the ACA-Pro ${\mu }_a$ quantification process applied to the MSI system in combination with Non-Contact DRS, according to the Dual-Step technique. The ACA-Pro algorithm makes use of an optimized number of reference phantoms having known optical properties (${\mu }_{s,\mathrm{ref}}^{\prime }$, ${\mu }_{a,\mathrm{ref}}$) that cover the optical range of study. The measured and treated signals $S_{N,\mathrm{ref}}$ of these phantoms are compared with their corresponding modeled diffuse reflectance $R_{\mathrm{ref}}^{\mathrm{LUT}}$ and their ratio is calculated to create the MSI CF reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$.

Fig. 4

Flowchart describing the absorption quantification ($\widehat{{\mu }_a}$) method of MSI with the ACA-Pro approach (in blue), based on the Non-Contact DRS scattering estimation $\widehat{{\mu }_s^{\prime }}$.

The aim of ACA-Pro is to calculate the optimal CF, noted as ${\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$, achieving optimal MSI estimation of absorption property $\widehat{{\mu }_a}$. This process starts with the estimation of $\widehat{{\mu }_s^{\prime }}$ with Non-Contact DRS.²¹

The DRS $\widehat{{\mu }_s^{\prime }}$ estimation defines a reduced set (group) of possible CFs from the entire MSI reference base $\mathrm{CF}(\widehat{{\mu }_s^{\prime }},{\mu }_{a,\mathrm{ref}})$, i.e., $\widehat{{\mu }_s^{\prime }}$ defines the CF for all the couples $\widehat{{\mu }_s^{\prime }}-{\mu }_{a,\mathrm{ref}}$. If $\widehat{{\mu }_s^{\prime }}$ is included in the reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$, i.e., $\widehat{{\mu }_s^{\prime }}={\mu }_{s,\mathrm{ref}}^{\prime }$, then the $\mathrm{CF}(\widehat{{\mu }_s^{\prime }},{\mu }_{a,\mathrm{ref}})$ set is directly selected. In practice, this is not possible due to estimation error of Non-Contact DRS.²¹ In other words, even if the theoretical scattering property of the unknown phantom, ${\mu }_{s,\mathrm{theo}}^{\prime }$, is equal to ${\mu }_{s,\mathrm{ref}}^{\prime }$, real Non-Contact DRS measurements of the unknown phantom will estimate $\widehat{{\mu }_s^{\prime }}\ne {\mu }_{s,\mathrm{ref}}^{\prime }$. As $\widehat{{\mu }_s^{\prime }}$ is not included in the reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$, i.e., $\widehat{{\mu }_s^{\prime }}\ne {\mu }_{s,\mathrm{ref}}^{\prime }$, the $\mathrm{CF}(\widehat{{\mu }_s^{\prime }},{\mu }_{a,\mathrm{ref}})$ set is found through interpolation.

The selected CFs from the $\mathrm{CF}(\widehat{{\mu }_s^{\prime }},{\mu }_{a,\mathrm{ref}})$ set are used to calculate the corresponding MSI absorption estimations $\widetilde{{\mu }_a}$ and related estimation difference $|\widetilde{{\mu }_a}-{\mu }_{a,\mathrm{ref}}|$. ACA-Pro selects the optimal CF, ${\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$, that estimates optimal $\widetilde{{\mu }_a}$, noted $\widehat{{\mu }_a}$, with minimal difference.

This method supposes an ideal situation in which the theoretical absorption property of the unknown phantom, noted as ${\mu }_{a,\mathrm{theo}}$, is equal to one of the ${\mu }_{a,\mathrm{ref}}$ of the $\mathrm{CF}(\widehat{{\mu }_s^{\prime }},{\mu }_{a,\mathrm{ref}})$ set, i.e., ${\mu }_{a,\mathrm{theo}}={\mu }_{a,\mathrm{ref}}$. This ideal case is used to test the ability of ACA-Pro in estimating $\widetilde{{\mu }_a}$ corresponding to a given CF and selecting ${\mathrm{CF}}_{\text{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$ that achieves minimal difference $|\widetilde{{\mu }_a}-{\mu }_{a,\mathrm{ref}}|$ and hence, minimal estimation error $|\widehat{{\mu }_a}-{\mu }_{a,\mathrm{theo}}|$ (Sec. 4.1.1). This resulting minimal estimation error is only due to measurement acquisition noise and corresponds to the ideal ACA-Pro performance.

