Theoretical evaluation of the impact of finite intervals in the measurement of the bidirectional reflectance distribution function
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Abstract
The bidirectional reflectance distribution function (BRDF) is a well-defined quantity that allows the bidirectional reflectance of surfaces to be described. However, the light propagation in a specific direction cannot be perfectly realized in practice, because the physical apertures are not infinitesimal but finite. Consequently, a BRDF measurement cannot be considered fully bidirectional, although the measure coincides with the BRDF within a certain confidence level. To properly understand the impact of the finite apertures on the BRDF measures, the deviation between the real BRDF and the BRDF to be obtained using real experimental conditions with finite apertures was theoretically studied for surfaces with realistic BRDFs. The biconical reflectance factor was used to estimate these “measured BRDFs” in different geometrical conditions, and a family of negative exponential functions was defined to assess the impact on surfaces with different angular scattering distributions. An expression for estimating the relative error from finite apertures is given, which considers the angular variation of the BRDF and the different solid angles involved in the measurement.
Keywords
Reflection BSDF, BRDF, and BTDF Scattering measurements MetrologyIntroduction
- 1.
L_{r} is linear with respect to L_{i}: For given incidence and reflection directions, L_{r} is proportional to L_{i} (L_{r} ∝ L_{i}). It is fulfilled as long as the irradiation is not so high to produce nonlinear effects.
- 2.
L_{r} is linear with respect to dΩ_{i}: For given incidence and reflection directions, L_{r} is proportional to dΩ_{i} (L_{r} ∝ dΩ_{i}), as long as “dΩ_{i} is chosen small enough that there is no significant difference in the radiant absorptance for all rays incident on dA within the solid angle dω_{i}.”1
- 3.
ρ is invariant with respect to L_{i} and dΩ_{i}: As a consequence of the two previous considerations, for given incidence and reflection directions, ρ, as defined in equation (3), is invariant with respect to L_{i} and dΩ_{i}. Since L_{r} = f_{r}L_{i} dΩ_{i} (from i and ii, where f_{r} is a constant for a given geometry), then ρ = f_{r} dΩ_{r}.
- 4.
ρ is linear with respect to dΩ_{r}: For given incidence and reflection directions, ρ is proportional to dΩ_{r} (ρ ∝ dΩ_{r}), which “holds only for values of dΩ_{r} small enough so that the partial reflectance does not change significantly with direction for rays within the solid angle dω_{r}.”1
Nicodemus preferred to use the differential term dL_{r} instead of L_{r} because “other incident radiation (from other sources in other directions) may also be diffusely reflected (scattered) into the same reflected directions along with that from the source under consideration.”1 What Nicodemus actually proposed was to avoid the dependence of the bidirectional reflectance ρ on the reflection solid angle dΩ_{r} by normalizing ρ by dΩ_{r}. This new quantity, denoted by f_{r} in equation (5), is the well-known bidirectional reflectance distribution function (BRDF), and it is widely used nowadays in diverse fields such as remote sensing, radiometry, or 3D-rendering as the scattering function. The Optics Classification and Indexing Scheme (OCIS) has reserved a code for it, which proves its general acceptance. The monograph on reflectance published by Nicodemus et al. in 1977,2 where the concept of BRDF is thoroughly explained, has been cited more than 1600 times by scientific publications in this century.
However, there are some aspects of the BRDF that are difficult to understand. Perhaps the most obvious is why the BRDF can reach any positive value, even an infinite value for perfectly specular reflectors. The reason is that f_{r} is defined as a division by a solid angle element [equation (5)]. The magnitude of these solid angle elements must be small enough to contain only directions at which the involved quantities (radiance or bidirectional reflectance) do not change significantly. For glossy surfaces, for which the reflectance changes rapidly approaching the specular peak, dω_{r} needs to be very small in order to meet this requirement, resulting in very high BRDF values near the specular peak [equation (5)]. This partial reflectance is a Dirac delta in the case of perfectly specular reflectors, so the BRDF has no defined value in specular directions.
According to the terminology introduced by Judd3 and later adopted by Nicodemus2, three adjectives are used to describe the reflectance under general geometrical conditions of irradiation and collection: directional (flux confined to an infinitesimal solid angle), conical (flux distributed over a finite solid angle), and hemispherical (flux distributed over the full hemisphere). These forms of reflectances are obtained by integration of f_{r} (r_{i};r_{r}).
