Thesis No 1857 presented at the computer science department of the

Ecole Polytechnique Fédérale de Lausanne, September 1998

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.

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