In a more realistic context, the unknown phantom does not have the same absorption properties as any of the reference phantoms in the $\mathrm{CF}(\widehat{{\mu }_s^{\prime }},{\mu }_{a,\mathrm{ref}})$ set, i.e., ${\mu }_{a,\mathrm{theo}}\ne {\mu }_{a,\mathrm{ref}}$. Therefore, ACA-Pro includes an iterative interpolation method of the CF. Given a single random CF from the $\mathrm{CF}(\widehat{{\mu }_s^{\prime }},{\mu }_{a,\mathrm{ref}})$ set estimating an initial $\widetilde{{\mu }_{a,0}}$, the CF is interpolated (within the set) to $\mathrm{CF}(\widehat{{\mu }_s^{\prime }},\widetilde{{\mu }_{a,0}})$, estimating $\widetilde{{\mu }_{a,1}}$. Hence, the $\mathrm{CF}(\widehat{{\mu }_s^{\prime }},\widetilde{{\mu }_{a,i+1}})$ is iteratively updated according to the previous estimation $\widetilde{{\mu }_{a,i}}$ until convergence, i.e., $\widetilde{{\mu }_{a,i+1}}=\widetilde{{\mu }_{a,i}}$. We note the final interpolated CF as ${\mathrm{CF}}_{\mathrm{interp}}$, which corresponds to ${\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$ and estimates $\widehat{{\mu }_a}$ with minimal estimation error $|\widehat{{\mu }_a}-{\mu }_{a,\mathrm{theo}}|$. Experimental results are shown in Sec. 4.1.2.

4.1.

Validation of ACA-Pro for Absorption Quantification with the Dual-Step Technique on Homogeneous Phantoms

A 10-reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$ is built with measurements taken on 10 different reference phantoms (described in Sec. 2.3.1) and four wavelengths $\lambda$ (500, 550, 600, and 700 nm). Figure 5 shows the resulting 10-reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$ according to $\lambda$ and reference optical properties. Remark the dependence of $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$ on ${\mu }_{a,\mathrm{ref}}$, originating from the lower accuracy of the diffusion model at high absorption values. This dependence justifies the use of the ${\mu }_a$-based ACA-Pro calibration strategy.

Fig. 5

Ten-reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$ according to illumination $\lambda$ built with ten-reference phantoms having two different scattering properties (${\mathrm{IL}}_{\mathrm{ref}}=1\%$ and ${\mathrm{IL}}_{\mathrm{ref}}=1.5\%$) and five different absorption properties ${\mu }_{a,\mathrm{ref}}=0.44,1.15,3.12,6.61$, and $11.45{\mathrm{cm}}^{-1}$ at 550 nm.

4.1.1.

Ideal case: choice of ${\mathsf{CF}}_{\mathsf{\text{opt}}}(\widehat{{\mu }_{\mathsf{s}}^{\prime }},\widehat{{\mu }_{\mathsf{a}}})$, for unknown phantoms included in the CF reference base

In this section, we analyze the MSI ${\mu }_a$-based ACA-Pro performance in the absorption estimation $\widetilde{{\mu }_a}$ and corresponding estimation error for any given CF. We also aim at defining the minimal error achievable with ACA-Pro applied to our measurement set-up. For this, we estimate the absorption of five unknown phantoms having theoretical properties equal to those of five reference phantoms used for the 10-reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$: ${\mathrm{IL}}_{\mathrm{theo}}=1\%$, ${\mu }_{a,\mathrm{theo}}=0.44,1.15,3.12,6.61$, and $11.45{\mathrm{cm}}^{-1}$ at 550 nm. The absorption coefficient of all five measured unknown phantoms is estimated with the entire 10-reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$, i.e., 10 CFs.