In practice, ω_{i} and ω_{r} are not completely zero, and a biconical quantity, instead of a bidirectional one, is measured. Therefore, these concepts will be used in the next section to establish the relation between the BRDF and the quantity actually measured. Some of the assumptions in the BRDF definition, as described above (“dΩ_{i} chosen small enough that there is no significant difference in the radiant absorptance for all rays incident on dA within the solid angle dω_{i},” or “dΩ_{r} small enough so that the partial reflectance does not change significantly with direction for rays within the solid angle dω_{r}”), need to be considered in the measurement of the BRDF, in order to avoid systematic errors or to account for them in the uncertainty budget. In practice, since physical irradiation and collection apertures are finite (neither infinitely big nor infinitesimal), and the measurement area on the surface has a certain size, it is irradiated with radiant fluxes from different directions, and the collected radiant flux comes from different directions too. To understand properly the limitations of a BRDF measurement, the deviation between the real BRDF and the BRDF to be obtained using real experimental conditions with finite apertures needs to be assessed.
The consequence of using a finite measurement area and finite irradiation and collection solid angles is that the BRDF is evaluated as a weighted average over a set of pairs of irradiation and collection directions, which can be modeled with the expression in equation (7). As long as this weighted average coincides with the real BRDF with a negligible deviation, the measurement conditions can be considered adequate.
The upper left plot shows a BRDF distribution without curvature. In this case, the average BRDF coincides with the value of the BRDF at θ_{asp} = 0°. For this distribution, the chosen experimental condition regarding the solid angle does not limit the accuracy of the measurement. This would not be true for the distributions shown in the upper right plot and the lower left plot, which are very representative of the BRDFs of matte materials at large incidence angles.4 In both cases, there is a curvature in the distribution, and a slight deviation is observed between the average BRDF and the actual value of the BRDF at θ_{asp} = 0°. This deviation is almost three times larger for the distribution in the upper right plot, which can be explained by a steeper slope of the distribution. For both cases, the deviation is always smaller with a smaller collection angle. Finally, the distribution shown in the lower right plot is not monotonic and has a sharp peak. This example may represent specular reflection. In this case, the deviation is quite large, and the measurement would be meaningless. Again, the deviation would always be lower using a smaller collection solid angle.
From this discussion, it can be asserted that the usage of finite apertures limits the BRDF measurements when its distribution has a curvature in the measuring angular range, being more critical when the relative variation of the distribution is higher. The maximum finite sizes of involved intervals that would give an acceptable error are what this article intends to provide. Specular reflection is an important case, but it is not the only one. Any reflectance highly nonlinear with respect to the geometry must be evaluated.
It is an important metrological issue. The International Commission on Illumination (CIE) has recently created a technical committee “Recommendation on the geometrical parameters for the measurement of the bidirectional reflectance distribution function (BRDF)” to “provide geometrical recommendations for the BRDF measurement according to the type of sample under investigation, in order to allow better comparison between the different instruments, to improve the traceability of the measurements, and to help the user to choose the right angular configuration.” A description of the metrological issues to be considered in the BRDF measurement around the specular peak was published,5 and it is a very good reference to understand the problem in practice.
This work examines the impact of using finite intervals in the measurement of BRDF, for BRDFs with distributions with different relative variations, and it is proposed as a practical approach to evaluate the limitations of instruments designed for the measurement of the BRDF, and the errors produced by these limitations. Only the impact due to the geometry is taken into account in this article. Out of scope are other effects from using finite areas, such as nonuniformity, which need to be separately considered.
Estimation of the measured BRDF
Only the pair of axial directions of the solid angles in Fig. 2 reproduces exactly the bidirectional geometry at which the BRDF is intended to be measured. The inclusion of the other pairs of directions has an impact on the measurement which depends on the sensitivity of the BRDF to geometry and to the size of the solid angles. This impact is what we want to examine here.
R_{A} is the radius of the measurement area, r and γ are, respectively, the radial and polar coordinates used to integrate over the measurement area, κ_{i} and κ_{r} are the angles subtended by the irradiation and collection apertures (see Fig. 2), and θ and φ, respectively, refer to the polar and azimuth spherical coordinates defining corresponding directions denoted as r. Notice that the angles subtended by the measurement area κ_{ma,i} and κ_{ma,r} (see Fig. 2) are used in the area integral \(\int_{0}^{2\pi } {\text{d}}\gamma \int_{0}^{{R_{\text{A}} }} {\text{dr}} F\).
