# Measuring the Optical Properties of Highly Diffuse Materials

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**object appearance representation**, and it focuses on capturing appearance features coupled to a specific geometry of an object. Therefore, these properties are intrinsic to the object considered and cannot be exported directly to a wider scope. It is practical to focus on the object appearance when it comes to assessing the quality of specific objects, or if the object geometry is original in itself. The second approach is the

**material appearance representation**. It consists of studying the properties which define the appearance of the material regardless of the geometrical shape of the object it may compose. Then, they can define the material at an industrial production level. Features such as colour, texture, gloss, translucency, or transparency can be utilized to describe the appearance of a material [12]. These features are frequently correlated with optical properties of the material such as absorption, scattering, transmittance or reflectance at specific wavelengths. They represent how the light interacts with the material surface or how it propagates inside and through the medium. We aim to estimate the optical properties related to absorption and scattering of highly diffuse materials, such as milk, prone to scattering and subsurface scattering effects. Thus, in this work, we consider the material appearance over the object appearance representation.

## 2. Method

#### 2.1. Acquisition

#### 2.2. Inversion

## 3. Results and Validation on Milk

#### 3.1. Q-Dataset Results

#### 3.2. Mix-Dataset Results

#### 3.3. Correlation with the Fat Content

#### 3.4. Comparison with the Literature

#### 3.5. Repeatability of Measurements

#### 3.6. Discussing the Inversion Method

## 4. Other Highly Diffuse Materials

#### 4.1. Optical Properties of Paper

#### 4.1.1. Vertical Dataset Results

#### 4.1.2. Horizontal Dataset Results

#### 4.1.3. Differences in Estimates between the Two Datasets

#### 4.1.4. Correlation with the Whiteness Index

#### 4.2. Optical Properties of White Paint Mixed with Water

#### 4.2.1. Results

#### 4.2.2. Correlation with the Concentration of White Pigments

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Sun, C.C.; Chien, W.T.; Moreno, I.; Hsieh, C.T.; Lin, M.C.; Hsiao, S.L.; Lee, X.H. Calculating model of light transmission efficiency of diffusers attached to a lighting cavity. Opt. Express
**2010**, 18, 6137–6148. [Google Scholar] [CrossRef] - Leyre, S.; Leloup, F.; Audenaert, J.; Durinck, G.; Hofkens, J.; Deconinck, G.; Hanselaer, P. Determination of the bulk scattering parameters of diffusing materials. Appl. Opt.
**2013**, 52, 4083–4090. [Google Scholar] [CrossRef] [Green Version] - Cheong, W.; Prahl, S.; Welch, A.J. A review of the optical properties of biological tissues. IEEE J. Quantum Electron.
**1990**, 26, 2166–2185. [Google Scholar] [CrossRef] [Green Version] - Gobin, L.; Blanchot, L.; Saint-Jalmes, H. Integrating the digitized backscattered image to measure absorption and reduced-scattering coefficients in vivo. Appl. Opt.
**1999**, 38, 4217–4227. [Google Scholar] [CrossRef] [PubMed] - Hyde, D.E.; Farrell, T.J.; Patterson, M.S.; Wilson, B.C. A diffusion theory model of spatially resolved fluorescence fromdepth-dependent fluorophore concentrations. Phys. Med. Biol.
**2001**, 46, 369. [Google Scholar] [CrossRef] [PubMed] - Zhang, R.; Verkruysse, W.; Choi, B.; Viator, J.A.; Jung, B.; Svaasand, L.O.; Aguilar, G.; Nelson, J.S. Determination of human skin optical properties from spectrophotometric measurements based on optimization by genetic algorithms. J. Biomed. Opt.
**2005**, 10, 024030. [Google Scholar] [CrossRef] [Green Version] - Qin, J.; Lu, R. Measurement of the Absorption and Scattering Properties of Turbid Liquid Foods Using Hyperspectral Imaging. Appl. Spectrosc.