Moreover, we look at the influence of $\widehat{{\mu }_s^{\prime }}$ on the absorption estimation $\widetilde{{\mu }_a}$ by comparing the results obtained with the “wrong” $\mathrm{CF}(\widehat{{\mu }_s^{\prime }}={\mu }_{s,\mathrm{ref}\mathrm{IL}=1.5\%}^{\prime }\ne {\mu }_{s,\mathrm{theo}\mathrm{IL}=1\%}^{\prime },{\mu }_{a,\mathrm{ref}})$ and “correct” $\mathrm{CF}(\widehat{{\mu }_s^{\prime }}={\mu }_{s,\mathrm{ref}\mathrm{IL}=1\%}^{\prime }={\mu }_{s,\mathrm{theo}\mathrm{IL}=1\%}^{\prime },{\mu }_{a,\mathrm{ref}})$ reduced sets.

For each measured unknown phantom, the resulting absorption difference $|\widetilde{{\mu }_a}-{\mu }_{a,\mathrm{ref}}|$ of $\widetilde{{\mu }_a}$ estimated with all 10 CFs of the 10-reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$ is shown in Fig. 6 for 550 nm. Notice the expected influence of $\widehat{{\mu }_s^{\prime }}$ on $\widetilde{{\mu }_a}$: the absorption difference is higher for the “wrong” $\mathrm{CF}(\widehat{{\mu }_s^{\prime }}={\mu }_{s,\mathrm{ref}\mathrm{IL}=1.5\%}^{\prime }\ne {\mu }_{s,\mathrm{theo}\mathrm{IL}=1\%}^{\prime },{\mu }_{a,\mathrm{ref}})$ than for the “correct” $\mathrm{CF}(\widehat{{\mu }_s^{\prime }}={\mu }_{s,\mathrm{ref}\mathrm{IL}=1\%}^{\prime }={\mu }_{s,\mathrm{theo}\mathrm{IL}=1\%}^{\prime },{\mu }_{a,\mathrm{ref}})$. The ${\mu }_{a,\mathrm{ref}}$ values of the CF also have a coherent effect on the absorption difference: the closer ${\mu }_{a,\mathrm{ref}}$ is to ${\mu }_{a,\mathrm{theo}}$, the lower the difference is. As expected, the minimal absorption difference is obtained with the CF corresponding to the optical properties of the unknown phantom $\mathrm{CF}(\widehat{{\mu }_s^{\prime }}={\mu }_{s,\mathrm{ref}\mathrm{IL}=1\%}^{\prime }={\mu }_{s,\mathrm{theo}\mathrm{IL}=1\%}^{\prime },{\mu }_{a,\mathrm{ref}}={\mu }_{a,\mathrm{theo}})$. ACA-Pro converges on this CF and defines it as ${\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$ that estimates $\widehat{{\mu }_a}$ with minimal error $|\widehat{{\mu }_a}-{\mu }_{a,\mathrm{theo}}|$ corresponding to measurement acquisition noise only.

Fig. 6

Spatially averaged $|\widetilde{{\mu }_a}-{\mu }_{a,\mathrm{ref}}|$ difference at 550 nm of unknown phantoms having common ${\mathrm{IL}}_{\mathrm{theo}}=1\%$ and different ${\mu }_{a,\mathrm{theo}}$ (shown in the $x$-axis), calibrated with the 10-reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$ (with reference optical properties stated in the legend) and the optimal ${\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$ chosen by MSI ACA-Pro.

This optimized estimation procedure, i.e., definition of ${\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$ is performed for each pixel value in the image to estimate wide field $\widehat{{\mu }_a}$. Figure 7 shows an example of wide-field $\widehat{{\mu }_a}$ for the lowest absorbing unknown phantom considered (${\mu }_{a,\mathrm{theo}}=0.44{\mathrm{cm}}^{-1}$ at 550 nm). Notice that all $\widehat{{\mu }_a}$ pixels are in accordance with the theoretical value ${\mu }_{a,\mathrm{theo}}$ and present an average error of 4% for all considered $\lambda$.

Fig. 7

Wide-field multispectral $\widehat{{\mu }_a}$ quantification for an unknown phantom having ${\mu }_{a,\mathrm{theo}}$ specified by the central figure and marked by the black arrows on the colorbars. These results are obtained with $T_x=T_y=10\text{pixels}$ [see Eq. (2)]. The image size is $33\times 33\text{pixels}$ corresponding to a FOV of $34.4\times 34.4{\mathrm{mm}}^2$.

