Isotropic Phase Function Research Articles

Redefining A in RGBA

Advances in multimaterial 3D printing have the potential to reproduce various visual appearance attributes of an object in addition to its shape. Since many existing 3D file formats encode color and translucency by RGBA textures mapped to 3D shapes, RGBA information is particularly important for practical applications. In contrast to color (encoded by RGB), which is specified by the object’s reflectance, selected viewing conditions, and a standard observer, translucency (encoded by A ) is neither linked to any measurable physical nor perceptual quantity. Thus, reproducing translucency encoded by A is open for interpretation. In this article, we propose a rigorous definition for A suitable for use in graphical 3D printing, which is independent of the 3D printing hardware and software, and which links both optical material properties and perceptual uniformity for human observers. By deriving our definition from the absorption and scattering coefficients of virtual homogenous reference materials with an isotropic phase function, we achieve two important properties. First, a simple adjustment of A is possible, which preserves the translucency appearance if an object is rescaled for printing. Second, determining the value of A for a real (potentially non-homogenous) material, can be achieved by minimizing a distance function between light transport measurements of this material and simulated measurements of the reference materials. Such measurements can be conducted by commercial spectrophotometers used in graphic arts. Finally, we conduct visual experiments employing the method of constant stimuli, and we derive from them an embedding of A into a nearly perceptually uniform scale of translucency for the reference materials.

Experimental evaluation of scattering phase function and optimization of radiation absorption in solar photocatalytic reactors

Radiation absorption in photocatalytic reactors is usually simulated using the Monte Carlo (MC) stochastic method. In this article, MC is coupled with three different scattering phase functions: isotropic (ISO) phase function, diffuse reflecting large sphere (DR) phase function and Henyey–Greenstein (HG) phase function. Here, the model accuracy is examined via adequate experiments. The results show that phase function plays an important role in MC method. HG phase function is superior to ISO and DR phase functions because of small errors (less than 5%), and the value of the characteristic parameter g should be 0.64. The simulate results show that the absorption of solar radiation relates strongly to geometry configuration of reactors and catalyst concentration. The solar surface uniform concentrator (SUC) has a 9.94 times higher radiation absorption than the tubular reactor and realizes a uniform radiation distribution. The optimal catalyst concentrations of the SUC and the tubular reactor are 0.53g/L and 0.68g/L respectively.

Ejecta curtain radiative transfer modeling for probing its geometry and dust optical properties

An ejecta curtain is produced as a result of impact cratering of celestial bodies that excavates and ejects their surface and underground material. Since ejecta particles are thought to reflect the interior materials and structures of celestial bodies, ejecta and ejecta curtains provide us with valuable information about the interior of the bodies. Following the Deep Impact mission, more and more in situ data on ejecta curtains are expected to become available in future space missions. To draw valuable information from such in situ measurements, a radiative-transfer model of the ejecta curtain plays an important role. This urges us to carefully assess the model parameters with experimental results as well as understand the contribution of each model parameter to the predicted intensities of the radiative calculations. In this work, we study the dependences of the projected intensity images of the ejecta on the scattering phase function and the geometry of an ejecta curtain produced by impact on an airless-body's surface, as a first step toward a correct interpretation of ejecta observations. Using the Monte-Carlo multiple scattering method, we have calculated the scattered intensities for three different orientations of the ejecta curtain. We have also computed the intensities for ejecta material using different phase functions. We find that the scattered intensities are highest for the isotropic phase function irrespective of the orientation of the ejecta cone for the scattering angles and geometry considered here. Observations from the other side could lead to higher intensities for forward scattering grains that are typical for asteroids. Therefore in situ observations of ejecta-curtains at different angles will enable us to use our model predictions to extract the phase functions as well as column densities of its component dust grains.