**2007**, 61, 388–396. [Google Scholar] [CrossRef] - Qin, J.; Lu, R. Measurement of the optical properties of fruits and vegetables using spatially resolved hyperspectral diffuse reflectance imaging technique. Postharvest Biol. Technol.
**2008**, 49, 355–365. [Google Scholar] [CrossRef] - Hu, D.; Fu, X.; Wang, A.; Ying, Y. Measurement Methods for Optical Absorption and Scattering Properties of Fruits and Vegetables. Trans. ASABE
**2015**, 58, 1387–1401. [Google Scholar] [CrossRef] - Guarnera, G.C.; Ghosh, A.; Hall, I.; Glencross, M.; Guarnera, D. Material Capture and Representation with Applications in Virtual Reality. In Proceedings of the SIGGRAPH’17: Special Interest Group on Computer Graphics and Interactive Techniques Conference, Los Angeles, CA, USA, 30 July–3 August 2017. [Google Scholar] [CrossRef]
- Tong, X.; Wang, J.; Lin, S.; Guo, B.; Shum, H.Y. Modeling and Rendering of Quasi-Homogeneous Materials. ACM Trans. Graph.
**2005**, 24, 1054–1061. [Google Scholar] [CrossRef] - ASTM E284-22; Standard Terminology of Appearance. ASTM International: West Conshohocken, PA, USA, 2022. [CrossRef]
- Nicodemus, F.E.; Richmond, J.C.; Hsia, J.J.; Ginsberg, I.W.; Limperis, T. Geometrical Considerations and Nomenclature for Reflectance; National Bureau of Standards: Gaithersburg, MD, USA, 1977. [Google Scholar]
- Doornbos, R.; Lang, R.; Aalders, M.; Cross, F.; Sterenborg, H. The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy. Phys. Med. Biol.
**1999**, 44, 967–981. [Google Scholar] [CrossRef] [PubMed] - Igarashi, T.; Nishino, K.; Nayar, S.K. The Appearance of Human Skin: A Survey. Found. Trends. Comput. Graph. Vis.
**2007**, 3, 1–95. [Google Scholar] [CrossRef] - Farrell, T.J.; Patterson, M.S.; Wilson, B. A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo. Med. Phys.
**1992**, 19, 879–888. [Google Scholar] [CrossRef] - Frisvad, J.R.; Jensen, S.A.; Madsen, J.S.; Correia, A.; Yang, L.; Gregersen, S.K.S.; Meuret, Y.; Hansen, P.E. Survey of Models for Acquiring the Optical Properties of Translucent Materials. Comput. Graph. Forum
**2020**, 39, 729–755. [Google Scholar] [CrossRef] - Kubelka, P.; Munk, F. Ein Beitrag zur Optik der Farbanstriche. Z. Tech. Phys.
**1931**, 12, 593–601. [Google Scholar] - Maheu, B.; Gouesbet, G. Four-flux models to solve the scattering transfer equation: Special cases. Appl. Opt.
**1986**, 25, 1122–1128. [Google Scholar] [CrossRef] - Prahl, S.A.; van Gemert, M.J.C.; Welch, A.J. Determining the optical properties of turbid media by using the adding-doubling method. Appl. Opt.
**1993**, 32, 559–568. [Google Scholar] [CrossRef] - Bashkatov, A.N.; Genina, E.A.; Tuchin, V.V. Optical Properties of Skin, Subcutaneous, and Muscle tissues: A Review. J. Innov. Opt. Health Sci.
**2011**, 4, 9–38. [Google Scholar] [CrossRef] - Tuchin, V. Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnostics; SPIE Press: Bellingham, WA, USA, 2015. [Google Scholar] [CrossRef]
- Munoz, A.; Masia, B.; Tolosa, A.; Gutierrez, D. Single-image appearance acquisition using genetic algorithms. Proc. Comput. Graph. Vis. Comput. Vis. Image Process.
**2009**, 24–32. [Google Scholar] - Munoz, A.; Echevarria, J.I.; Seron, F.J.; Lopez-Moreno, J.; Glencross, M.; Gutierrez, D. BSSRDF Estimation from Single Images. Comput. Graph. Forum
**2011**, 30, 455–464. [Google Scholar] [CrossRef] [Green Version] - Peers, P.; vom Berge, K.; Matusik, W.; Ramamoorthi, R.; Lawrence, J.; Rusinkiewicz, S.; Dutré, P. A Compact Factored Representation of Heterogeneous Subsurface Scattering. ACM Trans. Graph.
**2006**, 25, 746–753. [Google Scholar] [CrossRef] - Kienle, A.; Lilge, L.; Patterson, M.S.; Hibst, R.; Steiner, R.; Wilson, B.C. Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue. Appl. Opt.
**1996**, 35, 2304–2314. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kienle, A.; Patterson, M.S. Determination of the optical properties of turbid media from a single Monte Carlo simulation. Phys. Med. Biol.
