Matrix method to predict the spectral reflectance of stratified surfaces including thick layers and thin films

The most convenient way to assess the color rendering of a coated, painted, or printed surface in various illumination and observation configurations is predict its spectral, angular reflectance using an optical model. Most of the time, such a surface is a stack of layers having different scattering properties and different refractive indices. A general model applicable to the widest range of stratified surfaces is therefore appreciable. This is what we propose in this paper by introducing a method based on light transfer matrices: the transfer matrix representing the stratified surface is the product of the transfer matrices representing the different layers and interfaces composing it, each transfer matrix being expressed in terms of light transfers (e.g. diffuse reflectances and transmittances in the case of diffusing layers). This general model, inspired of models used in the domain of thin films, can be used with stacks of diffusing or nonscattering layers for any illumination-observation geometry. It can be seen, in the case of diffusing layers, as an extension of the Saunderson-corrected Kubelka-Munk model and Kubelka's layering model. We illustrate the through an experimental example including a thin coating, a thick glass plate and a diffusing background. 2. Introduction For a long time, the variation of the spectral properties of surfaces and objects by application of coatings has been a wide subject of investigation for physicians who proposed several models based on specific mathematical formalisms according to the type of physical components and the application domain. In the domain of paints, papers, and other diffusing media, a classical approach is to use the Kubelka-Munk system of two coupled differential equations to describe the propagation of diffuse fluxes in the medium [1,2]. The extension of this model by Kubelka to stacks of paint layers is based on geometrical series describing the multiple reflections and transmissions of these diffuse fluxes between the different layers [3,4]. Geometrical series were also used by Saunderson [5] when deriving his correction of the Kubelka-Munk model in order to account for the internal reflections of light between the paint layer and the paint-air interface, by Clapper and Yule [6] in their reflectance mode

Color and Spectral Mixings in Printed Surfaces

International audienceThe present paper discusses the concept of subtractive color mixing widely used in color hardcopy applications and shows that a more realistic concept would be " spectral mixing " : the physical description of the coloration of light by printed surfaces comes from the mixing of light components selectively absorbed by inks or dyes during their patch within the printing materials. Some classical reflectance equations for continuous tone and halftone prints are reviewed and considered as spectral mixing laws. The challenge of extending these models to new inkless printing processes based on laser radiation is also addressed. Color mixing is a key-concept in color reproduction, either by painting, printing, or displaying. It refers to the observation that a large panel of colors (the color gamut) can be achieved by varying the amount of a limited set of base colors, called primaries. With light emitting systems, the primaries are light sources, often with red, green and blue color, that are either superposed or juxtaposed with a shorter period than the visual acuity. Since the tristimulus values of the produced colors is a linear, additive combination of the tristimulus values of the three primaries, this type of color mixing has been called additive color mixing. This concept, based on Grassman's additivity law, enabled the color matching experiments at the basis of colorimetry [1]. In opposition to the light emitting systems, paintings and printed hardcopies selectively attenuate the incident white light in different proportions according to the wavelength. Layers of primaries, paints or inks, are coated on a reflecting support and play a role of spectral filtering of light. This type of color mixing is improperly called subtractive color mixing [2], by reference to the fact that part of the incident light is removed by filtering, but the tristimulus values of paint or ink mixtures cannot be obtained by combining the tristimulus values of the primaries; it is therefore not a color mixing in the sense of colorimetry. However, the subtractive color mixing is also related to a physical experience, which consists in producing many colors by mixing nonscattering dyes, usually of cyan, magenta and yellow color. According to the Beer-Lambert-Bouguer law [1], the spectral absorption coefficient of the dye mixture, () K λ , is a linear, additive combi-The final publication is available at http://link.springer.co

Assessing the capacity of two-flux models to predict the spectral properties of layered materials

International audienceA classical way of coloring a surface in order to create a still image is the application of a colored coating. The more recent digital printing systems enable depositing thick coatings or successive ink layers. The color rendering of the surface depends on the optical properties of the coated materials (optical index, spectral scattering and absorption coefficients) and their thickness. In order to predict its spectral reflectance as a function of these parameters, the so-called two-flux approach is to be tested in first since the model is simple and relies on analytical equations. It has a good chance to provide accurate predictions for coatings made of solid layers of strongly scattering or nonscattering media, or even complex stratified coatings obtained by stacking nonsymmetrical components such as printed films. The generalized Kubelka-Munk model summarized in this paper enables treating all these configurations with a unified mathematical formalism. But it has limitations and may provide poor color predictions for certain types of layered materials. We therefore propose a simple method based on parameters of the model to check the precision of the two-flux model for a given type of coating

Two-flux transfer matrix model for predicting the reflectance and transmittance of duplex halftone prints

