Mathematical modeling and numerical tools for simulation and design of light scattering in paper and print

Abstract: This work starts with a real industrial problem - the perceived need for a moredetailed and more accurate model for light scattering in paper and print than theKubelka‐Munk model of today. A careful analysis transfers this problem into aphysical description of the phenomena involved. This is then given a mathematicalformulation, and a detailed analysis leads to numerical solution procedures forspecific sub problems. Methods from scientific computing make it possible to meetindustrial demands made on speed and stability, and implementation in computercode is then followed by analysis of accuracy and stability.A problem formulation and a solution method are outlined for the forwardradiative transfer problem. First, all necessary steps to arrive at a numericallystable solution procedure are treated, and then methods are introduced to increasethe speed by a factor of several thousands or millions compared to a naiveapproach. The method is shown to be unconditionally stable, though the problemwas previously considered numerically intractable, and systematic studies ofnumerical performance are presented.The inverse radiative transfer problem is given a least‐squares formulation, anddifferent solution methods are analyzed and compared. Specifically, a two‐phasemethod for estimation of the scattering and absorption coefficients and theasymmetry factor (σs, σa and g) is presented. A sensitivity analysis is given, and it isshown how it can be used for designing measurements with minimal impact frommeasurement noise.It is shown how the standardized use of Kubelka‐Munk and the d/0°instrument leads to errors, and that the errors arising from an over‐idealized viewof the instrument - due to the fact that instrument readings are incorrectlyinterpreted - can be larger than any errors inherent in the Kubelka‐Munk modelitself. It is argued that the measurement device and the simulation model cannot beviewed as separate instances, which is a widespread implicit practice in appliedreflectance measurements. Rather, given a measurement device, measurement datashould be interpreted through a model that takes into consideration the actualgeometry, function and calibration of the instrument.The resulting tool, DORT2002, is in all aspects the Next Generation Kubelka‐Munk, and provides a greater range of applicability, higher accuracy and increasedunderstanding. It offers better interpretation of measurement data, and facilitatesthe exchange of data between the paper and graphical arts industries. It opens forunderstanding of anisotropic reflectance and for the utilization of the asymmetryfactor to design anisotropy, and thereby for the design of different visualappearance or optical performance in new printed or paper products.

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