**1996**, 41, 2221. [Google Scholar] [CrossRef] [Green Version] - Kienle, A.; Patterson, M.S. Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to the source. Phys. Med. Biol.
**1997**, 42, 1801. [Google Scholar] [CrossRef] [PubMed] - Stam, J. Multiple scattering as a diffusion process. In Rendering Techniques’ 95, Proceedings of the Eurographics Workshop, Dublin, Ireland, 12–14 June 1995; Patrick, M., Purgathofer, W., Eds.; Springer: Vienna, Austria, 1995; pp. 41–50. [Google Scholar] [CrossRef]
- Jensen, H.W.; Marschner, S.R.; Levoy, M.; Hanrahan, P. A Practical Model for Subsurface Light Transport. In Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, Association for Computing Machinery, Los Angeles, CA, USA, 12–17 August 2001; pp. 511–518. [Google Scholar] [CrossRef]
- Donner, C.; Jensen, H.W. Light Diffusion in Multi-Layered Translucent Materials. ACM Trans. Graph.
**2005**, 24, 1032–1039. [Google Scholar] [CrossRef] - Nichols, M.G.; Hull, E.L.; Foster, T.H. Design and testing of a white-light, steady-state diffuse reflectance spectrometer for determination of optical properties of highly scattering systems. Appl. Opt.
**1997**, 36, 93–104. [Google Scholar] [CrossRef] - Stocker, S.; Foschum, F.; Krauter, P.; Bergmann, F.; Hohmann, A.; Happ, C.S.; Kienle, A. Broadband Optical Properties of Milk. Appl. Spectrosc.
**2017**, 71, 951–962. [Google Scholar] [CrossRef] - Aernouts, B.; Van Beers, R.; Watté, R.; Huybrechts, T.; Jordens, J.; Vermeulen, D.; Van Gerven, T.; Lammertyn, J.; Saeys, W. Effect of ultrasonic homogenization on the Vis/NIR bulk optical properties of milk. Colloid Surf. B Biointerfaces
**2015**, 126, 510–519. [Google Scholar] [CrossRef] [Green Version] - Wabnitz, H.; Rinneberg, H. Imaging in turbid media by photon density waves: Spatial resolution and scaling relations. Appl. Opt.
**1997**, 36, 64–74. [Google Scholar] [CrossRef] - Abildgaard, O.H.A.; Kamran, F.; Dahl, A.B.; Skytte, J.L.; Nielsen, F.D.; Thomsen, C.L.; Andersen, P.E.; Larsen, R.; Frisvad, J.R. Non-Invasive Assessment of Dairy Products Using Spatially Resolved Diffuse Reflectance Spectroscopy. Appl. Spectrosc.
**2015**, 69, 1096–1105. [Google Scholar] [CrossRef] [Green Version] - Bahadi, M.; Ismail, A.A.; Vasseur, E. Fourier Transform Infrared Spectroscopy as a Tool to Study Milk Composition Changes in Dairy Cows Attributed to Housing Modifications to Improve Animal Welfare. Foods
**2021**, 10, 450. [Google Scholar] [CrossRef] [PubMed] - Blinov, A.; Siddiqui, S.; Blinova, A.; Khramtsov, A.; Oboturova, N.; Nagdalian, A.; Simonov, A.; Ibrahim, S. Analysis of the dispersed composition of milk using photon correlation spectroscopy. J. Food Compos. Anal.
**2022**, 108, 104414. [Google Scholar] [CrossRef] - Farrell, T.J.; Wilson, B.C.; Patterson, M.S. The use of a neural network to determine tissue optical properties from spatially resolved diffuse reflectance measurements. Phys. Med. Biol.
**1992**, 37, 2281. [Google Scholar] [CrossRef] [PubMed] - Bevilacqua, F.; Depeursinge, C. Monte Carlo study of diffuse reflectance at source–detector separations close to one transport mean free path. J. Opt. Soc. Am. A
**1999**, 16, 2935–2945. [Google Scholar] [CrossRef] - Fabritius, T.; Saarela, J.; Myllyla, R. Determination of the refractive index of paper with clearing agents. In International Conference on Lasers, Applications, and Technologies 2005: High-Power Lasers and Applications; SPIE: Bellingham, WA, USA, 2006; Volume 6053, p. 60530X. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of the process operated by the TLS850. The LED sends light vertically towards the surface of the material, then the light is scattered in it. The photodiode array collects the light coming out of the material at various distances from the entry point in the material.

