Fundamentals of Optics and Radiometry for Color Reproduction

Hébert, Mathieu; Hersch, Roger D.; Emmel, Patrick

doi:10.1002/9781118798706.hdi062

Cited by 14 publications

(23 citation statements)

References 61 publications

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“…The Kubelka–Munk theory is a mathematical model which predicts the spectral reflectance of a layer of diffusing material on a background knowing the spectral reflectance of the background and the physical parameters of the diffusing material, namely the spectral scattering coefficient S( λ ) and spectral absorption coefficient K( λ ) . The intrinsic reflectance

italicρ

and the intrinsic transmittance

italicτ

of a layer of thickness h are given by the following formulas, applicable to each wavelength of light:

ρ = \frac{sinh (bSh)}{asinh (bSh) + bcosh (bSh)}

and

τ = \frac{b}{asinh (bSh) + bcosh (bSh)}

with

a = \frac{K + S}{S}

and

b = \sqrt{a^{2} - 1}

.…”

Section: Methodsmentioning

confidence: 99%

“…The predicted spectral reflectance

R_{normalKM} false(italicλ false)

was obtained by the following mathematical formula derived from the Kubelka–Munk theory with a Saunderson correction in order to take into account the reflections and transmissions of light at the polish–air interface :

R_{normalKM} false(italicλ false) = \frac{T_{normalin} T_{normalout} {italicρ}_{n + p} false(italicλ false)}{1 - r_{i} {italicρ}_{n + p} false(italicλ false)}

where

{italicρ}_{n + p} false(italicλ false)

is the intrinsic spectral reflectance of the system {nail + polish} which depends on the colour of the nail and on the physical parameters of the nail polish (i.e. intrinsic reflectance, intrinsic transmittance and thickness) and where

r_{i}

T_{in}

and

T_{out}

are constant values corresponding to the reflectances and transmittances of the polish–air interface, issued from the Fresnel formulae , depending on the polish refractive index (assumed to be close to 1.47) and the measuring geometry. For the d:8° geometry that has been considered in the measurements, we have:

r_{i} = 0.58

T_{normalin} = 0.91

and

T_{normalout} = \frac{0.96}{{1.47}^{2}}

.…”

Section: Methodsmentioning

confidence: 99%

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Prediction of a nail polish colour applied on a nail

Monpeurt

Cinotti

Razafindrakoto

et al. 2017

Intern J of Cosmetic Sci

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OBJECTIVE: The colour of a nail polish varies according to the nail on which it is applied. The objective of this study was to predict the colour of the nail polish on a given nail and to study how the colour varies depending on the nail polish thickness. METHODS: Six nail polishes were applied in one, two and three layers on the nails of one subject, thus forming eighteen samples. The spectral reflectances of the eighteen nail polishes applied on the nails with different thicknesses were obtained by spectrophotometry. The spectral reflectances of the nails without polish were also measured using the same technique. The thicknesses of nail polishes were measured by high-definition optical coherence tomography (HD-OCT). Then, to determine the physical parameters of the nail polish itself, we applied the six nail polishes on an opacity drawdown chart and we measured the spectral reflectance and the thickness of each patch using spectrophotometry and HD-OCT, respectively. The Kubelka-Munk theory was used to get the predicted spectral reflectance of the nail polish applied on the nail according to the polish thickness by knowing the parameter of the polish itself and the spectral reflectance of the nail. The predicted spectral reflectances were finally compared with those measured directly on the nails. RESULTS: The predicted spectral reflectances were rather close to measured ones. Consequently, knowing the colour of the nail without polish and the optical parameters of the nail polish itself, we can estimate the colour of the nail polish applied on the nail depending on its thickness. CONCLUSION: Our study showed that the Kubelka-Munk theory can be used to predict the nail polish colour. The ability to predict the real colour of a nail polish applied on a nail could help a nail polish manufacturer to improve his polish formulae in order to obtain a precise colour.R esum e OBJECTIF: La couleur d'un vernis a ongles peut varier selon l'ongle sur lequel celui-ci est appliqu e. L'objectif de cette etude etait de pr edire la couleur d'un vernis a ongles lorsque ce dernier est appliqu e sur un ongle donn e et d' etudier comment la couleur varie en fonction de l' epaisseur du vernis. METHODES: Six vernis a ongles ont et e d epos es en une, deux et trois couches sur les ongles d'un individu, formant ainsi dix-huit echantillons. Les r eflectances spectrales des dix-huit couches de vernis a ongles d' epaisseurs diff erentes appliqu ees sur les ongles ont et e obtenues par spectrophotom etrie. Les r eflectances spectrales des ongles sans vernis ont egalement et e mesur ees par la même technique. Les epaisseurs des vernis a ongles ont et e mesur ees par tomographie par coh erence optique haute d efinition (HD-OCT). Ensuite, afin d'acqu erir les param etres physiques propres du vernis a ongles, nous avons appliqu e les six vernis a ongles sur un papier diffusant et nous avons mesur e la r eflectance spectrale et l' epaisseur de chaque echantillon en utilisant respectivement un spectrophotom etre et un HD-OCT. Les coule...

