Abstract

In this paper, we present a model for BRDF that can be developed from a limited set of experimentally observed data, and which then can be used for predictive purposes in scene-generation or sensor-performance applications. The model is physics based and can be as detailed as desired, depending on the scope of experimental data available. Basic input parameters required are the complex refractive index of the material, or the directional hemispherical reflectivity (DHR) for s- and p-polarized radiation. At least one BRDF measurement is needed to determine the angular spread function. Incorporating BRDF measurements at several angles of incidence into the model yields better accuracy for describing behavior such as forward scatter, depolarization, the participation of volumetric and surface-scattering mechanisms in layered surfaces, diffuse-scatter coherence properties, and narrowing of the angular spread function which provides an indication of optical depth. Illustrations and demonstrations of the methodology are drawn from data sets measured on bead-blasted aluminum, automotive paint incorporating a clear-coat layer, and green low-gloss vehicle paint.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Surfaces can be observed because incident radiation is scattered over a relatively broad solid angle. Reflections are often referred to as specular or diffuse. Most surfaces are in between these two ideals and perfectly specular or perfectly Lambertian surfaces rarely exist in practice. It is therefore of considerable interest to obtain a general model that can handle both scattering from rough surfaces and volume scattering. The scattered radiation can be described by a series expansion of several scatter distributions.

Surface and volume scattering of electromagnetic radiation are due to amplitude and phase variations of the reflected wavefront as it propagates. The scatter phenomenology is discussed in detail by Mandel and Wolf [1] and at an introductory level by Wolf [2]. It is highly dependent on the statistical properties of the scattering surface or the volume close to the surface. These properties vary with wavelength and the radiation’s depth of penetration into the material, as well as the angle of incidence and scatter angle.

Surface scatter behavior is commonly described by two historical approaches [3]. For smooth surfaces, the Rayleigh-Rice approach is often adopted. For rougher surfaces, the Beckmann-Kirchhoff approximation is often assumed for a paraxial treatment. Multiple scattering is difficult to deal with in this approximation. Analytical solutions to scattering phenomenology are very difficult to obtain since many factors are involved such as input and output directions, surface roughness, material composition and multiple scattering statistics. Surfaces are often described by height probability density function and by the surface correlation function. These parameters are often indicative of the surface scattering properties, but real surfaces also exhibit sub-surface volume scattering and intricate coherence behavior between the scattering sites.

Basic radiometric concepts, especially the concept of bidirectional reflectance distribution function [4] (BRDF) and its connection to electromagnetism are discussed by Greffet and Nieto-Vesperinas [5]. An empirical linear-systems formulation has been discussed by Krywonos et al. [6]. In their treatment, both the reflectance and the scattering parameter are constant and independent of polarization and incident and scattering angles. This model applies to moderately rough surfaces.

In spectral remote sensing, the microfacet model for rough surfaces is often selected [7-8]. The underlying assumption is that the microfacets are large compared to the wavelength and that diffraction is thus not important in the scattering process. In practice, the model is also used in cases where diffractive phenomenology does play a role, and in those cases typically shadowing and masking are handled empirically.

Due to the complexity of BRDF measurement, the computer-graphics community does not have sufficient data readily available. With growing understanding of the scatter phenomenology, simpler measuring devices can possibly be developed to improve on the availability of data.

There is a need to model rough surfaces such as painted surfaces where the reflectance varies with angle of incidence. A problem of empirical models is that it is difficult to obtain adequate justification of the selected approach. The approximation made here has been discussed in a previous paper [9] and assumes that the reflectance properties of the material can be separated from the phase variation of the reflected electromagnetic radiation. Both are angle and wavelength dependent. The reflectance is governed by the modified Fresnel equations where the modification is based on experimental observations. The angular distribution caused by the phase variations is described by a generalized Gaussian or Lorentzian function. In the model presented here, the reflectance is governed by the effective complex index of refraction of the material while the scatter pattern is governed by the angular spread function (ASF) of the material.

Compared to reference 9, the most significant advance is that the angular spread functions (ASFs) used in the model are now normalized. The Gaussian and Lorentzian ASFs are normalized numerically. The Lambertian ASF is normalized analytically. The normalization allows a proper interpretation of integrated reflectance, so that the complex index of refraction obtained from DHR measurements can be used directly in the BDRF model. This eliminates the need for ad hoc shadow/masking functions to obtain agreement between model predictions and experimental data, particularly at large angles. The adjustment of scattering amplitude as seen in Fig. 10 of reference [9] is no longer needed, yielding good agreements between model predictions and experimental data. This reduces the number of free parameters required to obtain a given accuracy in the fitting, and more closely connects the model parameters to measured material characteristics.

The polarimetric model presented here can be efficiently implemented in simulation of geometric objects. Each surface element can be described by an object polygon, as is common practice. Knowing the orientation of each element, Stokes vectors can be obtained from the BRDF model and from the surface orientations relative to the sun and viewing directions. For thermal radiation, the solar heating must also be estimated. Efficient modeling and simulation based on the generalized BRDF presented here will be accounted for in future work.

For the sake of convenience, the basic definition of BRDF from radiometry is presented next. A generalized BRDF related to a planar, secondary source is defined. The concept of primary and secondary sources is discussed in reference [1] sections 5.2 and 5.3. This secondary source, which is the scattering region illuminated by the radiation source, is assumed to be a wide-sense statistically stationary scattering potential. The scattering potential is characterized by the source’s position-dependent electric susceptibility or equivalently, its complex refractive index.