**Figure 2.**Contour plot of an arbitrary study of the cost function. The lines show the potential (${\mu}_{a},{\mu}_{s}^{\prime}$) solutions minimizing the cost function. An ideal solution would be to find an ellipse made of these contours, which is not the case in this study.

**Figure 3.**Reflectance profile for the milk sample containing 0.5% of fat in the Q-dataset. ${R}^{2}$ values for the fit of R, G, and B channels are all 0.90.

**Figure 4.**Reflectance profile for the milk sample containing 1.2% fat in the mix-dataset. ${R}^{2}$ values for the fit of R, G, and B channels are, respectively, 0.91, 0.90 and 0.89.

**Figure 5.**Linear regression for the absorption coefficient of milk versus the fat content with 95% confidence intervals (dashed lines). Star symbols (*) represent the value of the coefficient with the uncertainty bar.

**Figure 6.**Linear regression for the reduced scattering coefficient of milk versus the fat content with 95% confidence intervals (dashed lines). Star symbols (*) represent the value of the coefficient with the uncertainty bar.

**Figure 7.**Reflectance profile for the paper 120 g/m${}^{2}$ in the Vertical dataset. ${R}^{2}$ values for the fit of R, G, and B channels are, respectively, 0.70, 0.72, and 0.73.

**Figure 8.**Reflectance profile for the paper 80 g/m${}^{2}$ in the Horizontal dataset. ${R}^{2}$ values for the fit of R, G, and B channels are, respectively, 0.73, 0.75, and 0.75.

**Figure 9.**Linear regression for the absorption coefficient of white paint versus the drops number with 95% confidence intervals. Star symbols (*) represent the value of the coefficient with the uncertainty bar.

**Figure 10.**Linear regression for the reduced scattering coefficient of white paint versus the drops number with 95% confidence intervals. Star symbols (*) represent the value of the coefficient with the uncertainty bar.

Fat Content | ${\mathit{\mu}}_{\mathit{a}}$ (mm${}^{-1}$) | ${\mathit{\mu}}_{\mathit{s}}^{\prime}$ (mm${}^{-1}$) | ||||
---|---|---|---|---|---|---|

R | G | B | R | G | B | |

0.1% | 3.2 × 10${}^{-9}$ | 2.4 × 10${}^{-3}$ | 1.0 × 10${}^{-2}$ | 0.62 | 0.74 | 0.81 |

0.5% | 1.4 × 10${}^{-3}$ | 3.7 × 10${}^{-3}$ | 1.2 × 10${}^{-2}$ | 0.62 | 0.75 | 0.84 |

1% | 3.0 × 10${}^{-3}$ | 7.8 × 10${}^{-3}$ | 1.6 × 10${}^{-2}$ | 0.69 | 0.84 | 0.95 |