Modeling radiative transfer in photobioreactors for algal growth

Simulations of radiative transfer within an air-lift photobioreactor (PBR) are demonstrated by coupling it to the fluid hydrodynamics and employing wavelength dependant properties for the participating media. The radiative properties of the algal media are determined by matching the numerical predictions against measurements of radiative intensity distributions. To assist towards the design, scale-up and optimization of such reactors, a parametric investigation of the relative importance of the angular resolution of the radiation calculations, scattering phase functions of the bubbles, air mass flow rate and the bubble size are investigated by examining the 3D distributions of radiation within the PBR. Two hundred and eighty eight solid angles in the finite volume (FV) formulation of the radiative transfer equation (RTE) provide an optimum combination of speed and accuracy in the resulting calculations. While scattering results in a more effective redistribution of the energy, the results from employing isotropic and forward-scattering phase functions for the bubbles are found to be similar. The importance of bubble scattering diminishes at algal concentrations (>0.5g/L) where the radiation attenuation is dominated by the absorption coefficient of the algal media. While 1μm sized bubbles more effectively redistribute the radiation downstream of the radiators compared to larger sized bubbles, the differences in the radiation profiles obtained from 10μm and 100μm-sized bubbles were small within this reactor. The radiation distributions are also influenced by the mass flow rate of air. The calculations demonstrate the need to rigorously account for the air flow rate, bubble size and the scattering effects of bubbles through fully coupled numerical solutions to the fluid flow and radiative transfer equations and provide some best practice guidelines for increasing the fidelity of radiation simulations in PBR’s.

Efecto de la Dispersión de Radiación sobre el Rendimiento Cuántico en la Degradación Fotocatalítica de Ácido Dicloroacético

Different models of radiation scattering used to valuate its effect on the amount of radiant energy absorbed by a photocatalyst in the determination of quantum yields were analyzed. The radiation field in a heterogeneous photodecomposition of dichloroacetic acid catalyzed by particles of titanium dioxide was simulated using a one-dimensional model to evaluate the absorption and scattering of radiation in the reaction space. The results obtained by the different models of nonisotropic phase function with the isotropic phase function were compared. It was found that the scattering of isotropic radiation is a reasonable approximation for scattering produced by suspended particles of titanium dioxide.

Efficient Visibility Encoding for Dynamic Illumination in Direct Volume Rendering

We present an algorithm that enables real-time dynamic shading in direct volume rendering using general lighting, including directional lights, point lights, and environment maps. Real-time performance is achieved by encoding local and global volumetric visibility using spherical harmonic (SH) basis functions stored in an efficient multiresolution grid over the extent of the volume. Our method enables high-frequency shadows in the spatial domain, but is limited to a low-frequency approximation of visibility and illumination in the angular domain. In a first pass, level of detail (LOD) selection in the grid is based on the current transfer function setting. This enables rapid online computation and SH projection of the local spherical distribution of visibility information. Using a piecewise integration of the SH coefficients over the local regions, the global visibility within the volume is then computed. By representing the light sources using their SH projections, the integral over lighting, visibility, and isotropic phase functions can be efficiently computed during rendering. The utility of our method is demonstrated in several examples showing the generality and interactive performance of the approach.

A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters

Several numerical and analytical solutions of the radiative transfer equation (RTE) were compared for plane albedo in a problem of solar light reflection by sea water. The study incorporated the simplest case—a semi-infinite one-dimensional plane—parallel absorbing and scattering homogeneous layer illuminated by a monodirectional light beam. Inelastic processes (such as Raman scattering and fluorescence), polarization and air–water surface refraction–reflection effects, were not considered. Algorithms were based on the invariant imbedding method and two different variants of the discrete ordinate method (DOM). Calculations were performed using parameters across all possible ranges (single-scattering albedo ω 0 and refracted solar zenith angle θ 1), but with a special emphasis on natural waters. All computations were made for two scattering phase functions, which included an almost isotropic Rayleigh phase function and strongly anisotropic double-peaked Fournier–Forand–Mobley phase function. Models were validated using quasi-single-scattering (QSSA) and exponential approximations, which represent the extreme cases of ω 0→0 and ω 0→1, respectively. All methods yielded relative differences within 1.8% for modeled natural waters. An analysis of plane albedo behavior resulted in the development of a new extended QSSA approximation, which when applied in conjunction with the extended Hapke approximation developed earlier, resulted in a maximum relative error of 2.7%. The study results demonstrated that for practical applications, the estimation of inherent optical properties from observed reflectance can best be achieved using an extended Hapke approximation.