2. Radiometry

The model is developed for materials with isotropic reflection, i.e. the BRDF is invariant to rotations around the local surface normal that is along the z-axis. Direction cosines of a unit vector are the cosines of the angles between the vector and the three coordinate axes. These are given by:

α=sin(θ)cos(φ)
β=sin(θ)sin(φ)
γ=cos(θ),
where direction is defined by polar angle θ and azimuthal angle φ in the spherical coordinate system as commonly used in physics. The BRDF is a function of both incident and outgoing directions, and the direction cosines are interdependent because of the usual constraints for unit vectors.

The directions of propagation are defined by the unit vectors s for the scattered radiation and S for the incident radiation, where s = {αo,β o,γ o } and S = {αi,0,γ i }. We assume that the input direction is defined to be in the x,z-plane and the output direction can be in any arbitrary direction. Assuming a linear and reciprocal medium, the direction of propagation is reversible.

The bidirectional reflectance distribution function (BRDF) is here denoted as fBRDF{αi,αo,β o;λ} [sr−1] and defined as:

Lr(αo,βo;λ)=fBRDF(αi,αo,βo;λ)Li(αi;λ)γidΩ,
where dΩ is a solid angle in a direction defined by the corresponding polar and azimuthal angle and the integration is over the 2π hemisphere. It is assumed that the definition is local and therefore the reflectivity depends on the local position. Li (αI ;λ) is the incident radiance, Lr (αo,β o;λ) is the radiance reflected into the viewing direction and λ is the wavelength.

3. Surface scattering theory

The theory is limited by the conditions that apply to a quasi-homogeneous secondary source. Some of the consequences can be stated as follows. If the object is illuminated with a laser, speckles are created that are not described by the present theory since they depend on the shape of the illuminated object. Speckle patterns therefore have to be averaged and if the properties of the object also fulfill some basic requirements discussed below, the equations given here can be applied. If the object is illuminated with a broadband source or radiates thermally, the signal being detected is most likely spectrally limited by the sensor. Speckles are automatically integrated, and the model given here applies as long as the scattering parameters do not change within the spectral bandwidth of the sensor. The illuminated object must be large enough not to cause a diffraction pattern that is of similar angular width to the ASF.

The angular spread function (ASF) G is given by the Fourier transform of the correlation function g, G = F{g}. The function g(x,yio) is the correlation function of the linear, isotropic, statistically stationary scattering potential. The integrated angular spread function (ASF) G is normalized.

It is assumed that the BRDF is independent of position and can be described by the reflectance F, the ASF G, and a function H that adjusts the reflectance parameter and which depends on whether the irradiance is subject to surface scatter or volume scatter below the dielectric surface. In most cases, only a small number of functions are needed. Three types of functions are defined including diffuse scatter and both polarizations. The same number of functions are defined for the angular spread function, i.e. generalized Lambertian, generalized Gaussian and generalized Lorentzian. The H function has also three types, surface scatter and s- and p-polarized volume scatter. The resulting BRDF is given by:

fBRDFpol(αi,αo,βo;λ)=typesatypepolHtypepol(αi,αo,βo;λ)Ftypepol(αi,αo,βo;λ)Gtype(αi,αo,βo;λ),
where atypepol denotes the weights of the different types of contributions in the expansion and Σtypes indicate a sum over all combinations of types of functions. This general form is simplified in actual use since only a limited number of terms are needed in order to obtain a high-quality fit to observed BRDFs.

Equation (3) has some similarity with the model introduced in reference [9]. The normalization procedure in the present model has advantages that will become clearer when discussing the individual types of functions. This is especially obvious at large angle of incidence where a shadow/masking function earlier had to be invoked. It is also further exemplified in the interpretation of the DHR approximation given later on.

Normalization of the (ASF) function G now leads to a proper interpretation of integrated reflectance. The generalized Lambertian now also contains an analytical normalization. Both angular appearance and polarization are indicators for surface/volume scattering. This phenomenology is easily tested using the new model. As a first step, the functions in the series expansion are defined.

3.1 Reflectance

The first reflectance function is of the Lambertian type and independent of any angles:

FDiffuse=1.

The following two functions describe the reflectance based on modified Fresnel equations for s- and p-polarized radiation:

FFresnels(αi,αo,βo;λ)=Rp(βo2;λ)Rp(0;λ)Rs(αi+αo2;λ)
FFresnelp(αi,αo,βo;λ)=Rs(βo2;λ)Rs(0;λ)Rp(αi+αo2;λ)

In our development, the BRDF is modelled at the average of the direction cosines, (αi + αo)/2. This is close to using the average angle (θi + θo)/2 as in the facet model but yields a much simpler final expression. The Fresnel equations are:

Rs(α,n,k)=|(nik)1α2(nik)21α2(nik)1α2(nik)2+1α2|2
Rp(α,n,k)=|(nik)1α21α2(nik)2(nik)1α2+1α2(nik)2|2.

For βo = 0 and αi = αo, FFresnelpol is equal to the Fresnel equations. Although strictly speaking, the Fresnel formulas are defined for plane waves implying an infinite surface, we generalize the concept here and use them in a locally defined context. Observe that the direction cosine α in Eq. (7 and 8) is replaced by (αi + αo)/2 in Eq. (5 and 6).

3.2 Surface/volume effect

The diffuse surface scattering effect is independent of angles and described by:

HSurface=1.