4% | 5.3 × 10${}^{-3}$ | 1.5 × 10${}^{-2}$ | 3.6 × 10${}^{-2}$ | 1.05 | 1.35 | 1.52 |

Fat Content | ${\mathit{\mu}}_{\mathit{a}}$ (mm${}^{-1}$) | ${\mathit{\mu}}_{\mathit{s}}^{\prime}$ (mm${}^{-1}$) | ||||
---|---|---|---|---|---|---|

R | G | B | R | G | B | |

0.1% | 1.7 × 10${}^{-5}$ | 3.4 × 10${}^{-3}$ | 8.6 × 10${}^{-3}$ | 0.60 | 0.70 | 0.78 |

0.5% | 1.2 × 10${}^{-3}$ | 4.0 × 10${}^{-3}$ | 1.3 × 10${}^{-2}$ | 0.61 | 0.75 | 0.83 |

0.7% | 1.9 × 10${}^{-3}$ | 5.0 × 10${}^{-3}$ | 1.4 × 10${}^{-2}$ | 0.67 | 0.82 | 0.91 |

1% | 3.3 × 10${}^{-3}$ | 6.6 × 10${}^{-3}$ | 1.9 × 10${}^{-2}$ | 0.68 | 0.84 | 0.92 |

1.2% | 4.3 × 10${}^{-3}$ | 5.8 × 10${}^{-3}$ | 1.9 × 10${}^{-2}$ | 0.71 | 0.90 | 0.98 |

3.5% | 5.1 × 10${}^{-3}$ | 1.2 × 10${}^{-2}$ | 2.7 × 10${}^{-2}$ | 0.92 | 1.26 | 1.40 |

4% | 5.3 × 10${}^{-3}$ | 1.1 × 10${}^{-2}$ | 3.5 × 10${}^{-2}$ | 0.98 | 1.36 | 1.51 |

Channel | R | G | B | |
---|---|---|---|---|

${\mu}_{a}$ | ${R}^{2}$ | 0.86 | 0.94 | 0.95 |

p-value | 2.8 × 10${}^{-3}$ | 3.5 × 10${}^{-4}$ | 2.2 × 10${}^{-4}$ | |

${\mu}_{s}^{\prime}$ | ${R}^{2}$ | 0.98 | 1.00 | 1.00 |

p-value | 1.6 × 10${}^{-6}$ | 7.1 × 10${}^{-8}$ | 3.2 × 10${}^{-8}$ |

**Table 4.**Relative difference of estimated optical values between the two datasets. The Q-dataset is considered as reference value.

Fat Content | Relative Difference of ${\mathit{\mu}}_{\mathit{a}}$ | Relative Difference of ${\mathit{\mu}}_{\mathit{s}}^{\prime}$ | ||||
---|---|---|---|---|---|---|

R | G | B | R | G | B | |

0.1% | 46.9% | −41.7% | 14% | 3.2% | 5.4% | 3.7% |

0.5% | 14.3% | −8.1% | −8.3% | 1.6% | 0% | 1.2% |

1% | −10% | 15.4% | −18.8% | 1.4% | 0% | 3.2% |

4% | 0% | 26.7% | 2.8% | 6.7% | −0.7% | 0.7% |

Weight (g/m${}^{2}$) | ${\mathit{\mu}}_{\mathit{a}}$ (mm${}^{-1}$) | ${\mathit{\mu}}_{\mathit{s}}^{\prime}$ (mm${}^{-1}$) | ||||
---|---|---|---|---|---|---|

R | G | B | R | G | B | |

80 | 6.62 × 10${}^{-2}$ | 8.51 × 10${}^{-2}$ | 8.38 × 10${}^{-2}$ | 3.13 | 2.61 | 2.69 |

100 | 4.83 × 10${}^{-2}$ | 7.96 × 10${}^{-2}$ | 7.65 × 10${}^{-2}$ | 3.45 | 2.86 | 2.91 |