Figure 8 shows the minimal spatially averaged ${\mu }_a$ errors of $\widehat{{\mu }_a}$ estimated with ${\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$ for all unknown phantoms ${\mu }_{a,\mathrm{theo}}$ and $\lambda$. Notice that all minimal mean errors are $<5.5\%$ (see 600 nm). These results comprehensively validate the wide-field ${\mu }_a$ calibration procedure of MSI ACA-Pro for the range of optical properties considered.

Fig. 8

Spatially averaged $|\widehat{{\mu }_a}-{\mu }_{a,\mathrm{theo}}|$ estimation errors of unknown phantoms having common ${\mathrm{IL}}_{\mathrm{theo}}=1\%$ and different ${\mu }_{a,\mathrm{theo}}$ calibrated with corresponding ${\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$ for all considered $\lambda$. The error is inferior to 5.5% (see 600 nm). The green circle shows the error to be compared with that achieved with the interpolation strategy (Sec. 4.1.2).

4.1.2.

General case: choice of ${\mathsf{CF}}_{\mathsf{\text{opt}}}(\widehat{{\mu }_{\mathsf{s}}^{\prime }},\widehat{{\mu }_{\mathsf{a}}})$, for unknown phantoms not included in the CF reference base. Interpolation strategy

We consider a general case to illustrate and analyze the iterative interpolation strategy of MSI ${\mu }_a$-based ACA-Pro that calculates ${\mathrm{CF}}_{\mathrm{interp}}={\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$.

In this case, ACA-Pro uses a reduced two-reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$, built only with two reference phantoms having ${\mathrm{IL}}_{\mathrm{ref}}=1\%$ and ${\mu }_{a,\mathrm{ref}}=1.15$ and $6.61{\mathrm{cm}}^{-1}$ at 550 nm. The measured unknown phantom has ${\mathrm{IL}}_{\mathrm{theo}}=1\%$ and ${\mu }_{a,\mathrm{theo}}=3.12{\mathrm{cm}}^{-1}$ at 550 nm. We use the real Non-Contact DRS ${\mu }_s^{\prime }$ estimation, $\widehat{{\mu }_s^{\prime }}\ne {\mu }_{s,\mathrm{ref}}^{\prime }$ (refer to Sec. 3.3). Notice that the ${\mu }_{a,\mathrm{theo}}=3.12{\mathrm{cm}}^{-1}$ of the unknown phantom does not correspond to any of the ${\mu }_{a,\mathrm{ref}}$ in the two-reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$, i.e., ${\mu }_{a,\mathrm{theo}}\ne {\mu }_{a,\mathrm{ref}}$.

A first random CF from the $\mathrm{CF}(\widehat{{\mu }_s^{\prime }},{\mu }_{a,\mathrm{ref}})$ set estimates $\widetilde{{\mu }_{a,0}}$ that defines the next CF, $\mathrm{CF}(\widehat{{\mu }_s^{\prime }}\widetilde{{\mu }_{a,0}})$, through interpolation of the two-reference base/set. $\mathrm{CF}(\widehat{{\mu }_s^{\prime }}\widetilde{{\mu }_{a,0}})$ then estimates $\widetilde{{\mu }_{a,1}}$. After various iterations $\widetilde{{\mu }_{ai+1}}$ stabilizes and converges to $\widetilde{{\mu }_{a,i}}$, becoming $\widehat{{\mu }_a}$, estimated with ${\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$ (refer to Fig. 4).