There are materials where not only the specular scattering but also the diffuse scattering shows a polarization dependence. This is understandable for e.g. clear-coated surfaces. If the radiation is passing through a dielectric surface, scattered by pigment particles in the volume and scattered back through the dielectric surface, these effects can be included as shown below. It is observed that the degree of depolarization is strongly dependent on the type of pigmentation in the material. A parameter was introduced in order to be able to adjust the amount of depolarization. The parameter a governs the degree of depolarization with a = 0 for no depolarization and a = 1 for complete depolarization:

HFresnels(αi,αo,βo;λ)=(1Rs(αi;λ))((1a2)(1Rs(αo;λ))+a2(1Rp(αo;λ)))
HFresnelp(αi,αo,βo;λ)=(1Rp(αi;λ))((1a2)(1Rp(αo;λ))+a2(1Rs(αo;λ))).

The surface transmission of the irradiance and the surface transmission of the sub-surface scattered radiation is moderated by the HFresnel function. The resulting attenuation depends on both the angle of incidence, the index of refraction of the surface and the degree of depolarization of the scattered radiation. This is illustrated in Fig. 1.

 figure: Fig. 1

Fig. 1 The attenuation due to passing through the dielectric surface twice is illustrated with no depolarization in the diffuse scattering shown in A) and complete depolarization shown in B). Red is s-polarized, and blue is p-polarized radiation. The index of refraction was set to n = 1.5.

Download Full Size | PPT Slide | PDF

The induced polarization can also be due to plasmonic polariton effects, e.g. when structured metallic surfaces are involved. This situation would result in different constants atypepol for the two states of polarization.

3.3 Angular spread function

With respect to diffuse scattering, often Lambertian sources is assumed, e.g. in connection to blackbody radiation, resulting in a constant BRDF that is independent of the scattering angle. The angular distribution of the radiant intensity generated by a quasi-homogeneous source such as a Lambertian source is independent of the shape of the source. It primarily depends on the two-dimensional spatial Fourier transform of the spectral degree of coherence of the radiation across the source. To account for variations in coherence, a generalized Lambertian ASF is defined as:

GDiffuse(αo,βo,νD;λ)=1+νD/2π(1αo2βo2)νD/2,
where the parameter νD varies as −1 < νD ≤ 1; νD = 0 corresponds to a Lambertian surface and νD = 1 corresponds to an incoherent source. The normalization was obtained by integration after converting the direction cosines back to trigonometric functions.

The phase shift theorem of the Fourier transform theory is applied on the surface transfer function considering that the non-normal incident wavefront causes a linear phase variation across the secondary source. The advantage with this scattering function is that it seems to have the flexibility needed for describing a variety of surfaces. The drawback is that it is not possible to find an analytical Fourier transform or analytical solution to the integrated scattering for all values of the shape factor ν. Numerical normalization is therefore required.

A generalized Gaussian distribution function (GGD) and a generalized Lorentzian distribution function (GLD) also called a K-correlation model [4] has been selected for the description of the scattering due to the geometric structure of the material. These two generalized ASFs allows the corresponding correlation function g to vary approximately between exponential Gaussian shape and Lorentzian shape. This variation covers many observed surfaces.

The angular spread function GGaussian is defined by:

GGaussian(αi,αo,βo,ρ^,ν;λ)=normG(αi,ρ^,ν;λ)Exp(|αoαi|ν+|βo|ν((1αi2+1αo2βo2)ρ^2)ν),
where

normG(αi,ρ^,ν;λ)=1/αo=1αo=1βo=1αo2βo=1αo2Exp(|αoαi|ν+|βo|ν((1αi2+1αo2βo2)ρ^2)ν)dαodβo

the angular spread function GLorentzian is defined by:

GLorentzian(αi,αo,βo,ρ^,ν;λ)=normL(αi,ρ^,ν;λ)(1+(αoαi)2+βo2(1αi2+1αo2βo2)ρ^2)(ν+1)/2,
where

normL(αi,ρ^,ν;λ)=1/αo=1αo=1βo=1αo2βo=1αo2(1+(αoαi)2+βo2(1αi2+1αo2βo2)ρ^2)(ν+1)/2dαodβo.

As can be seen in Fig. 2, the functions GGaussian and GLorentzian are complementary with respect to shape and tail distribution.

 figure: Fig. 2

Fig. 2 The generalized Gaussian distribution A) and the generalized Lorentzian distribution B) are shown. The scattering parameter is ρ^ = 0.2 and the shape parameter is set to ν = (1,2,3).

Download Full Size | PPT Slide | PDF

The parameter ρ^ represents the scattering width that is both wavelength dependent and dependent on the angle of incidence as shown in the equation above. In volume scattering, the penetration depth will depend on the angle of incidence causing an extra angular dependence, i.e. ρ^(αi). It is experimentally observed that this variation generally scales proportionally with the direction cosine of the incident radiation, i.e. ρ^(αi)=1αi2ρ^0. In those cases, the function is renamed to GGaussianVolume.

For some rough surfaces an increased forward scattering can be observed for intermediate angles of incidence. This is modelled by formally increasing the value of the direction cosine, αi, of the incident radiation by including a multiplicative factor that is empirically determined.