120 | 5.70 × 10${}^{-2}$ | 7.75 × 10${}^{-2}$ | 8.21 × 10${}^{-2}$ | 3.37 | 2.81 | 2.86 |

160 | 3.86 × 10${}^{-2}$ | 5.41 × 10${}^{-2}$ | 5.44 × 10${}^{-2}$ | 3.59 | 3.08 | 3.19 |

200 | 3.51 × 10${}^{-2}$ | 5.90 × 10${}^{-2}$ | 5.27 × 10${}^{-2}$ | 3.63 | 3.00 | 3.16 |

250 | 4.20 × 10${}^{-2}$ | 7.14 × 10${}^{-2}$ | 5.45 × 10${}^{-2}$ | 3.51 | 2.77 | 3.09 |

Weight (g/m${}^{2}$) | ${\mathit{\mu}}_{\mathit{a}}$ (mm${}^{-1}$) | ${\mathit{\mu}}_{\mathit{s}}^{\prime}$ (mm${}^{-1}$) | ||||
---|---|---|---|---|---|---|

R | G | B | R | G | B | |

80 | 6.34 × 10${}^{-2}$ | 8.06 × 10${}^{-2}$ | 8.00 × 10${}^{-2}$ | 2.97 | 2.55 | 2.56 |

100 | 5.07 × 10${}^{-2}$ | 8.28 × 10${}^{-2}$ | 7.76 × 10${}^{-2}$ | 3.33 | 2.80 | 2.80 |

120 | 5.43 × 10${}^{-2}$ | 6.41 × 10${}^{-2}$ | 8.15 × 10${}^{-2}$ | 3.29 | 2.94 | 2.88 |

160 | 4.45 × 10${}^{-2}$ | 5.67 × 10${}^{-2}$ | 5.97 × 10${}^{-2}$ | 3.43 | 3.04 | 3.08 |

200 | 4.33 × 10${}^{-2}$ | 6.77 × 10${}^{-2}$ | 6.83 × 10${}^{-2}$ | 3.48 | 2.97 | 3.04 |

250 | 4.26 × 10${}^{-2}$ | 7.07 × 10${}^{-2}$ | 7.05 × 10${}^{-2}$ | 3.52 | 2.87 | 3.07 |

**Table 7.**Relative difference of reduced scattering values between the Vertical and Horizontal datasets. The Vertical dataset is taken as a reference.

Weight (g/m${}^{2}$) | Relative Difference of ${\mathit{\mu}}_{\mathit{s}}^{\prime}$ | ||
---|---|---|---|

R | G | B | |

80 | 5.1% | 2.3% | 4.8% |

100 | 3.5% | 2.1% | 3.8% |

120 | 2.4% | −4.6% | −0.7% |

160 | 4.5% | 1.3% | 3.4% |

200 | 4.1% | 1.0% | 3.8% |

250 | −0.3% | −3.6% | 0.6% |

Weight (g/m${}^{2}$) | 80 | 100 | 120 | 160 | 200 | 250 |
---|---|---|---|---|---|---|

WI CIE | 133.4 | 135.1 | 137.4 | 140.1 | 141.2 | 142.6 |

WI E313 | 23.06 | 25.92 | 29.13 | 32.32 | 34.02 | 34.45 |

WI CIE | WI E313 | ||||||
---|---|---|---|---|---|---|---|

R | G | B | R | G | B | ||

${\mu}_{a}$ (ver) | ${R}^{2}$ | 0.72 | 0.43 | 0.79 | 0.76 | 0.49 | 0.77 |

p-value | 0.03 | 0.16 | 0.02 | 0.02 | 0.12 | 0.02 | |

${\mu}_{a}$ (hor) | ${R}^{2}$ | 0.84 | 0.58 | 0.24 | 0.88 | 0.63 | 0.29 |

p-value | 0.01 | 0.08 | 0.33 | 0.006 | 0.06 | 0.27 | |

${\mu}_{s}^{\prime}$ (ver) | ${R}^{2}$ | 0.67 | 0.30 | 0.79 | 0.75 | 0.39 | 0.84 |