International audienceWe introduce a model allowing convenient calculation of the spectral reflectance and transmittance of duplex prints. It is based on flux transfer matrices and enables retrieving classical Kubelka-Munk formulas, as well as extended formulas for non-symmetric layers. By making different assumptions on the flux transfers, we obtain two predictive models for the duplex halftone prints: The "duplex Clapper-Yule model" which is an extension of the classical Clapper-Yule model, and the "duplex primary reflectance-transmittance model". The two models can be calibrated from either reflectance or transmittance measurements; only the second model can be calibrated from both measurements, thus giving optimal accuracy for both reflectance and transmittance predictions. The conceptual differences between the two models are deeply analyzed, as well as their advantages and drawbacks in terms of calibration. According to the test carried out in this study with paper printed in inkjet, their predictive performances are good provided appropriate calibration options are selected

Flux transfer spectral models for predicting colors of duplex halftone prints

La protection des documents fiduciaires et identitaires contre la fraude exige le développement d’outils de contrôle fondés sur des effets visuels sans cesse renouvelés, difficiles à contrefaire (même pour un expert ... de la contrefaçon !). Ce projet de recherche s’inscrit dans cette problématique et vise à apporter des solutions originales via l’impression de supports diffusants d’une part, et le développement de modèles de rendu visuel d’autre part. Les effets visuels recherchés sont des ajustements de couleurs entre les deux faces d’un imprimé lorsque celui-ci est observé par transparence devant une source lumineuse. Pour obtenir facilement des ajustements de couleurs quelles que soient les couleurs visées, il est capital d’avoir un modèle à disposition, permettant de calculer les quantités d’encre à déposer. Un modèle doit être capable de prédire les facteurs spectraux de réflexion et de transmission du support imprimé en décrivant les phénomènes de diffusion optique présents en pratique dans les couches d’encre et le support. Nous nous intéressons plus particulièrement aux imprimés translucides contenant des couleurs en demi-ton des deux côtés de la surface avec pour objectif de prédire le rendu visuel pour diverses configurations d’observation. Pour cela, nous proposons une nouvelle approche basée sur l’utilisation de matrices de transfert de flux pour prédire les facteurs spectraux de réflexion et de transmission des imprimés lorsqu’ils sont éclairés simultanément des deux côtés. En représentant le comportement optique des différents composants d’un imprimé par des matrices de transfert, la description des transferts de flux entre ces composantes s’en trouve simplifiée. Ce cadre mathématique mène à la construction de modèles de prédiction de couleurs imprimées en demi-ton sur des supports diffusants. Nous montrons par ailleurs que certains modèles existants, comme le modèle de Kubelka-Munk ou encore le modèle de Clapper-Yule, peuvent également être formulés en termes de matrices de transfert. Les résultats obtenus avec les modèles proposés dans ce travail mettent en évidence des qualités de prédiction équivalentes, voire supérieures, à celles qu’on retrouve dans l’état de l’art, tout en proposant une simplification de la formulation mathématique et de la description physique des échanges de flux. Cette simplification fait de ces modèles des outils de calcul qui s’utilisent très facilement, notamment pour la détermination des quantités d’encre à déposer sur les deux faces de l’imprimé afin d’obtenir des ajustements de couleursThe protection of banknotes or identity documents against counterfeiting demands the development of control tools based on visual effects that are continuously renewed. These visual effects become thus difficult to counterfeit even by an expert forger ! This research tries to deal with that issue. Its objective is to bring new solutions using on the one side, the printing of diffusing materials, and on the other side the development of visual rendering models that can be observed. The visual effects that are sought-after are the color matching on both sides of a printed document when observed against thelight. To easily obtain a color matching, whatever the colors that are aimed for, it is essential to have a model that helps in calculating the quantity of ink to be left on the document. A model must be used to predict the spectral reflectance and the transmittance factors of the printed document by describing the phenomena of optical diffusion really present in the ink layers and in the document. We shall focus our interest especially on translucent printed documents that have halftone colors on both sides. Our goal here is to predict the visual rendering in different configurations of observation. To that end, we are offering a new approach based on the use of flux transfer matrices to predict the spectral reflectance and transmittance factors of prints when they are simultaneously lit up on both sides. By representing with transfer matrices the optical behavior of the different components present in a printed document, we see that the description of flux transfer between these elements is thus simplified. This mathematical framework leads to the construction of prediction models of halftone printed colors on diffusing materials. We also show that some existing models, such as the Kubelka-Munk or the Clapper-Yule models, can also be formulated in transfer matrices terms. The results that we get with the models used in this work make apparent identical prediction quality and in some cases even better ones to the ones found in the state of the art, while offering a simplification of the mathematical formulation and the physical description of the flux transfer. This simplification thus transforms these models into calculation tools that can easily be used especially for the choice of quantities of ink that must be left on both sides of the document in order to obtain color matchin

Modèles spectraux à transferts de flux appliqués à la prédiction de couleurs sur des surfaces imprimées en demi-ton