The final BRDF function is described by a series expansion of BRDF functions encompassing each of the separate scattering phenomenologies. For example, scattering from a rough surface with diffusely scattering sub-surface pigments, could be modeled as:

fBRDFpol=aF,D,Ds,pHFresnelpolFDiffuseGDiffuse+aS,F,Gs,pHSurfaceFFresnelpolGGaussian,
or
fBRDFpol=a1HFresnelpolGDiffuse+a2FFresnelpolGGaussian,
where as,p is the same for s- and p-polarized radiation. The parameter a1 is the diffuse reflectance below the dielectric surface and a2 is the reflectance coefficient for the rough surface and models the attenuation compared to Fresnel reflectance.

4. DHR

The directional hemispherical reflectance (DHR) is defined by:

fDHRpol(αi;λ)=αo=1αo=1βo=1αo2βo=1αo2fBRDFpoldαodβo
the DHR can be compared to Fresnel reflectance by numerical integration of a test case, e.g. a diffuse paint. In doing so, it can be observed that the Fresnel equation describes also the DHR at angles of incidence close to normal. This is shown in Fig. 3 where an index of refraction of n = 1.5 and an ASF parameter ρ = 0.1 are adopted.

 figure: Fig. 3

Fig. 3 Illustration of the similarity of Fresnel equation and DHR simulations for a diffuse paint.

Download Full Size | PPT Slide | PDF

In order to obtain an estimate of the effective complex index of refraction, this integral has successfully been approximated by the function:

fDHRpol(αi;λ)=1+σ1αi21+σFFresnelpol(αi,αi,0;λ)+σD,
where the numerical parameter σ is a scaling factor due to the surface roughness and σD is a bias. For σ = 0, the original Fresnel equations are obtained. With increasing ASF parameter ρ, the σ parameter increases through positive values toward a saturation value. For angles of incidence close to the surface normal, the Fresnel reflectance is approximately constant over the scattering angles. Since the ASF is normalized, the scaling parameter in front of the Fresnel equation should be close to one as can be observed in Eq. (19). For large angles of incidence, the variation is more complex and can be scaled by adjusting the parameter σ to match the observed DHR variation. This model performs well and an effective complex index of refraction is obtained that fits well with the BRDF model.

5. Experimental observations

The implementation of the model can be understood from simple examples. Three examples are given. The first one is dominated by surface scattering, the second one is characterized by surface and volume scattering and the third one is a clear coated car paint also showing volume scattering. The model is discussed from the perspective that different amounts of data might be available to characterize the material. In the first example, it is assumed that the complex index of refraction is available from a database and a series of BRDF measurements have been performed. In the second example, the complex index of refraction including an attenuation factor and bias is obtained from a DHR measurement. The BRDF parameters are in this case taken from normal incidence data. Accuracy of the resulting model is judged in both cases from a set of detailed BRDF measurements. It is also shown that minor tuning of the parameters involved can substantially improve the fitting accuracy. In the third case, the index of refraction is assumed to be typical for a binder and the diffuse scattering is modeled with a Lorentzian distribution.

5.1 Effective complex index of refraction

An initial value of the refractive index can be obtained from a material database, if that is available, or from DHR measurements. In many applications, values from a refractive index database [10] are good enough to use for this purpose. If needed, the refractive index values can be refined by fitting to observed BRDF measurements. The reflectance FFresnelpol is determined by the complex index of refraction {n,k} used in the Fresnel equations. The first step is therefore to obtain at least an approximate value of these parameters.

An approximate value of the refractive index can also be obtained from the directional hemispherical reflectance measurements using Eq. (19). For a flat, extended surface, the parameter σ is expected to be close to zero while the bias is close to zero. For a diffuse surface, the DHR is expected to be close to the constant σD. The refractive index obtained in this way is reasonably close to the true material value. Further refinements can be obtained by fitting Eq. (3) to BRDF measurements.

5.2 Scatter parameters

More information is needed to determine the scatter parameter ρ^. The type of measurement selected depends on the accuracy needed and instrumentation available. The most accurate measurements are from the bidirectional reflectance distribution measurements. Those are however very expensive both in terms of equipment and in labor and it is sometimes difficult to obtain appropriately sized sample pieces for laboratory measurements. With a reliable model, simpler in situ measurements might suffice. If BRDF data is available, a detailed model can be obtained containing an accurate description on scatter contributions.

BRDF data at normal incidence gives information on whether there is a strong specular component, the magnitude of relatively diffuse contribution and some indication of the more or less Lambertian type scattering. The data will not be very sensitive to the refractive index except in its relation to the scatter amplitude.

BRDF data at a quasi-Brewster angle will be rather sensitive to the refractive index. These measurements can be used for an accurate determination of the refractive index. At larger angles of incidence, small model deviations start to play a role. The amplitude correction can be left as a preliminary correction parameter at this stage. Similarly, smaller values of ρ^ might be allowed for by using e.g. a correcting multiplier in Eq. (13). Extending the model as described above by invoking a series expansion of the G-distribution, the desired accuracy can be obtained over a wide range of incident and reflected angles.

5.3 Example I: bead-blasted aluminum

The need of a series expansion is first illustrated below for the bead-blasted aluminum BRDF. There are two “specular” contributions, one narrow peak emanating from parts that have only been partly affected by the blasting and a diffuse part that comes from portions of the surface that have been more strongly modified. Only the diffuser component shows forward scattering. Apart from the Lambertian-like diffuse scattering, two components are here needed to describe the observation.

In this example, it is assumed that the refractive index is obtained from a database while detailed BRDF measurements are available at the wavelength of 3.39 µm. From the database, the refractive index at 3.39 µm is n = 5.131 and k = 32.911.