p-value | 0.05 | 0.26 | 0.02 | 0.03 | 0.18 | 0.01 | |

${\mu}_{s}^{\prime}$ (hor) | ${R}^{2}$ | 0.78 | 0.58 | 0.86 | 0.82 | 0.69 | 0.91 |

p-value | 0.02 | 0.08 | 0.007 | 0.01 | 0.04 | 0.003 |

Drops | ${\mathit{\mu}}_{\mathit{a}}$ (mm${}^{-1}$) | ${\mathit{\mu}}_{\mathit{s}}^{\prime}$ (mm${}^{-1}$) | ||||
---|---|---|---|---|---|---|

R | G | B | R | G | B | |

2 | 6.1 × 10${}^{-12}$ | 2.8 × 10${}^{-11}$ | 1.5 × 10${}^{-11}$ | 0.45 | 0.46 | 0.50 |

4 | 2.0 × 10${}^{-12}$ | 1.4 × 10${}^{-13}$ | 1.4 × 10${}^{-11}$ | 0.40 | 0.43 | 0.44 |

6 | 2.3 × 10${}^{-5}$ | 6.4 × 10${}^{-5}$ | 9.3 × 10${}^{-5}$ | 0.43 | 0.48 | 0.48 |

8 | 1.1 × 10${}^{-10}$ | 3.8 × 10${}^{-5}$ | 6.5 × 10${}^{-5}$ | 0.44 | 0.49 | 0.49 |

10 | 1.3 × 10${}^{-4}$ | 3.5 × 10${}^{-4}$ | 5.5 × 10${}^{-4}$ | 0.47 | 0.53 | 0.52 |

12 | 5.7 × 10${}^{-4}$ | 1.0 × 10${}^{-3}$ | 1.4 × 10${}^{-3}$ | 0.51 | 0.56 | 0.56 |

14 | 1.0 × 10${}^{-3}$ | 1.1 × 10${}^{-3}$ | 1.8 × 10${}^{-3}$ | 0.53 | 0.60 | 0.59 |

16 | 1.5 × 10${}^{-3}$ | 1.1 × 10${}^{-3}$ | 2.3 × 10${}^{-3}$ | 0.55 | 0.64 | 0.63 |

18 | 2.0 × 10${}^{-3}$ | 1.7 × 10${}^{-3}$ | 3.0 × 10${}^{-3}$ | 0.56 | 0.68 | 0.65 |

20 | 2.7 × 10${}^{-3}$ | 2.0 × 10${}^{-3}$ | 3.9 × 10${}^{-3}$ | 0.57 | 0.72 | 0.68 |

Channel | R | G | B | |
---|---|---|---|---|

${\mu}_{a}$ | ${R}^{2}$ | 0.93 | 0.96 | 0.97 |

p-value | 1.2 × 10${}^{-4}$ | 2.6 × 10${}^{-5}$ | 9.2 × 10${}^{-6}$ | |

${\mu}_{s}^{\prime}$ | ${R}^{2}$ | 0.97 | 0.99 | 0.99 |

p-value | 9.8 × 10${}^{-6}$ | 3.1 × 10${}^{-7}$ | 7.7 × 10${}^{-8}$ |

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**MDPI and ACS Style**

Nguyen, M.; Thomas, J.-B.; Farup, I.
Measuring the Optical Properties of Highly Diffuse Materials. *Sensors* **2023**, *23*, 6853.
https://doi.org/10.3390/s23156853

**AMA Style**

Nguyen M, Thomas J-B, Farup I.
Measuring the Optical Properties of Highly Diffuse Materials. *Sensors*. 2023; 23(15):6853.
https://doi.org/10.3390/s23156853

**Chicago/Turabian Style**

Nguyen, Mathieu, Jean-Baptiste Thomas, and Ivar Farup.
2023. "Measuring the Optical Properties of Highly Diffuse Materials" *Sensors* 23, no. 15: 6853.
https://doi.org/10.3390/s23156853