The protection of banknotes or identity documents against counterfeiting demands the development of control tools based on visual effects that are continuously renewed. These visual effects become thus difficult to counterfeit even by an expert forger ! This research tries to deal with that issue. Its objective is to bring new solutions using on the one side, the printing of diffusing materials, and on the other side the development of visual rendering models that can be observed. The visual effects that are sought-after are the color matching on both sides of a printed document when observed against thelight. To easily obtain a color matching, whatever the colors that are aimed for, it is essential to have a model that helps in calculating the quantity of ink to be left on the document. A model must be used to predict the spectral reflectance and the transmittance factors of the printed document by describing the phenomena of optical diffusion really present in the ink layers and in the document. We shall focus our interest especially on translucent printed documents that have halftone colors on both sides. Our goal here is to predict the visual rendering in different configurations of observation. To that end, we are offering a new approach based on the use of flux transfer matrices to predict the spectral reflectance and transmittance factors of prints when they are simultaneously lit up on both sides. By representing with transfer matrices the optical behavior of the different components present in a printed document, we see that the description of flux transfer between these elements is thus simplified. This mathematical framework leads to the construction of prediction models of halftone printed colors on diffusing materials. We also show that some existing models, such as the Kubelka-Munk or the Clapper-Yule models, can also be formulated in transfer matrices terms. The results that we get with the models used in this work make apparent identical prediction quality and in some cases even better ones to the ones found in the state of the art, while offering a simplification of the mathematical formulation and the physical description of the flux transfer. This simplification thus transforms these models into calculation tools that can easily be used especially for the choice of quantities of ink that must be left on both sides of the document in order to obtain color matchingLa protection des documents fiduciaires et identitaires contre la fraude exige le développement d’outils de contrôle fondés sur des effets visuels sans cesse renouvelés, difficiles à contrefaire (même pour un expert ... de la contrefaçon !). Ce projet de recherche s’inscrit dans cette problématique et vise à apporter des solutions originales via l’impression de supports diffusants d’une part, et le développement de modèles de rendu visuel d’autre part. Les effets visuels recherchés sont des ajustements de couleurs entre les deux faces d’un imprimé lorsque celui-ci est observé par transparence devant une source lumineuse. Pour obtenir facilement des ajustements de couleurs quelles que soient les couleurs visées, il est capital d’avoir un modèle à disposition, permettant de calculer les quantités d’encre à déposer. Un modèle doit être capable de prédire les facteurs spectraux de réflexion et de transmission du support imprimé en décrivant les phénomènes de diffusion optique présents en pratique dans les couches d’encre et le support. Nous nous intéressons plus particulièrement aux imprimés translucides contenant des couleurs en demi-ton des deux côtés de la surface avec pour objectif de prédire le rendu visuel pour diverses configurations d’observation. Pour cela, nous proposons une nouvelle approche basée sur l’utilisation de matrices de transfert de flux pour prédire les facteurs spectraux de réflexion et de transmission des imprimés lorsqu’ils sont éclairés simultanément des deux côtés. En représentant le comportement optique des différents composants d’un imprimé par des matrices de transfert, la description des transferts de flux entre ces composantes s’en trouve simplifiée. Ce cadre mathématique mène à la construction de modèles de prédiction de couleurs imprimées en demi-ton sur des supports diffusants. Nous montrons par ailleurs que certains modèles existants, comme le modèle de Kubelka-Munk ou encore le modèle de Clapper-Yule, peuvent également être formulés en termes de matrices de transfert. Les résultats obtenus avec les modèles proposés dans ce travail mettent en évidence des qualités de prédiction équivalentes, voire supérieures, à celles qu’on retrouve dans l’état de l’art, tout en proposant une simplification de la formulation mathématique et de la description physique des échanges de flux. Cette simplification fait de ces modèles des outils de calcul qui s’utilisent très facilement, notamment pour la détermination des quantités d’encre à déposer sur les deux faces de l’imprimé afin d’obtenir des ajustements de couleur

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

International audienceThe 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

Model-based Design of Recto-Verso Prints Displaying Different Images According to the Illuminated Face

International audiencePredicting simultaneously the spectral reflectance and transmittance of halftone prints is now possible thanks to a recently developed model based flux-transfer matrices, called Duplex Primary Reflectance-Transmittance model, valid for single-face printing as well as duplex printing. The model can be calibrated from either spectral reflectance measurements or spectral transmittance measurements; but it can also be calibrated from both measurements by minimizing the distance between the theoretical transfer matrices and experimental transfer matrices. According to the test carried out with paper printed in inkjet, the predictive performances of DPRT model, coupled with the new calibration method, are good enough to permit interesting applications in graphical arts, such as the display of multiple images depending on whether the light source is in front of the duplex color print or beside it

Revisited Yule–Nielsen model without fitting of the n parameter

International audienceWe introduce a so-called " mean-path-defined Yule–Nielsen " (MPD-YN) model for predicting the color of half-tone prints in reflectance or transmittance modes, inspired by the Yule–Nielsen modified spectral Neugebauer model, where the empirical n value is replaced with a spectral parameter different for each halftone, directly calculated thanks to a closed-form formula, a function of the measured spectral reflectances (or accordingly trans-mittances) of full-tone calibration patches and the surface coverages of the Neugebauer primaries in the halftone. This parameter is based on the average number of internal reflections undergone by light between two half layers of the print, whose expression derives from a flux transfer model between the two half layers. According to the tests carried out in this study with paper printed in inkjet, the predictive performances of the MPD-YN model are rather good and very close to those obtained with the Yule–Nielsen model