The BRDF model for bead-blasted aluminum is given by:

fBRDFpol=aS,D,DpolHSurfaceFDiffuseGDiffuse+aS,F,G1s,pHSurfaceFFresnelpolG1,Gaussian+aS,F,G2s,pHSurfaceFFresnelpolG2,Gaussian,
or
fBRDFpol=a1polGDiffuse+a2FFresnelpolG1,Gaussian+a3FFresnelpolG2,Gaussian,
whereG1,Gaussian and G2,Gaussian have different sets of parameters. The parameter apol is different for s- and p-polarized radiation because of plasmon polariton effects.

The parameters in the series expansion are determined from fitting the model to the observed BRDF measurements. Software for solving non-linear least-square problems are available from several suppliers. Here, a nonlinear fitting model in Mathematica is used. Initial parameter values were obtained as described above. Both s- and p-polarized BRDF measurements were fitted simultaneously. There are two contributions with different scatter parameters plus a diffuse contribution of the generalized Lambertian type. The sharp specular peak contributes approximately 7 per cent to the total scattering while the more diffuse parts contributes with approximately 90 per cent. The fractional scattering is given by a2 = 0.07 and a3 = 0.90. The diffuse part is estimated to be 3 per cent. The corresponding shape parameters of the ASF in Eq. (13) are ν1 = 1.0 and ν2 = 1.5.

Based on these parameters, the polarimetric BRDF is predicted and compared to observations in Fig. 4.

 figure: Fig. 4

Fig. 4 BRDF measurements of bead-blasted Aluminum and initial model predictions based on limited amount of data. The difference between s-polarized (red) and p-polarized (blue) radiation is small. Data is missing ± 2 degrees around the angle of incidence due to obscuration effects.

Download Full Size | PPT Slide | PDF

A forward scattering is observed at intermediate and large angles of incidence. The direction cosine for the second scattering parameter corresponding to the 90 per cent contribution is increased by 5 per cent for angles of incidence at 40 degrees and 60 degrees and 1 per cent for the angle of incidence of 80 degrees. The result of this fine tuning of the scattering angle is shown in Fig. 5.

 figure: Fig. 5

Fig. 5 BRDF measurements of bead-blasted Aluminum and model predictions based on initial data but a small correction for increased forward scattering is introduced.

Download Full Size | PPT Slide | PDF

The diffuse Lambertian type scattering is regarded as a surface scatter and the corresponding model is used.

The model is compared with observation in more detail at a 60 degree angle of incidence. On a logarithmic scale in Fig. 6, it is observed that the diffuse scatter varies somewhat with angle of incidence. Backscatter is small and not included in this model for bead-blasted aluminum.

 figure: Fig. 6

Fig. 6 BRDF measurements and model predictions at a logarithmic scale. The diffuse scattering is here assumed to be Lambertian.

Download Full Size | PPT Slide | PDF

An initial guess for the diffuse scattering was that this accounted for 3 percent of the scattered radiation. The shape was assumed to be Lambertian, i.e. νD = 0. It can be observed in Fig. 6 that there is a difference in scatter magnitude for s- and p-polarization in the diffuse part of the angular spread function.

An increased scatter for p-polarized radiation is observed. Different values for s- and p-polarization can e.g. appear due to plasmon-polariton interaction. Parameters adopted for diffuse scatter are as = 0.02 and ap = 0.04 with the average of as,p = 0.03. The corresponding shape factor is νD = −0.1. The result is shown in Fig. 7. The difference between model and observation is attributed to a fat-tail contribution not included in the model. Fat-tail distributions are encountered in description of surface scattering and also in many areas of physics. The Gaussian and Lorentzian ASFs exhibit different behavior, especially with respect to scattering at large angles. The Lorentzian distribution shows higher scattering at large angles compared to the Gaussian distribution. This is sometimes called a “fat tail” distribution. In some cases, there is a need to add a “fat tail” distribution by adding a power law decay in the tail of the distribution. The deviation observed in Fig. 7 is an indication of the need of such an addition.

 figure: Fig. 7

Fig. 7 Comparison of model with adjusted diffuse scattering parameters to observed polarimetric BDRF measurements of bead-blasted aluminum at the angle of incidence of 60 degrees. Different s-polarized and p-polarized diffuse scatter parameters has been added to the model. Red is s-polarized, and blue is p-polarized.

Download Full Size | PPT Slide | PDF

The model parameters vary with respect to the treatment of the aluminum surface. A shiny surface will be dominated by a1 and the corresponding narrow ρ^1. For a strongly blasted surface, the parameters a2 will dominate with little contribution from a1. The specular peak might be observed at large angles of incidence due to the narrowing of the angular scatter distribution.

5.4 Example II: diffuse green paint

The BRDF model for diffuse green paint is given by:

fBRDFpol=aF,D,Ds,pHFresnelpolFDiffuseGDiffuse+aS,F,Gs,pHSurfaceFFresnelpolGGaussian+aS,F,G1s,pHSurfaceFFresnelpolG1,GaussianVolume+aS,F,G2s,pHSurfaceFFresnelpolG2,GaussianVolume,

or

fBRDFpol=a1HFresnelpolGDiffuse+a2FFresnelpolGGaussian+a3FFresnelpolG1,GaussianVolume+a4FFresnelpolG2,GaussianVolume

Often only a couple of terms dominate the contribution to the observed behavior. In this case, terms one and three are dominant. The second term is a narrow weak peak that shows up only at large angles of incidence. The third term is the dominant term and causes the broad glossy peak. The forth term is used only to indicate the appearance of a small retro-effect.