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italicρ

and the intrinsic transmittance

italicτ

of a layer of thickness h are given by the following formulas, applicable to each wavelength of light:

ρ = \frac{sinh (bSh)}{asinh (bSh) + bcosh (bSh)}

and

τ = \frac{b}{asinh (bSh) + bcosh (bSh)}

with

a = \frac{K + S}{S}

and

b = \sqrt{a^{2} - 1}

.…”

Section: Methodsmentioning

confidence: 99%

“…The predicted spectral reflectance

R_{normalKM} false(italicλ false)

R_{normalKM} false(italicλ false) = \frac{T_{normalin} T_{normalout} {italicρ}_{n + p} false(italicλ false)}{1 - r_{i} {italicρ}_{n + p} false(italicλ false)}

where

{italicρ}_{n + p} false(italicλ false)

r_{i}

T_{in}

and

T_{out}

r_{i} = 0.58

T_{normalin} = 0.91

and

T_{normalout} = \frac{0.96}{{1.47}^{2}}

.…”

Section: Methodsmentioning

confidence: 99%

Prediction of a nail polish colour applied on a nail

Monpeurt

Cinotti

Razafindrakoto

et al. 2017

Intern J of Cosmetic Sci

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“…By integrating these differential equations [31], one obtains the following expressions for the reflectance R and the transmittance T of the layer:…”

Section: Kubelka-munk Model and Saunderson Correctionmentioning

confidence: 99%

“…When the layer is on top of an opaque background with reflectance ρg, the reflectance becomes [31] ( )…”

Section: Kubelka-munk Model and Saunderson Correctionmentioning

confidence: 99%

Model-based Skin Pigment Cartography by High-Resolution Hyperspectral Imaging

Séroul¹,

Hbert²,

Cherel³

et al. 2016

jist

Self Cite

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“…The four-flux model can be expressed by using a matrix formalism as suggested by Rozé et al in the case of multilayer [21]. Recent formulations [22,23] also enable predicting interface effects. The four-flux model can be used for various scattering systems as illustrated by recent publications [24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Multilayer four-flux matrix model accounting for directional-diffuse light transfers

et al. 2015

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The four-flux model is a method to solve light radiative transfer problems in planar, possibly multilayer structures. The light fluxes are modeled as two collimated and two diffuse beams propagating forwards and backwards perpendicularly to the layer stack. In the present contribution, we develop a four-flux model relying on a matrix formalism to determine the reflectance and transmittance factors of stacks of components by knowing those of each individual component. This model is also extended to generate the bidirectional scattering distribution function (BSDF) of the stack by considering an incoming collimated flux in any direction, and by taking into account the directionality of the diffuse fluxes exiting from the material at the border components of the stack. The model is applied to opaque Lambertian backgrounds with flat or rough interface, for which analytical expressions of the BSDF are obtained.

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Fundamentals of Optics and Radiometry for Color Reproduction

Cited by 14 publications

References 61 publications

Prediction of a nail polish colour applied on a nail

Prediction of a nail polish colour applied on a nail

Model-based Skin Pigment Cartography by High-Resolution Hyperspectral Imaging

Multilayer four-flux matrix model accounting for directional-diffuse light transfers

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