Light transfer in bubble sparged photobioreactors for H 2 production and CO 2 mitigation

This paper presents a parametric study simulating light transfer in a photobioreactor containing gas bubbles and filamentous cyanobacteria Anabaena variabilis suspended in water. To the best of our knowledge, this paper presents for the first time a model for such system: (i) using a consistent set of radiation characteristics of the medium derived from experimental data and from Mie theory; (ii) accounting for anisotropic scattering by both the bubbles and the filamentous microorganisms; (iii) considering the spectral dependency of radiation characteristics in the spectral range from 400 to 700 nm using a box model, and (iv) evaluating light transfer in a photobioreactor containing genetically engineered microorganisms with reduced pigment content. The steady-state one-dimensional radiation transfer equation is solved using the modified method of characteristics and a quadrature with 24 directions per hemisphere adapted to forward scattering media. The parameters investigated include the bacteria concentration, the bubble radius, and the void fraction, as well as the approximate scattering phase function. It was established that the strongly forward scattering by the bubbles must be accounted for and the truncated phase function (TPF) is recommended. In the absence of bubbles, ignoring in-scattering by the bacteria leads to errors as high as 20%. On the other hand, accounting for in-scattering with isotropic phase function gives acceptable results. Moreover, genetically reducing the pigment content of the microorganisms by an order of magnitude increases the significance of forward scattering of light by the microorganisms. This in turn, increases the penetration depth and can be accounted for by either the Henyey–Greenstein or the TPF approximations. Finally, the model presented can also be applied to (i) other types of microorganisms such as unicellular green algae or photosynthetic bacteria, (ii) different photobioreactor processes such as food product or pharmaceutical production, or (iii) photochemical reactors.

Experimental and RDFI calculated radiative properties of a mullite foam

The solid phase of a mullite foam has been experimentally characterized as a mainly scattering semi transparent medium with an isotropic phase function. In a second step, from the corresponding determined data, the Radiative Distribution Function Identification (RDFI) method of Zeghondy et al. has been applied to predict the bi-directional reflectance of a mullite foam sample. A direct experimental determination of this reflectance agrees with the model results.

Beam attenuation coefficient retrieval by inversion of airborne lidar-induced chromophoric dissolved organic matter fluorescence I Theory

It is shown that the oceanic beam attenuation coefficient can be retrieved from airborne laser-induced and depth-resolved chromophoric dissolved organic matter (CDOM) fluorescence. The radiative transfer equation (RTE) retrieval methodology does not require a laser beam spread function model since two CDOM fluorescence bands are used in conjunction with a beam attenuation spectral model, is self-normalizing since the CDOM absorption coefficient and laser beam irradiance are common to both fluorescence observational channels, and is enabled by the known isotropic phase function for CDOM fluorescence. Although this RTE analytical inversion theory is exact, the retrieval uncertainty is reduced by configuring the proposed lidar in the multiple-field-of-view beam attenuation mode to significantly diminish observation of multiple scattering. The theory can be applied over wide regions of the ocean's continental margins, estuaries, lakes, and rivers that are known to have sufficient CDOM.

All-frequency interactive relighting of translucent objects with single and multiple scattering

We present a technique, based on precomputed light transport, for interactive rendering of translucent objects under all-frequency environment maps. We consider the complete BSSRDF model proposed by Jensen et al. [2001]. which includes both single and diffuse multiple scattering components. The challenge is how to efficiently precompute all-frequency light transport functions due to subsurface scattering. We apply the two-pass hierarchical technique by Jensen et al. [2002] in the space of non-linearly approximated transport vectors, which allows us to efficiently evaluate transport vectors due to diffuse multiple scattering. We then include an approximated single scattering term in the precomputation, which previous interactive systems have ignored. For an isotropic phase function, this approximation produces a diffuse transport vector per vertex, and is combined with the multiple scattering component. For a general phase function, we introduce a technique from BRDF rendering to factor the phase function using a separable decomposition to allow for view-dependent rendering. We show that our rendering results qualitatively match the appearance of translucent objects, achieving a high level of realism at interactive rates.