DHR and BRDF measurements of diffuse green paint have been performed at 3.39 µm. From the DHR measurements, a value of σ = 0.64(11) and an effective index of refraction n = 1.499(29) and k = 0.0 (indeterminate) was obtained using Eq. (19). The bias was found to be equal to 0.0301(38). Numbers in parentheses are standard errors. These are approximate values, useful as a starting point for the modelling of polarimetric BRDF measurements. If a less accurate model is satisfactory, this result is already useful in itself. From BRDF measurements at an angle of incidence of 5 degrees, the following parameters are obtained, ρ^ = 0.13 and ν = 1.5. As pointed out above, the angular spread parameter is assumed to decrease proportionally with the direction cosine of the angle of incidence in volume scattering. The model now encompasses five fitting parameters in total. The amplitude was allowed to vary when fitting the parameters to the BRDF measurement, resulting in Fig. 8.

 figure: Fig. 8

Fig. 8 Angular spread function parameters are determined from scaled comparison to the BRDF measurement at an angle of incidence of 5 degrees.

Download Full Size | PPT Slide | PDF

Using the three parameters determined from the DHR measurement and the two parameters determined from the BRDF measurement, the results shown in Fig. 9 are obtained.

 figure: Fig. 9

Fig. 9 Polarimetric BRDF measurements and initial model predictions. Red is s-polarized and blue is p-polarized radiation.

Download Full Size | PPT Slide | PDF

The largest sensitivity to changes in complex index of refraction can be observed close to Brewster angle. In order to emphasize the low-level scatter magnitudes, the measurements at an angle of incidence of 60 degrees are presented below on a logarithmic scale. The shape parameters have been adjusted somewhat. Backscatter is also observed, and a backscatter component is introduced by simply adding one term in the series expansion and by changing the sign of the angle of incidence. This means that the backscattered peak coincides with the angle of incidence.

From the measured BRDF data corrected values of k = 0.2 with the correspondingly adjusted value n = 1.448 are used in order to keep reflectance constant at small angles. The shape parameter of the main peak is decreased to ν = 1.0. The parameters for the Lambertian type scattering are ρD = 0.0085, νD = 0.2. The angular spread parameter for the retro-scattering is ρ = 0.02 and the scale factor is set to a very small number. These values are used in the illustration in Fig. 10.

 figure: Fig. 10

Fig. 10 Comparison of measured and predicted BRDF curves at a log-scale at an angle of incidence of 60 degrees. Parameters have been fine tuned to fit observations.

Download Full Size | PPT Slide | PDF

5.5 Example III: red and green automotive paint at 633 nm

Automotive paints often share the common property of consisting of strongly scattering pigments overcoated with a clear coat of the same index of refraction as the pigment binder. The specular reflectance from the clear coat is therefore often very similar and independent of the color of the car. An index of refraction of neff = 1.5 at visible wavelengths is often adopted. Due to the narrowness of this peak, the peak value of the BRDF is high. Not surprisingly, the diffuse pigment scattering varies drastically with the wavelength, depending on the color of the pigment.

The BRDF model for car paint is given by:

fBRDFpol=aF,D,Ds,pHFresnelpolFDiffuseGDiffuse+aS,F,Gs,pHSurfaceFFresnelpolGGaussian+aS,F,Ls,pHSurfaceFFresnelpolGLorentzian+aF,F,Ls,pHFresnelpolFFresnelpolGLorentzian+aF,D,Ls,pHFresnelpolFDiffuseGLorentzian,
or
fBRDFpol=a1HFresnelpolGDiffuse+a2FFresnelpolGGaussian+a3FFresnelpolGLorentzian+a4HFresnelpolFFresnelpolGLorentzian+a5FFresnelpolGLorentzian,
where the first term is the diffuse sub-surface scattering, the second term is the specular clear coat reflection and the third term is a scattering of low magnitude while the fourth and fifth terms together constitutes the quite complex low magnitude scattering with a fat tail observed for the green paint. This is illustrated in Fig. 11.

 figure: Fig. 11

Fig. 11 BRDF measurements of red A) and green B) car paint at 633 nm. Observe the much higher diffuse scattering level of the red paint. The diffuse distribution was modelled using the GLorentzian distribution.

Download Full Size | PPT Slide | PDF

6. Discussion

The basic functional form of the BRDF presented here is based on physical optics, and is still as flexible and as simple in form as BRDF models based on geometrical optics. The function is separated into two main parts, one describing the reflectance and one describing the angular spread function. The reflectance is determined by the material, i.e. the effective complex index of refraction. The angular spread function is described by a generalized Gaussian or Lorentzian distribution. More complex scatter distributions are described by series expansions of these basic generalized distributions. A generalized Lambertian distribution is introduced to model the diffuse Lambertian type scatter. This model has two main aspects. The first one describes surface scatter while the second one describes volume scatter and includes surface transmission. The detailed microscopic interactions and corresponding coherence properties have a role in determining the scatter distribution. The shape parameter of the generalized Gaussian function typically varies between 1 and 2 in the examples shown here.

There is a narrowing of the angular spread parameter beyond what is described by geometric interference for surface scatter. This is interpreted as a volume scatter effect and is a function of the angle of incidence.