Figure 9 shows the wide-field $\widetilde{{\mu }_a}$ estimations at 550 nm with the two CFs of the reference base: $\mathrm{CF}{\mu }_{a,\mathrm{ref}}=1.15{\mathrm{cm}}^{-1}$ and $\mathrm{CF}{\mu }_{a,\mathrm{ref}}=6.61{\mathrm{cm}}^{-1}$. These $\widetilde{{\mu }_a}$ estimations are compared with the $\widehat{{\mu }_a}$ estimated with the interpolated ${\mathrm{CF}}_{\mathrm{interp}}={\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$. The average wide-field $|\widetilde{{\mu }_a}-{\mu }_{a,\mathrm{theo}}|$ estimation errors obtained with all three CFs are shown at the bottom right of Fig. 9. Remark that ${\mathrm{CF}}_{\mathrm{interp}}$ achieves an averaged minimal wide-field $|\widehat{{\mu }_a}-{\mu }_{a,\mathrm{theo}}|$ estimation error of 2.6%, which is comparable with that obtained with the ideal $\mathrm{CF}(\widehat{{\mu }_s^{\prime }},{\mu }_{a,\mathrm{ref}}={\mu }_{a,\mathrm{theo}})$: $\text{error}=1.7\%$ (see Sec. 4.1.1, green circle of Fig. 8 at 550 nm for the same ${\mu }_{a,\mathrm{theo}}$). This result validates the iterative interpolation strategy of ${\mu }_a$-based ACA-Pro applied to MSI.

Fig. 9

Wide-field ${\mu }_a$ estimation at 550 nm of an unknown phantom having ${\mu }_{a,\mathrm{theo}}=3.12{\mathrm{cm}}^{-1}$ calibrated with the two-reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$ built with two-reference phantoms only having ${\mu }_{a,\mathrm{ref}}=1.15$ and $6.61{\mathrm{cm}}^{-1}$. (a) ${\tilde{\mu }}_a$ estimation with CF corresponding to ${\mu }_{a,\mathrm{ref}}=1.15{\mathrm{cm}}^{-1}$. (b) ${\tilde{\mu }}_a$ estimation with CF corresponding to ${\mu }_{a,\mathrm{ref}}=6.61{\mathrm{cm}}^{-1}$. (c) $\widehat{{\mu }_a}$ estimation with ${\mathrm{CF}}_{\mathrm{interp}}={\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$. (d) Mean $|\widetilde{{\mu }_a}-{\mu }_{a,\mathrm{theo}}|$ estimation errors with $\mathrm{CF}{\mu }_{a,\mathrm{ref}}=1.15{\mathrm{cm}}^{-1}$, $\mathrm{CF}{\mu }_{a,\mathrm{ref}}=6.61{\mathrm{cm}}^{-1}$, and $|\widehat{{\mu }_a}-{\mu }_{a,\mathrm{theo}}|$ for ${\mathrm{CF}}_{\mathrm{interp}}$. The results given in this figure are obtained with $T_x=T_y=10\text{pixels}$ [see Eq. (2)]. The image size is $33\times 33\text{pixels}$ corresponding to an FOV of $34.4\times 34.4{\mathrm{mm}}^2$.

An optimal choice of the minimal amount of reference phantoms used to build the general reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$ is necessary to optimally benefit from the interpolation strategy of ACA-Pro. This choice directly depends on the range of interest of optical properties.

Contrarily to what is expected, Figs. 7 and 9 show that the estimated absorption map is not spatially homogeneous for the measured homogeneous unknown phantoms. This artifact is due to the fact that, for the given results, a unique CF, computed in the middle of the image, is used to correct the entire image. Therefore, a spatially local CF correction will be implemented in prospective work to correct this drawback.

4.2.

Influence of the Depth of Field on the Absorption Estimation Accuracy

We mentioned in Sec. 3.1 that an error of 4.1% on $\widehat{{\mu }_s^{\prime }}$ estimation is achieved for a DOF of 1.2 mm with Non-Contact DRS. Furthermore, we quantified the error on ${\mu }_a$ (27%) and concluded that Non-Contact DRS is only quantitative at the focal plane for both optical parameters (1% error for ${\mu }_s^{\prime }$ and 2.4% for ${\mu }_a$).²⁶

