We 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 nonsymmetric 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.

The 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.