Modeling Light Scattering in Paper for Halftone Print

This paper is concerned with the study and description of light scattering in turbid material such as paper. We present a new modeling approach, which is based on the transport theory. A statistical description for the scattering and absorption behavior inside the material is used to obtain spatial distributions of wavelength dependent photon number density and fluxes. Opposed to previous methods, the new approach can account for complex multi-point statistics of the material properties in an efficient and general way. This is achieved by solving a model equation for photon number density and propagation direction distribution. The present method is able to account for the spacial distribution of specific, wavelength dependent scattering and absorption characteristics. Thereby, it offers a general framework which allows to predict for example the appearance of colors in halftone prints. Introduction The study of light scattering in paper is of major concern and importance for the description of phenomena related to halftone printing. However, often it is not possible to predict or explain experimental data with existing models. There are two fundamentally different approaches [2, 3]. The first one, the analytical theory [2, 4, 9], is rigorous, but computationally very expensive. More adequate for practical problems [2] is the second approach, the transport theory [1], which was developed on a heuristic basis dealing with transport of energy through turbid media directly [10]. In this paper we present a new modeling approach, which is based on the transport theory. A statistical description for the scattering and absorption behavior inside the paper is used to obtain spatial distributions of wavelength dependent photon number density and fluxes. Opposed to previous methods, our approach allows to account for the typically complex paper structure and multi-point statistics in a general, but still efficient way. This is achieved by solving a modeled evolution equation for photon number density and joint probability density function (PDF) of propagation direction. In order to solve the high dimensional PDF equation efficiently, a particle method is employed. In this framework, each computational particle represents a number of photons indicated by a weight. Additional particle properties are position in physical space, propagation direction and wavelength. To account for absorption, the particle weight decreases with time. We first describe the transport equation for the photon density and direction distribution. Then a particle solution algorithm is devised. With numerical results we demonstrate that the method shows the correct tendencies in terms of absorption coefficient and correlation length scales in the paper. Finally, we discuss how the new model can be extended to account for more specific scattering behavior in a very general way. Since the model honors wavelength dependency and spatial distribution of the coefficients, it can be used to predict the effect of optical dot gain on the colors in halftone prints. Moreover, it is interesting that in the 1D case our model is consistent with the existing random walk [7, 8] and Kubelka and Munk [5, 6] models. PDF Transport Equation First, a brief outline of the transport theory is given. The most important quantity considered is the radiance, I(x,s), which is the average energy flux per solid angle at location x in direction s. The change of I(x,s) experienced along the path from x in direction s is expressed by the differential equation dI(x,s) ds =−γt I(x,s)+ γt 4π ∫ 4π p(s,s′)I(x,s′)dω ′, (1) where γt is the extinction coefficient (which is composed of the absorption and scattering coefficients γa and γs, respectively), p(s,s′) is the phase function describing the part of photon flux scattered from the direction s′ into the direction s, and dω ′ is the elementary solid angle about the direction s. Knowing the photon number density, ρ , and the PDF, fŝ(s;x, t), of photon propagation direction, s, one can extract all statistics of interest (s is the sample space variable of ŝ). The objective of the following modeling approach is to compute ρ and fŝ(s;x, t) by solving the PDF evolution equation ∂ρ fŝ ∂ t + ∂ ∂xi {〈 dx̂i dt ∣∣s;x, t 〉 ρ fŝ } + ∂ ∂ si {〈 dŝi dt ∣∣s;x, t 〉 ρ fŝ } =− fŝ τa , (2) where c is the speed of light and following the Einstein summation convention we sum over the index i. The first term describes the change of photon number density in the x-s-space with time, the second term accounts for transport in physical space, the third term for evolution in direction sample space, and the term on the right-hand side for absorption. The conditional expectations, 〈dŝi/dt|s;x, t〉 and 〈dŝi/dt|s;x, t〉, require modeling. Note that in general one is only interested in a steady state solution of Eq. (2). Stochastic Model for Photon Scattering We propose a modeling framework, which is based on solving Eq. (2). A particular difficulty is the high dimensional x-sspace, in which ρ(x) fŝ(s;x, t) evolves. Therefore, and due to an easier approach to modeling photon propagation, a Lagrangian particle method is employed. In fact, such Monte Carlo particle methods are widely used in computational physics to solve highdimensional problems, since the computational cost increases only linearly with the number of dimensions. Particle Method In our framework, we consider a cloud of computational particles in the x-s-space such that ρ(x) fŝ(s;x, t) is represented by the particle number density. The computational particles have a weight w∗, a position x̂∗ in physical space, a propagation direction ŝ∗ (position in s-space) and possibly further properties, e.g. a wavelength. Position, x̂∗ , and propagation direction, ŝ∗ n+1 , at the new time tn+1 = tn+ dt are modeled in terms of x̂∗ and ŝ∗ n at the previous time tn. Moreover, the weight w∗ is modified depending on the absorption time scale τa as w∗ n+1 = w∗ n e−dt/τa . (3) Scattering behavior as well as absorption time scale depend on the material properties and are therefore position dependent. Next, we describe how x̂∗ and ŝ∗ n+1 are modeled as functions of x̂∗ and ŝ∗ n , while we assume that the medium is isotropic. We consider a local, orthogonal coordinate system with its origin at x̂∗ and the unit vectors e1 = ŝ∗ n , e2 ⊥ e1 and e3 = e1×e2. In this coordinate system, the new particle location and propagation direction, x̂′∗ and ŝ′∗ n+1 , are determined by random lookup from pre-computed evolution tables. These tables depend on wavelength, material properties (and therefore on the position x̂∗ ) and time increment dt. Each entry represents an equally possible new state (x̂∗ , ŝ∗ n+1 ) and the new particle properties in the reference coordinate system are obtained by the transformations x̂∗ n+1 = x̂∗ n +T · x̂′∗ (4)