Common behavior can be observed for similar materials such as paints varying only in pigmentation. This observation allows for a more physically intuitive interpretation and might also lead to simplified measurements in order to characterize materials of similar type.

The model can be used in efficient and accurate polarimetric scene modelling and simulations. The Stokes vectors can easily be obtained for the object polygons. The simulation is further simplified if only unpolarized sources are considered.

Acknowledgments

The authors would like to thank Tomas Hallberg at FOI - Swedish Defence Research Agency for making BRDF and DHR measurements available.

References and links

1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

2. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

3. E. L. Church and P. Z. Takacs, “Surface Scattering,” in Handbook of Optics, M. Bass, E. W. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), pp. 7.1–7.14.

4. J. C. Stover, Optical Scattering Measurement and Analysis (McGraw-Hill, New York, 1990).

5. J.-J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchoff’s law,” J. Opt. Soc. Am. A 15(10), 2735–2744 (1998). [CrossRef]  

6. A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28(6), 1121–1138 (2011). [CrossRef]   [PubMed]  

7. M. T. Eismann, Hyperspectral Remote Sensing, SPIE Press, Bellingham, Washington (2012).

8. R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002). [CrossRef]  

9. I. G. E. Renhorn, T. Hallberg, and G. D. Boreman, “Efficient polarimetric BRDF model,” Opt. Express 23(24), 31253–31273 (2015). [CrossRef]   [PubMed]  

10. M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info. Accessed on 2017–12–05.

References

  • View by:
  • |
  • |
  • |

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  2. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
  3. E. L. Church and P. Z. Takacs, “Surface Scattering,” in Handbook of Optics, M. Bass, E. W. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), pp. 7.1–7.14.
  4. J. C. Stover, Optical Scattering Measurement and Analysis (McGraw-Hill, New York, 1990).
  5. J.-J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchoff’s law,” J. Opt. Soc. Am. A 15(10), 2735–2744 (1998).
    [Crossref]
  6. A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28(6), 1121–1138 (2011).
    [Crossref] [PubMed]
  7. M. T. Eismann, Hyperspectral Remote Sensing, SPIE Press, Bellingham, Washington (2012).
  8. R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
    [Crossref]
  9. I. G. E. Renhorn, T. Hallberg, and G. D. Boreman, “Efficient polarimetric BRDF model,” Opt. Express 23(24), 31253–31273 (2015).
    [Crossref] [PubMed]
  10. M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info . Accessed on 2017–12–05.

2015 (1)

2011 (1)

2002 (1)

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[Crossref]

1998 (1)

Boreman, G. D.

Choi, N.

Greffet, J.-J.

Hallberg, T.

Harvey, J. E.

Krywonos, A.

Meier, S. R.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[Crossref]

Nieto-Vesperinas, M.

Priest, R. G.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[Crossref]

Renhorn, I. G. E.

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[Crossref]

Opt. Express (1)

Other (6)

M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info . Accessed on 2017–12–05.

M. T. Eismann, Hyperspectral Remote Sensing, SPIE Press, Bellingham, Washington (2012).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

E. L. Church and P. Z. Takacs, “Surface Scattering,” in Handbook of Optics, M. Bass, E. W. Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, New York, 1995), pp. 7.1–7.14.

J. C. Stover, Optical Scattering Measurement and Analysis (McGraw-Hill, New York, 1990).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1 The attenuation due to passing through the dielectric surface twice is illustrated with no depolarization in the diffuse scattering shown in A) and complete depolarization shown in B). Red is s-polarized, and blue is p-polarized radiation. The index of refraction was set to n = 1.5.
Fig. 2
Fig. 2 The generalized Gaussian distribution A) and the generalized Lorentzian distribution B) are shown. The scattering parameter is ρ ^ = 0.2 and the shape parameter is set to ν = (1,2,3).
Fig. 3
Fig. 3 Illustration of the similarity of Fresnel equation and DHR simulations for a diffuse paint.
Fig. 4
Fig. 4 BRDF measurements of bead-blasted Aluminum and initial model predictions based on limited amount of data. The difference between s-polarized (red) and p-polarized (blue) radiation is small. Data is missing ± 2 degrees around the angle of incidence due to obscuration effects.
Fig. 5
Fig. 5 BRDF measurements of bead-blasted Aluminum and model predictions based on initial data but a small correction for increased forward scattering is introduced.
Fig. 6
Fig. 6 BRDF measurements and model predictions at a logarithmic scale. The diffuse scattering is here assumed to be Lambertian.
Fig. 7
Fig. 7 Comparison of model with adjusted diffuse scattering parameters to observed polarimetric BDRF measurements of bead-blasted aluminum at the angle of incidence of 60 degrees. Different s-polarized and p-polarized diffuse scatter parameters has been added to the model. Red is s-polarized, and blue is p-polarized.
Fig. 8
Fig. 8 Angular spread function parameters are determined from scaled comparison to the BRDF measurement at an angle of incidence of 5 degrees.
Fig. 9
Fig. 9 Polarimetric BRDF measurements and initial model predictions. Red is s-polarized and blue is p-polarized radiation.
Fig. 10
Fig. 10 Comparison of measured and predicted BRDF curves at a log-scale at an angle of incidence of 60 degrees. Parameters have been fine tuned to fit observations.
Fig. 11
Fig. 11 BRDF measurements of red A) and green B) car paint at 633 nm. Observe the much higher diffuse scattering level of the red paint. The diffuse distribution was modelled using the GLorentzian distribution.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

