Multilayer four-flux matrix model accounting for directional-diffuse light transfers

Matching target color in polyolefins by estimating pigment concentrations using a four-flux model

In applications of computer color formulation where color stimuli are optically thick (e.g., textiles, coatings, etc.), a simple single-constant or two-constant theory (e.g., Kubelka-Munk model) would suffice. To accurately predict reflectance and transmittance of materials with optical thickness ranging from optically thin to optically thick (e.g., plastics), mathematically complex radiative transfer theories (e.g., many-flux models) have been recommended. A many-flux model can even predict color formulation involving special-effect pigments (e.g., metallic, pearlescent, etc.), but implementation of such models is manyfold complicated. In the current study, applicability of a relatively simple Maheu-Letoulouzan-Gouesbet (MLG) four-flux radiative transfer model to optically varying pigmented polyolefins is thoroughly investigated. First, the MLG model was implemented to determine absorption and scattering coefficients of over 120 pigments where a new mean relative absolute spectral error (MRASE) between measured and calculated spectral reflectance and transmittance of the calibration samples was minimized as an objective function. Second, currently determined absorption and scattering coefficients were further validated by color recipe prediction of 350 historical product colors. Measured and predicted reflectance curves were compared in units of MRASE, CIEDE2000 color difference, metamerism index, root mean square error, and goodness-of-fit coefficient. Moreover, transmission matching was evaluated in units of percent difference between the required and predicted average transmittance. Results showed that with the current implementation of the MLG four-flux model, color recipes of at least 95% of the target colors can be predicted within the acceptability thresholds in units of different error metrics used in the study.

Photometric properties of piles of glass plates: retrospective

Stacked glass plates have discreetly accompanied the understanding of light since the origins of modern optics. They were studied by Bouguer, Lambert, Brewster, Arago, Stokes, Rayleigh, and many others, whose successive works progressively refined the predictive formulas of the reflectance and transmittance of piles of glass plates as a function of the number of plates and the angle of incidence by considering the decay of light flux by absorption, the multiple reflections between plates, the change in the degrees of polarization, and the possible interferential effects. Through this history of ideas about the optical properties of piles of glass plates, up to the mathematical formalisms from only a few years ago, we show that these successive works, and their subsequent errors and corrections, are inseparable from the evolution of the quality of the glass available each time, in particular its absorptance and its transparency, which strongly influence the quantities and the degree of polarization of the reflected and transmitted beams.

Multi-printer learning framework for efficient optical printer characterization

A high prediction accuracy of optical printer models is a prerequisite for accurately reproducing visual attributes (color, gloss, translucency) in multimaterial 3D printing. Recently, deep-learning-based models have been proposed, requiring only a moderate number of printed and measured training samples to reach a very high prediction accuracy. In this paper, we present a multi-printer deep learning (MPDL) framework that further improves data efficiency utilizing supporting data from other printers. Experiments on eight multi-material 3D printers demonstrate that the proposed framework can significantly reduce the number of training samples thus the overall printing and measurement efforts. This makes it economically feasible to frequently characterize 3D printers to achieve a high optical reproduction accuracy consistent across different printers and over time, which is crucial for color- and translucency-critical applications.

Unsupervised learning with a physics-based autoencoder for estimating the thickness and mixing ratio of pigments

Layered surface objects represented by decorated tomb murals and watercolors are in danger of deterioration and damage. To address these dangers, it is necessary to analyze the pigments' thickness and mixing ratio and record the current status. This paper proposes an unsupervised autoencoder model for thickness and mixing ratio estimation. The input of our autoencoder is spectral data of layered surface objects. Our autoencoder is unique, to our knowledge, in that the decoder part uses a physical model, the Kubelka-Munk model. Since we use the Kubelka-Munk model for the decoder, latent variables in the middle layer can be interpretable as the pigment thickness and mixing ratio. We conducted a quantitative evaluation using synthetic data and confirmed that our autoencoder provides a highly accurate estimation. We measured an object with layered surface pigments for qualitative evaluation and confirmed that our method is valid in an actual environment. We also present the superiority of our unsupervised autoencoder over supervised learning.

Deep learning models for optically characterizing 3D printers

Multi-material 3D printers are able to create material arrangements possessing various optical properties. To reproduce these properties, an optical printer model that accurately predicts optical properties from the printer's control values (tonals) is crucial. We present two deep learning-based models and training strategies for optically characterizing 3D printers that achieve both high accuracy with a moderate number of required training samples. The first one is a Pure Deep Learning (PDL) model that is essentially a black-box without any physical ground and the second one is a Deep-Learning-Linearized Cellular Neugebauer (DLLCN) model that uses deep-learning to multidimensionally linearize the tonal-value-space of a cellular Neugebauer model. We test the models on two six-material polyjetting 3D printers to predict both reflectances and translucency. Results show that both models can achieve accuracies sufficient for most applications with much fewer training prints compared to a regular cellular Neugebauer model.

Modeling, measuring, and using BRDFs: significant French contributions

The scattering of light by a surface is described by the bidirectional reflectance distribution function (BRDF). Unfortunately, this function cannot be straightforwardly acquired or modeled. French researchers have proposed interesting contributions to the field, with several models and accurate experimental systems. For instance, the National Metrological Institute (LNE-CNAM) has implemented the best angular resolution goniospectrophotometer (0.015°). Modeling the BRDF has also been deeply studied in France, especially with the microfacet theory in recent years, a better understanding of the shadowing-masking function, new general distribution functions, visible normals, interfaced Lambertian microfacets, and analysis concerning light multiple reflections. This paper presents the state of the art regarding some significant French contributions in these fields.