In this work, we quantify the influence of the DOF on the $\widehat{{\mu }_a}$ estimation accuracy by MSI. In this system, the measured reflectance depends linearly on the distance between the sample and the objective lens. We analyze this effect with a test unknown phantom having the same optical properties as the single reference phantom (${\mu }_{a,\mathrm{ref}}=0.44{\mathrm{cm}}^{-1}$ at 550 nm and ${\mathrm{IL}}_{\mathrm{ref}}=1\%$), by estimating $\widehat{{\mu }_a}$ according to the procedure described in Sec. 3.2. The scattering coefficient of the phantom, estimated by Non-Contact DRS ($\widehat{{\mu }_s^{\prime }}$) and the calculated MSI ${\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$ are obtained at the focal plane. This ${\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$ is then used to estimate $\widehat{{\mu }_a}$ with MSI on the same phantom placed at $+/-3.5\mathrm{mm}$ around the focal plane. The resulting MSI error $\widehat{{\mu }_a}-{\mu }_{a,\mathrm{theo}}$, due to the reflectance intensity variation related to the plane height difference, is shown in Fig. 10. As expected, the $\widehat{{\mu }_a}$ estimation decreases with height, since the reflectance intensity increases as the sample approaches the source. Notice that for an error of 4.1%, the DOF is around 2.5 mm. Thus, because the DOF is more constraining for $\widehat{{\mu }_s^{\prime }}$, we consider only a DOF of 1.2 mm.

Fig. 10

$\widehat{{\mu }_a}-{\mu }_{a,\mathrm{theo}}$ error originated from the reflectance variation at different planes ($+/-3.5\mathrm{mm}$) at 550 nm.

Therefore, the final DOF of the Dual-Step technique is determined by the most constraining modality, i.e., the Non-Contact DRS. The defined DOF of 1.2 mm obtains an error inferior to 4.1% for Non-Contact DRS $\widehat{{\mu }_s^{\prime }}$ estimation (see Fig. 2). The effect of the 4.1% Non-Contact DRS ${\mu }_s^{\prime }$ error on the $\widehat{{\mu }_a}$ MSI estimation (5.5% error), has been quantified as a maximum error variation of $\pm 4.03\%$ (for all studied wavelengths: 500, 550, 600, and 700 nm). Further improvement in the $\widehat{{\mu }_a}$ MSI estimation will be achieved in future work by integrating a height-calibration (based on Fig. 10) after a relief analysis of the sample.

4.3.

Optical Properties Quantification Procedure on Heterogeneous Phantoms

After validating with homogeneous phantoms, and to approach clinical measurement of heterogeneous biological samples, we tested our technique with a heterogeneous phantom. For this, we used a homemade gelatine-based phantom with different optical properties shown in Fig. 11.

Fig. 11

Homemade gelatine-based heterogeneous phantom composed of different unknown optical properties ${\mu }_a$ and ${\mu }_s^{\prime }$.

Absorption of each zone depends on the different concentrations of VAHINÉ red ink or HP-printer blue ink used. Scattering depends on the amount of scattering milk and edible gelatine used. Blue shapes (circle and corners) are made of the same milk-gelatine mixture and two different concentrations of blue ink defined as “dark” for the bottom left corner and the bottom right circle and as “light” for the top right corner (see Fig. 11). The pink background is made of a different milk-gelatine mixture and of a low concentration of red ink. Note that, because the blue shapes have been molded before the pink ones, some overlap of the blue shapes with the pink background is present at border zones.

The 2-D MSI absorption estimation relies on the preliminary DRS estimation of local $\widehat{{\mu }_s^{\prime }}$. In case of heterogeneous phantoms, this scattering coefficient has to be measured in each zone of common scattering properties. Figure 11 shows the different zones that have been measured (crossed black circles on blue corners and pink background) with our reference Contact DRS. Figure 12 shows the Contact DRS ${\mu }_s^{\prime }$ and ${\mu }_a$ estimations, specific to the different zones and used as the optimal scattering estimation $\widehat{{\mu }_s^{\prime }}$ and the theoretical absorption ${\mu }_{a,\mathrm{theo}}$ (see Fig. 12) in our MSI ACA-Pro algorithm. Note that the measured blue corners have the same $\widehat{{\mu }_s^{\prime }}$, which is lower to that of the pink background.

Fig. 12

Contact DRS estimated (a) ${\mu }_s^{\prime }$ and (b) ${\mu }_a$ properties of the different zones (background and light blue or dark blue shapes) on the heterogeneous gelatine-based phantom.