α=sin( θ )cos( φ )
β=sin( θ )sin( φ )
γ=cos( θ ),
L r ( α o , β o ;λ )= f BRDF ( α i , α o , β o ;λ ) L i ( α i ;λ ) γ i dΩ,
f BRDF pol ( α i , α o , β o ;λ )= types a type pol H type pol ( α i , α o , β o ;λ ) F type pol ( α i , α o , β o ;λ ) G type ( α i , α o , β o ;λ ),
F Diffuse =1.
F Fresnel s ( α i , α o , β o ;λ )= R p ( β o 2 ;λ ) R p ( 0;λ ) R s ( α i + α o 2 ;λ )
F Fresnel p ( α i , α o , β o ;λ )= R s ( β o 2 ;λ ) R s ( 0;λ ) R p ( α i + α o 2 ;λ )
R s ( α,n,k )= | ( nik ) 1 α 2 ( nik ) 2 1 α 2 ( nik ) 1 α 2 ( nik ) 2 + 1 α 2 | 2
R p ( α,n,k )= | ( nik ) 1 α 2 1 α 2 ( nik ) 2 ( nik ) 1 α 2 + 1 α 2 ( nik ) 2 | 2 .
H Surface =1.
H Fresnel s ( α i , α o , β o ;λ )=( 1 R s ( α i ;λ ) )( ( 1 a 2 )( 1 R s ( α o ;λ ) )+ a 2 ( 1 R p ( α o ;λ ) ) )
H Fresnel p ( α i , α o , β o ;λ )=( 1 R p ( α i ;λ ) )( ( 1 a 2 )( 1 R p ( α o ;λ ) )+ a 2 ( 1 R s ( α o ;λ ) ) ).
G Diffuse ( α o , β o , ν D ;λ )= 1+ ν D /2 π ( 1 α o 2 β o 2 ) ν D /2 ,
G Gaussian ( α i , α o , β o , ρ ^ ,ν;λ )=nor m G ( α i , ρ ^ ,ν;λ )Exp( | α o α i | ν + | β o | ν ( ( 1 α i 2 + 1 α o 2 β o 2 ) ρ ^ 2 ) ν ),
nor m G ( α i , ρ ^ ,ν;λ )=1/ α o =1 α o =1 β o = 1 α o 2 β o = 1 α o 2 Exp( | α o α i | ν + | β o | ν ( ( 1 α i 2 + 1 α o 2 β o 2 ) ρ ^ 2 ) ν )d α o d β o
G Lorentzian ( α i , α o , β o , ρ ^ ,ν;λ )=nor m L ( α i , ρ ^ ,ν;λ ) ( 1+ ( α o α i ) 2 + β o 2 ( 1 α i 2 + 1 α o 2 β o 2 ) ρ ^ 2 ) ( ν+1 )/2 ,
nor m L ( α i , ρ ^ ,ν;λ )=1/ α o =1 α o =1 β o = 1 α o 2 β o = 1 α o 2 ( 1+ ( α o α i ) 2 + β o 2 ( 1 α i 2 + 1 α o 2 β o 2 ) ρ ^ 2 ) ( ν+1 )/2 d α o d β o .
f BRDF pol = a F,D,D s,p H Fresnel pol F Diffuse G Diffuse + a S,F,G s,p H Surface F Fresnel pol G Gaussian ,
f BRDF pol = a 1 H Fresnel pol G Diffuse + a 2 F Fresnel pol G Gaussian ,
f DHR pol ( α i ;λ )= α o =1 α o =1 β o = 1 α o 2 β o = 1 α o 2 f BRDF pol d α o d β o
f DHR pol ( α i ;λ )= 1+σ 1 α i 2 1+σ F Fresnel pol ( α i , α i ,0;λ )+ σ D ,
f BRDF pol = a S,D,D pol H Surface F Diffuse G Diffuse + a S,F,G1 s,p H Surface F Fresnel pol G 1,Gaussian + a S,F,G2 s,p H Surface F Fresnel pol G 2,Gaussian ,
f BRDF pol = a 1 pol G Diffuse + a 2 F Fresnel pol G 1,Gaussian + a 3 F Fresnel pol G 2,Gaussian ,
f BRDF pol = a F,D,D s,p H Fresnel pol F Diffuse G Diffuse + a S,F,G s,p H Surface F Fresnel pol G Gaussian + a S,F,G1 s,p H Surface F Fresnel pol G 1,Gaussian Volume + a S,F,G2 s,p H Surface F Fresnel pol G 2,Gaussian Volume ,
f BRDF pol = a 1 H Fresnel pol G Diffuse + a 2 F Fresnel pol G Gaussian + a 3 F Fresnel pol G 1,Gaussian Volume + a 4 F Fresnel pol G 2,Gaussian Volume
f BRDF pol = a F,D,D s,p H Fresnel pol F Diffuse G Diffuse + a S,F,G s,p H Surface F Fresnel pol G Gaussian + a S,F,L s,p H Surface F Fresnel pol G Lorentzian + a F,F,L s,p H Fresnel pol F Fresnel pol G Lorentzian + a F,D,L s,p H Fresnel pol F Diffuse G Lorentzian ,
f BRDF pol = a 1 H Fresnel pol G Diffuse + a 2 F Fresnel pol G Gaussian + a 3 F Fresnel pol G Lorentzian + a 4 H Fresnel pol F Fresnel pol G Lorentzian + a 5 F Fresnel pol G Lorentzian ,

Metrics