Evaluation of the measured BRDF for different conditions
To understand the impact of finite angles, these representative scattering functions were used along with different realistic geometrical conditions, with combinations of the subtended angles κ_{i}, κ_{r}, κ_{ma,i}, and κ_{ma,r} (as defined in Fig. 2). To reduce the analysis, a negligible value of κ_{i} was used (0.0006°, very small irradiation aperture), which means that the effect produced by the values of κ_{r} prevails over the effect produced by κ_{i}. There is reciprocity between κ_{i} and κ_{r} since, according to the Helmholtz reciprocity principle, the result from the analysis is equal to the inverted case at which apertures and distances are interchanged between source and collector, as long as the measurement is for unpolarized reflectance13. Values of κ_{r}, κ_{ma,i}, and κ_{ma,r} were selected as 0.11°, 0.23°, 0.49°, or 0.92°, with the additional condition that κ_{ma,i} is always equal to κ_{ma,r}, equivalent to considering that the selected distances from source and receiver to the measurement area are identical. The selected κ values are realistic. If source and detector were located at a distance of 1 m from the sample, the diameter of their areas (assumed circular) would be 3.8 mm for the minimum (0.11°) and 32.1 mm for the maximum (0.92°). It must be reminded here that the solid angles subtended by the measurement area are projected under different incidence and collection directions and that the values of κ_{ma,i} and κ_{ma,r} reported in this work are those at normal irradiation or collection directions.
The measured BRDF was calculated using equation (11) for the four functions shown in Fig. 3, and for the different geometries, as described in the above paragraph.
As expected, the larger the β (related to sample’s scattering curvature) and κ values (subtended angles) are, the larger the deviation is. The relative deviation obtained from the evaluation is similar for the two selected irradiation angles, but slightly lower for the largest one. (Note that different irradiation angles are represented with lines of different thicknesses in Fig. 4.) The reason for this difference is that at larger angles the incidence angles θ_{i} on the measurement area are more closely distributed around the irradiation angle θ_{i0}, since the solid angle subtended by this area is projected.
This relative deviation is represented as a function of κ_{r}, and every curve corresponds to a different value of κ_{ma,i}, which is equal to κ_{ma,r} in this case. The curves are constant when the effect of κ_{ma,r} dominates that of κ_{r}.
General rule to estimate error
The error for any condition can be estimated as the difference between f_{r} and the result from equation (11). However, it may be of interest for the experimentalist to have a general rule or metric to estimate errors. The described methodology was used to obtain the following general rule to calculate the error introduced by finite intervals when measuring BRDFs varying with the incidence and collection directions within a small angular range. Equations (11)–(13) were used to estimate the “measured” BRDF under different measuring conditions of surfaces with the scattering distribution functions described by equation (14).
An estimation of a “measured” BRDF is defined by β, S/S_{max} [or S (β, θ_{asp})/S(β,0)], θ_{i0}, κ_{i}, κ_{r}, κ_{ma,i}, and κ_{ma,r}. Notice that θ_{asp} can be determined from β and S/S_{max} (θ_{asp} = -ln(S/S_{max})/β) and that θ_{r0} can be obtained from the values of θ_{asp} and θ_{i0}. In order to assess the impact of the different variables on the error due to finite intervals, two sets of estimations were defined, both of them for β = 1/2 degrees^{−1}, β = 1/4 degrees^{−1}, β = 1/8 degrees^{−1}, and β = 1/16 degrees^{−1}. The motivation of this strategy was to make the analysis more efficient in terms of computational time, while providing the most valuable information. Note that each estimation is the sum of around 50 millions terms used to calculate the integral in equation (11) with a reduced level of noise and that the calculation of the second set of estimations described below took 80 h.
Description of the two sets of estimations used to assess the impact of the different variables on the error due to finite intervals
Variables | Set 1 of estimations | Set 2 of estimations |
---|---|---|
θ _{i0} | 5° | From 15º to 75º (Δ = 15º) |
S/S_{max} | From 0.25 to 0.95 (Δ = 0.05) | 0.3, 0.6, 0.9 |
κ_{i}, κ_{r}, κ_{ma,i}, κ_{ma,r} | κ_{r} = 0.23°, 0.46°, 0.92°, 1.83° Relation κ_{i} : κ_{r} : κ_{ma,i} : κ_{ma,r} = 0.25:1:0.5:1 | All combinations with any κ taking 0.11°, 0.34°, and 0.92° |
Number of estimations | 240 | 1620 |
The second set of estimations was defined to assess the variation of the error at different values of θ_{i0} and to confirm the relationships observed after examining the estimations of the first set. In this set (see third column of Table 1), fewer values of S/S_{max} were used (0.3, 0.6, and 0.9). θ_{i0} ran from 15° to 75°, with steps of 15°, and estimations were done for all combinations of the values 0.11°, 0.34°, and 0.92° for κ_{i}, κ_{r}, κ_{ma,i}, and κ_{ma,r}.
By using the 1860 estimations, it was possible to obtain a function expressing the dependence of the 95th percentile of the estimated error on S × F_{κ}. After trying different types of functions, it was found that an exponential function was the most adequate to relate estimated errors to S × F_{κ}. The values of S used in the fitting were calculated as the product of the parameters (S/S_{max}) and β (see Table 1). The equation resulting from the fitting is given below in equation (18). This function is shown in the lower plot of Fig. 5 as a continuous line. This function allows the 95th percentile of the relative error [δ_{95}(f_{r})] to be estimated from the κ values and the curvature of S, and it can be used as metric for this kind of error.