Two filters at 550 and 600 nm were chosen for absorption quantification with MSI as they highlight a clear difference between the two chromophores to be analyzed (blue and red ink, see Fig. 12). Images with these filtered lights are acquired and treated as explained in Sec. 2.2. Furthermore, the small specular reflections from the original image are filtered with an algorithm that accords the minimal value of a mask to the central pixel. Filtered images are shown in Fig. 13. Notice the presence of holes (due to air bubbles during the preparation of the phantoms); they will not be considered for quantification.

Fig. 13

Treated images of the heterogeneous phantom under 550 and 600 nm filtered illumination. The image size is $330\times 330\text{pixels}$ corresponding to a FOV of $34.4\times 34.4{\mathrm{mm}}^2$.

The first step in the quantification of wide-field $\widehat{{\mu }_a}$ with MSI is to give a DRS $\widehat{{\mu }_s^{\prime }}$ estimation value to each pixel of the image. The $\widehat{{\mu }_s^{\prime }}$ wide-field map is built with an intensity-based segmentation procedure, based on the high contrast of the 600 nm image and the Otsu method. DRS $\widehat{{\mu }_s^{\prime }}$ values [Fig. 12(a)] are set for each segmented homogeneous zone (background and shapes) as shown in Fig. 14.

Fig. 14

Wide-field $\widehat{{\mu }_s^{\prime }}$ map for 550 and 600 nm. The image size is $330\times 330\text{pixels}$ corresponding to an FOV of $34.4\times 34.4{\mathrm{mm}}^2$.

This quantitative multispectral $\widehat{{\mu }_s^{\prime }}$ map together with the 10-reference base $\mathrm{CF}({\mu }_{s,\mathrm{ref}}^{\prime },{\mu }_{a,\mathrm{ref}})$ (refer to Fig. 5) is used to calibrate with ${\mathrm{CF}}_{\mathrm{interp}}={\mathrm{CF}}_{\mathrm{opt}}(\widehat{{\mu }_s^{\prime }},\widehat{{\mu }_a})$ single pixels of images MSI ACA-Pro (refer to Sec. 3.3). This results in the optimal wide-field $\widehat{{\mu }_a}$ quantification shown in Fig. 15. Notice that these quantitative $\widehat{{\mu }_a}$ maps present values that are in accordance with ${\mu }_{a,\mathrm{theo}}$ estimated with Contact DRS [Fig. 12(b)]. Therefore, these results ultimately validate the wide-field $\widehat{{\mu }_a}$ quantification of unknown heterogeneous samples, based on a priori built $\widehat{{\mu }_s^{\prime }}$ map and the MSI ACA-Pro calibration procedure.

Fig. 15

Wide-field $\widehat{{\mu }_a}$ at 550 and 600 nm shown with two different scales (left to right) to enhance visualization. The results given in this figure are obtained with $T_x=T_y=3\text{pixels}$ [refer to Eq. (2)]. The image size is $110\times 110\text{pixels}$ corresponding to an FOV of $34.4\times 34.4{\mathrm{mm}}^2$.

The analysis of quantitative $\widehat{{\mu }_a}$ maps at 550 and 600 nm is taken a step further with the calculation of blue and red ink concentrations. This is based on linear combinations of spectral unmixing that express the estimated $\widehat{{\mu }_a}$ at each wavelength (${\mu }_{a,550}$ and ${\mu }_{a,600}$) as a linear combination of the relative extinction coefficient ${ϵ}_R$ of red and blue inks multiplied by their corresponding concentrations $C_{\text{red}}$ and $C_{\text{blue}}$ [Eq. (4)]. In this specific case, no theoretical quantification of ${ϵ}_R$ is available so it has been replaced by the reference Contact DRS ${\mu }_a$ signatures of the background and the light blue zones [refer to Fig. 12(b)]

Fig. 16

Relative concentration of (a) red ink (b) blue ink, shown with two different scales for visual enhancement. The results are obtained with $T_x=T_y=3\text{pixels}$ [see Eq. (2)]. The image size is $110\times 110\text{pixels}$ corresponding to a FOV of $34.4\times 34.4{\mathrm{mm}}^2$.