Two-flux and multiflux matrix models for colored surfaces

International audienceThis paper presents various extensions of the so-called two-flux models for prediction of reflectance and transmittance of diffusing media, i.e. the ubelka-Munk model, and the extension of Kubelka-Munk for stacks of diffusing layers. A first matrix formulation of the Kubelka-Munk differential equations leads to a matrix framework based on transfer matrices, which can be extended to stacks of diffusing layers, stacks of nonscattering films, and stacks of scattering and non-scatterings films as a generalization of the Williams-Clapper model for prediction of the reflectance of paper photographs, each of these configurations being illustrated through various examples. This paper also exposes the limitsof the two flux approach and shows that the matrix formalism extends in a straightforward manner to multiflux models, where the size of the matrices is increased

Modèles de prédiction couleur appliqués à l'impression jet d'encre

In order to guarantee good colour reproduction, modern printers use very large calibration tables based on the measurement of large sample sets. In this thesis, we aim at establishing an accurate colour prediction model which is able to compute the spectra of theses samples without printing and measuring them. Knowing the halftoning algorithm and the physical properties of the inks and the paper, our model can predict the spectra of printed samples. It was applied to ink-jet printers. We set up a new mathematical formulation which expresses the problem in a more general framework that simplifies calculations and reasoning. Our model combines various phenomena which were treated separately until now. Furthermore, several particular cases lead to classical solutions known in the literature. The new mathematical framework simplifies the study of media composed of superposed uniform layers. We show that the classical Kubelka-Munk problem is solved by computing the exponential of a matrix, and that the case of stratified media with varying absorption and scattering coefficients is addressed using the perturbation method. A refractive surface crossing matrix modelizes multiple internal reflections caused by a change of the refractive index. The Saunderson correction formula can easily be derived from this matrix. Our formulation allows also to handle fluorescence and predicts the spectra of fluorescent inks applied on transparency or on paper. We demonstrate that the mixing of the inks and the sequence in which light passes through the layers have an important influence on the resulting spectrum. Considering halftoned samples, we have generalized Neugebauer's 7-primaries model in order to take an infinite number of primaries into account. This allowed us to split the prediction problem into a geometric aspect and a spectral aspect. The geometrical part is addressed by the use of a large pixel grid on which the ink impacts are simulated, and the spectral part results from the study of superposed uniform layers. In this framework, light scattering is expressed in a probabilistic way and concerns only the geometrical aspect. Using the pixel grid, the computer determines the probability for a photon entering the printed medium through a given ink combination to emerge through another ink combination. If light is scattered only over short distances the algebraic calculation leads to the Murray-Davis equation, and if light is scattered over long distances the calculation leads to the Clapper-Yule relation. The accuracy of our predictions is as good as that of existing models, but our new approach is better due to its generalized framework, its physical base and the elegance of its mathematical formulation

Spectral colour prediction model for a transparent fluorescent ink on paper

A new spectral colour prediction model for a fluorescent ink printed on paper is presented. It is based on our previous work on transparent support 4 and on a new mathematical formalism which generalizes the Kubelka-Munk theory. The printed paper is modelized by means of three matrices: an interface correction matrix, a matrix exponential modelizing the layer which contains the fluorescent ink, and a reflection matrix caracterising the substrate. The interface correction matrix allows to take multiple reflections into account by operating the Saunderson correction. These matrices are related to physical properties of ink and paper which must be measured: the transmittance spectra, the quantum yields, the absorption bands and the emission spectra of the fluorescent inks, and the reflection properties of the paper. Our new model can predict the reflection spectra of uniform samples for different ink concentrations and under different illuminants. It is applied successfully to predict the spectra of real samples with an average prediction improvement of about ΔE = 17 in comparison with Beer's law

A "one channel" spectral colour prediction model for transparent fluorescent inks on a transparent support