(θ_{m} being the measurement angle) can be used as model, which is symmetric to the previous one with respect to S = S(θ_{m}) and produces the same value of δ_{95}(f_{r}).
This expression is obtained from the fit shown in the lower plot in Fig. 5. The value of β must be given in degree^{−1}, and the values of the different κs, and any angular variable, in degrees.
Discussion
Equations (16)–(18) developed from the simulations described above can be used to evaluate the value of the 95th percentile of the relative finite-intervals-related error for a particular experimental setup. Equation (18) is composed of two important factors: \(\beta \times e^{{ - \beta \times \left| {\theta - \theta_{0} } \right|}}\) and F_{κ} which are, respectively, related to the scattering distribution function of the surface and the size of the finite intervals. Any of these two factors can make the error negligible if it is small enough, no matter the value of the other factor. For instance, a small β, which makes the first factor very small, represents a Lambertian surface and produces a negligible error regardless of the size of the finite intervals (the values of κ). On the other hand, making κ values small enough, the committed error can be made negligible even in the extreme case of very large values of \(\beta \times e^{{ - \beta \times \left| {\theta - \theta_{0} } \right|}}\), at geometries close to specular peaks.
In general, keeping similar values for κ_{i} and κ_{ma,i} (and for κ_{r} and κ_{ma,r}) is optimal in a goniospectrophotometer, since a lower value of only one of these variables does not reduce significantly the relative error due to finite intervals, and reduces the available light, which increases other uncertainty sources.
Equation (18) supports design or characterization of goniospectrophotometers or multiangle spectrophotometers, since it allows the applicability range of the instrument to be specified for a target uncertainty. For instruments with a fixed value of F_{κ}, as it is the case of commercially available portable multiangle spectrophotometers, it may help to estimate for different types of samples the finite intervals uncertainty at their measurement geometries, which usually have different aspecular angles. Aspecular angles would be used as |θ − θ_{0}| value in equation (18), and different values for β would be used to illustrate different types of samples, from very diffuse to very glossy. Instruments allowing the variation of the size of apertures, and consequently F_{κ}, can be adapted to the specific type of sample to be measured. For these instruments, the necessary F_{κ} is worked out from equation (18), using a value of β which, based on the experience, corresponds with the observed glossiness of the sample, and, as δ_{95}, the aimed maximum relative uncertainty. Those instruments designed to measure at the specular peak, as glossimeters, introduce an error which depends on their apertures. In such cases, that error cannot be estimated by equation (18), but by comparing the result from equation (11) with the BRDF. To illustrate the error for different types of samples, the BRDF can be modeled with the family of curves in equation (14). In general, equation (11) is recommended to simulate complex conditions for which the derived equation (18) is not suitable.
To estimate experimentally the relative error for a given specimen, the values of β and θ_{0} must be known for every measurement geometry. If the expected error due to finite intervals is relatively low, the “measured” BRDF itself can be used to estimate β and θ_{0}. However, if a high error is expected in the measurement, as for low aspecular angles in highly glossy specimens, some deconvolution algorithm needs to be applied to estimate β and θ_{0} from the deconvolved BRDF.14,15 In these cases, equation (18) is recommended to be used to characterize the limitations of the instrument to measure BRDFs with high angular dependence, and to redesign it if required.
Regarding κ values, the use of equation (18) is restricted to values below 1º.
Conclusions
The impact of using finite intervals (for the measurement area, irradiation and collection apertures, and solid angles) has been examined for the measurement of BRDF. A theoretical approach has been proposed to evaluate the limitations of experimental systems for the measurement of the BRDF and to account for the relative error exclusively due to finite intervals. This error, calculated as the relative difference between a biconical integration of the BRDF and the actual BRDF, is composed of two main factors, which are related to the scattering distribution function of the surface, and the geometries of the incident and collected radiant fluxes. Any of these factors can reduce to negligible the total error if it is set to a value small enough. The general rule for experimentalists provided here should be useful for the design of instruments for measuring BRDFs very dependent on incidence and collection geometries, such as those intended to measure gloss or iridescence.
Notes
Acknowledgments
This article was written within the EMPIR 16NRM08 Project BiRD “Bidirectional reflectance definitions.” The EMPIR is jointly funded by the EMPIR participating countries within EURAMET and the European Union. The author is also grateful to Comunidad de Madrid for funding the project SINFOTON-CM: S2013/MIT-2790.
Funding
The funding for this was obtained from EURAMET and the European Union (16NRM08 Project BiRD), and Comunidad de Madrid (S2013/MIT-2790).
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