Approaches for calibration and validation of near-infrared optical methods for oxygenation monitoring

Pulse oximetry for arterial oxygenation monitoring and tissue oximetry for monitoring of cerebral oxygenation or muscle oxygenation are based on quantitative in vivo diffuse optical spectroscopy. However, in both cases the information on absolute or relative concentration of human tissue constituents and especially on hemoglobin oxygenation can often not be retrieved by model-based analysis. An in vivo calibration against an accepted reference measurement can be a practical alternative. Pulse oximeters and most of commercial cerebral tissue oximeters rely on empirical calibration based on invasive controlled human desaturation studies. As invasive in vivo tests on healthy subjects are ethically disputable and should be limited to exceptional cases this calibration practice is unsatisfactory. We present the current status and problems of calibration and validation in pulse oximetry and cerebral tissue oximetry including the pros and cons of in vivo as well as in vitro methods. We emphasize various digital and physical phantom approaches and discuss the prospects of their application and possible further developments.

Physically meaningful Monte Carlo approach to the four-flux solution of a dense multilayered system

Due to the complexity of the radiative transfer equation, light transport problems are commonly solved using either models under restrictive assumptions, e.g., N-flux models where infinite lateral extension is assumed, or numerical methods. While the latter can be applied to more general cases, it is difficult to relate their parameters to the physical properties of the systems under study. Hence in this contribution we present, first, a review of a four-flux formalism to study the light transport problem in a plane-parallel system together with a derivation of equations to evaluate the different contributions to the total absorptance and, second, as a complementary tool, a Monte Carlo algorithm with a direct correspondence between its inputs and the properties of the system. The combination of the four-flux model and the Monte Carlo approach provides (i) all convergence warranties since the formalism has been established as a limit and (ii) new added capabilities, i.e., both temporal (transient states) and spatial (arbitrarily inhomogeneous media) resolution. The support between the theoretical model and the numerical tool is reciprocal since the model is utilized to set a Monte Carlo discretization criterion, while the Monte Carlo approach is used to validate the aforementioned model. This reinforces the parallel approach used in this work. Furthermore, we provide some examples to show its capabilities and potential, e.g., the study of the temporal distribution of a delta-like pulse of light.

Rendering Rough Opaque Materials with Interfaced Lambertian Microfacets

Specular microfacet distributions have been successfully employed by many authors for representing glossiness of materials. They are generally combined with a Lambertian term to account for the colored aspect. These representations make use of the Fresnel reflectance factor at the interface, but the transmission factor at the interface should also be managed. One solution is to employ a multi-layered model with a single layer for the rough interface, which requires a numerical simulation for handling the multiple reflections of light between the substrate and the interface. In this paper, we propose rather to use a representation corresponding to a Fresnel interface lying on a Lambertian substrate, for which the multiple reflections of light between the interface and the substrate can be expressed analytically. With this interfaced Lambertian model, we show how Fresnel transmission affects the material appearance for flat and rough surfaces with isotropic and anisotropic distributions, that produce light backscattering effects. We also propose a methodology for using such materials in any physically based Monte Carlo rendering system, as well as an approximate representation, suitable for GPU applications or measured data fitting. Our approach generalizes several previous models, including flat Lambertian materials as well as specular and Lambertian microfacets. Our results illustrate the wide range of materials that can be rendered with this representation.

Highly amplified light transmission in a parity-time symmetric multilayered structure

We propose a parity-time symmetric dielectric-nano-film-dielectric multilayered structure that could facilitate highly amplified transmission of optical power in the infrared spectrum. We have theoretically studied our model using the transfer-matrix formalism. The reflection and the transmission coefficients of the S-matrix are evaluated. The theoretical results are validated by FDTD numerical simulation. We have shown how the thickness of the layers and the gain/loss coefficient of the active layers could generate spectral singularities in the S-matrix and how these singularities could be exploited to achieve amplified transmission of a single wavelength through the structure.

Extending the applicability of the four-flux radiative transfer method

A generalized four-flux method capable of modeling and tuning the spectral reflectance of a diverse range of complex composite coatings is presented. An example application is exploring and maximizing the visible and near-infrared (IR) spectral reflectance available from the diverse structures arising from combinations of the many practical paint ingredients that are available or can be made when applied to different substrates. This requires consideration of scatterers that can differ in composition, particle size, size distribution, and fill factor, and are held in place by a variety of organic binders, which typically partially absorb in the near IR. This extended model is further enhanced by an explicit matrix algorithm that allows analysis of diverse multilayer stacks. This is applied to a multilayer and is designed to model useful changes that result from varying the pigment fill factor as a function of depth within a layer. What we believe is a novel feature is the way the scattering affects matrix absorptance. The model includes contributions to total absorptance from the scattering pigments and from the paint binder that can arise in different bands or simultaneously at the same wavelengths. Model accuracy is demonstrated by example results when compared to experimental data on dried single layer paint profiles using imaged cross sections. The model input covering the actual pigment and binder properties used are material, shape, size, and size distributions, mass added, and the measured optical constants from 400 nm to 2,500 nm of the undoped binder resin layer. One interesting result is the comparison of a two-layered stack, with bigger particles in the first layer and smaller ones in the second, to one with the opposite depth profile.

Transfer matrix method for four-flux radiative transfer

We develop a transfer matrix method for four-flux radiative transfer, which is ideally suited for studying transport through multiple scattering layers. The model predicts the specular and diffuse reflection and transmission of multilayer composite films, including interface reflections, for diffuse or collimated incidence. For spherical particles in the diffusion approximation, we derive closed-form expressions for the matrix coefficients and show remarkable agreement with numerical Monte Carlo simulations for a range of absorption values and film thicknesses, and for an example multilayer slab.