High-Quality BIMA-OVRO Images of Saturn and its Rings at 1.3 and 3 Millimeters

We present high-quality data of Saturn and its ring system from the Berkeley-Illinois-Maryland Association and Owens Valley Radio Observatory arrays. We observed at wavelengths of 1.3 and 3.0 mm in 2002 October and 2003 March, respectively. At that time Saturn's rings were near their maximum ring opening angle (|B| ~ 26°), which allowed us to map the brightness distribution on Saturn's disk, as well as from its rings, despite the moderate resolution achieved (~25–4''). Clear latitudinal structure is seen on Saturn's disk, indicative of an overall upwelling of gas on the southern hemisphere and subsidence of dry air in the north. The rings are brighter at 1.3 mm than at 3.0 mm, as expected from the increase in ring particles' thermal emission at shorter wavelengths. We use our radiative transfer code to model the relative contributions of the scattered and thermal radiation emanating from the rings. Unlike previous (centimetric) observations, the present (millimetric) data serve as an adequate test of the model's ability to account for the thermal contribution from the rings. A comparison between the model and data shows that Saturn's light transmitted and scattered through the B ring, as well as the radiation reflected from the A ring behind and just off the planet's south pole, is much brighter than predicted by our model. Hence, the ring particles appear to scatter more efficiently both in the forward and backward directions than implied by an isotropic phase function.

Isotropic scaling limits for one-dimensional radiative heat transfer with collimated incidence

As high-anisotropic scattering of materials induces difficulty in the resolution of radiative transfer equation, the isotropic assumption or approximate anisotropic phase functions are very useful. Isotropic scattering approximations need to be solved with a scaled optical depth and albedo. Isotropic scaling models involve the transformation of an anisotropic problem to an isotropic one. In this paper, the scaled optical depth and albedo are derived from the zero and first moment calculation of the scaled albedo and phase function product. Comparisons between the isotropic scaling, the anisotropic Henyey–Greenstein phase function approximations, and the exact solution are studied for one-dimensional steady state radiative heat transfer. The results show that both the isotropic scaling and the Henyey–Greenstein approximation are accurate for highly or weakly scattering media. For absorbing/scattering medium, the accuracy of the two approximations depends on the optical thickness and on the scattering function of the medium. The maximum errors between the two approximations and the benchmark solution are for albedo values around 0.5.

Scattered starlight contamination in the spectrum of reddened stars

The comparison, undertaken in preceding papers, of the UV observations of nebulae and of reddened stars reveals contradictory aspects of interstellar extinction. The aim of this paper is to understand the implications hidden behind the apparent contradictions. The questions treated will be: how can small grains with an isotropic phase function make an appreciable contribution in the UV spectrum of a star? Why are small grains not observed in the spectrum of a nebula? How much of starlight can be scattered by large grains in the forward direction?

Rotational relaxation in isotropic phase of some liquid crystals

Raman spectra of two liquid crystal compounds with differently substituted -CN groups were measured in isotropic phase and rotational autocorrelation functions were calculated in picosecond range both from deformation of phenyl ring and -CN stretching vibrations. The comparison of presented results shows the complicated nature of motions around long axes and the influence of the molecular structure.