A color prediction model meant for computing the transmittance spectra of uniform color samples was presented. The study involved the prediction of color spectra produced with one fluorescent ink at different concentrations. In regard with it, the measurements of quantum yields, absorbtion bands, and emission spectra were also presented

A Unified model for color prediction of halftoned prints

This study introduces a new model and a new mathematical formulation describing the light scattering and ink spreading phenomena in printing. The new model generalizes the classical Kubelka-Munk theory, and unifies it with the Neugebauer model within a single mathematical framework based on matrices. Results like the Saunderson correction, the Clapper-Yule equation, the Murray-Davis relation and the Williams-Clapper equation are shown to be particular cases of the new model. Using this new theoretical tool, the reflection spectra of 100 samples printed on high quality paper by two different ink-jet printers were computed with an average prediction error of about ΔE = 2.1 in CIELAB

Exploring ink spreading

This study aims at exploring ink spreading, which causes significant colour deviations in ink-jet printing. We present a method for investigating this phenomenon by considering only a limited number of cases. Using a combinatorial approach based on Polya's counting theory, we determine a small set of ink drop configurations which allows to deduce the ink spreading in all other cases. This improves the estimation of the area covered by each ink combination which is crucial in colour prediction models. Such models simplify the calibration of ink-jet printers

A model for colour prediction of halftoned samples incorporating light scattering and ink spreading

A model for color prediction of halftoned samples incorporating light scattering and ink spreading was presented. The spreading process was modeled by enlarging the drop impact according to the configuration of its neighbors and the state of the surface. The spectra of halftoned samples produced with one ink were predicted with an average prediction error or about δE=1.4 in CIELAB. For two halftoned ink layers, good spectral predictions were achieved with an average error of about δE=2.1 in CIELAB

Prediction of the Reflection Spectra of Three Ink Colour Prints

We have developed a colour prediction model and an ink-spreading model. The present study aims at confirming the validity of both models for the case of ink-jet prints using cyan, magenta and yellow inks. Our colour prediction model, augmented by the ink-jet spreading model, predicts accurately the reflection spectra of halftoned samples printed on an HP printer and on an Epson printer. For each printer, the reflection spectra of 125 samples uniformly distributed in the CMY colour cube were computed. The average prediction error between measured and predicted spectra is about ΔE = 2.5 in CIELAB. The model requires the estimation of a set of parameters which are deduced from a small set of measured samples. Such a model simplifies the calibration of ink-jet printers, as well as their recalibrations when ink or paper is changed

Predicting Monochrome Color Transmittance Spectra of Electrophotographic Prints

We create a computer based numerical model to predict the color spectra of printed patches on dry toner electrophotographic printers. The goal of this research is to obtain a simplified model describing the input-output behavior of the printers based on the physical characteristics of the different printing process steps and the interactions between them. This leads to a better understanding of the factors that have an impact on printing quality. Furthermore, by modeling the non-linearities of the electrophotographic process, the prediction model will allow the creation of device calibration data with a minimal effort. In order to avoid the additional optical non-linearities produced by light reflections on paper (dot-gain), we have limited the present investigation to transparency prints. In its current version, the proposed model is capable of predicting the transmittance spectra of a printed monochrome wedge down to a mean deviation less than CIELAB ΔE* ab = 1.5. The proposed simulation incorporates sub-models of the electrophotographic process shown in Fig. 1, such as the exposure of the photoreceptor, the generated electrostatic field, the toner's charge and diameter distributions, as well as the transfer and the fusing steps

Extending Kubelka-Munk's Theory with Lateral Light Scattering

Due to its simplicity, the theory of KUBELKA-MUNK [1] has found a wide acceptance for modeling the optical properties of light scattering materials. However, the concept is not explicitly adapted to predict halftone prints on paper. In this respect, a recent improvement was given by BERG. Our approach is an extension of BERG'S model in order to reduce the gap between the mathematical description of the paper's point spread function and the experimental results of simple reflectance measurements