Nature of light scattering in dental enamel and dentin at visible and near-infrared wavelengths

The light-scattering properties of dental enamel and dentin were measured at 543, 632, and 1053 nm. Angularly resolved scattering distributions for these materials were measured from 0° to 180° using a rotating goniometer. Surface scattering was minimized by immersing the samples in an index-matching bath. The scattering and absorption coefficients and the scattering phase function were deduced by comparing the measured scattering data with angularly resolved Monte Carlo light-scattering simulations. Enamel and dentin were best represented by a linear combination of a highly forward-peaked Henyey-Greenstein (HG) phase function and an isotropic phase function. Enamel weakly scatters light between 543 nm and 1.06 µm, with the scattering coefficient (µ(s)) ranging from µ(s) = 15 to 105 cm(-1). The phase function is a combination of a HG function with g = 0.96 and a 30-60% isotropic phase function. For enamel, absorption is negligible. Dentin scatters strongly in the visible and near IR (µ(s)≅260 cm(-1)) and absorbs weakly (µ(a) ≅ 4 cm(-1)). The scattering phase function for dentin is described by a HG function with g = 0.93 and a very weak isotropic scattering component (˜ 2%).

Two-dimensional anisotropic scattering in a semi-infinite cylindrical medium exposed to a laser beam

A modified Ambarzumian's method is used to develop a system of nonlinear integral equations for the source functions at the boundary of a two-dimensional cylindrical medium with a scattering phase function represented by a series of Legendre polynomials. The scattering medium is infinite in both the r- and z-directions. The incident radiation is collimated and normal to the surface of the medium and is Bessel-varying in the radial coordinate. Superposition is used to construct the solution for the Gaussian shaped laser beam from the Bessel-varying solution. The back-scattered intensity and flux are calculated for three and five term phase functions which are characterized by moderately strong forward scattering. Appreciable differences are observed between the results for these phase functions and the results for the isotropic, linear, and Rayleigh phase functions. A simple scaling procedure is introduced that collapses the results for different phase functions into the results for isotropic scattering at large optical distances from the laser beam.

Expansions for two-dimensional back-scattering from a semi-infinite medium with refractive index greater than unity

Two-dimensional scattering from a semi-infinite medium with a refractive index greater than unity is investigated. The incident radiation is collimated and normal to the surface of a pure scattering medium. The spatial variation of the incident radiation is characterized by a spatial frequency. Expansions valid for small spatial frequencies are developed and used to find solutions valid for large optical distances from an incident Gaussian-shaped laser beam. Expressions for the back-scattered intensity and flux are presented. Also, the solution for a strongly anisotropic phase function which is composed of a spike in the forward direction superimposed on an otherwise isotropic phase function is expressed in terms of the isotropic results.

Scaled Isotropic Results for Two-Dimensional Anisotropic Scattering Media

The full anisotropic scattering solutions of the radiative equation of transfer are compared with the scaled isotropic scattering solutions. Square enclosures with a collimated incidence, a diffuse incidence, or an isothermal emission are considered for comparison. The isotropic scaling approximation is found to predict accurately the radiative flux and the average incident radiation for the isothermal emission problem and for most diffuse incidence problems. For the collimated incidence problem, the isotropic scaling solutions are acceptable only for weakly scattering media. For large scattering albedo the error in the isotropic scaling is appreciable for the diffuse incidence problem and unacceptably large for the collimated incidence problem. The largest error in the y-direction net flux is found at the side wall regions when the medium is purely scattering. The isothermal emission problem or problems with symmetric boundary conditions can be accurately modeled by a scaled isotropic phase function, since the effect of the phase function anisotropy is negligible in such problems.

Radiative transfer in a semi-infinite absorbing/scattering medium with reflective boundary

A variation of Ambarzumian's method is used to formulate the equations for the source function, flux and intensity at the reflective boundary of a one-dimensional semi-infinite medium. Emission is neglected, while the medium is allowed to absorb energy and to scatter isotropically. The boundary conditions studied are collimated incident intensity and diffuse incident intensity. As in the case of unit refractive index, the solutions for the diffuse boundary condition are found to be integrals of the solutions for the collimated boundary condition. Equations are written for the desired functions just inside and outside the medium. Example numerical results are presented for an albedo of 0.7 and refractive indices of 1.1, 1.33, 1.5, and 2.0. It is also shown that the solutions for the isotropic scattering problem can be extended to the special case of anisotropic scattering, for which the phase function is made up of a forward spike superimposed upon an otherwise isotropic phase function.