Apr 27, 2021 Fundamental scattering quantities for the determination of

: The bidirectional reflectance distribution function (BRDF) and the bidirectional scattering - surface reflectance distribution function (BSSRDF), which relate radiance at the surface to irradiance and radiant flux, respectively, are regarded as the most fundamental scattering quantities used to determine the reflectance of objects. However, for materials where the optical radiation is transmitted under the surface, this radiance depends not only on irradiance and radiant flux, but also on the size of the irradiated area of the surface. This article provides insight into such dependence under the special condition in which the radiance is evaluated within the irradiated area and, consequently, is produced by both the insurface reflection and the subsurface scattering, in contrast to the situation in which the radiance is evaluated at non-irradiated areas and only subsurface scattering contributes. By explicitly considering both contributions, two other scattering quantities are defined: one that accounts exclusively for the insurface reflection and the other that accounts for subsurface scattering. In this regard, these quantities might be considered more fundamental than the BRDF and the BSSRDF, although they are coincident with these two functions apart from the above-mentioned special condition and for materials with negligible subsurface scattering. In this work, the relevance of the proposed scattering quantities is supported by experimental data, practical considerations are given for measuring them, and their relation to the bidirectional transmittance distribution function (BTDF) is discussed.


Introduction
According to the CIE International Lighting Vocabulary (ILV) [1], reflectance is defined as "the ratio of the reflected radiant flux or luminous flux to the incident flux in the given conditions", while transmittance is defined as "the ratio of the transmitted radiant flux or luminous flux to the incident flux in the given conditions". Both quantities describe the interaction of the radiant flux (Φ i ) with the objects, but for given conditions. These conditions include the irradiated area of the surface (A i ), the collection area of the surface for which the optical radiation is evaluated (A r ), the irradiation and collection solid angles (ω i and ω r ) (Fig. 1, Table 1), and other conditions not addressed in this work such as polarization. Therefore, when reporting the value of reflectance or transmittance, these conditions must be specified explicitly. Quantities such as diffuse reflectance, specular (regular [1]) reflectance, diffuse transmittance, regular transmittance or others [2] are defined to implicitly include well-established geometrical conditions. Many other quantities might be defined, that would provide very different results, making the measurement  Table 1 for a better understanding of the equations. of reflectance and transmittance in given conditions far from being a general characterization of the interaction of the radiant flux with objects.
In order to obtain a more general description, quantities must be defined that will allow the reflectance and the transmittance to be derived for any condition. Nicodemus et al. [3][4][5] defined the Bidirectional Reflectance Distribution Function (BRDF) with the purpose of describing such a fundamental quantity for reflectance. This function describes the bidirectional reflectance for infinitesimal solid angles and for any irradiation (r i ) and collection (r r ) directions ( Fig. 1, Table 1). The BRDF should allow reflectance to be calculated for any condition via integration, as thoroughly described in Ref. [5], where it is defined as: f r (r i ; r r ) = dL r (r i ; r r ) dE(r i ) where E is the irradiance (= dΦ i /dA i ), which must be uniform on the sample, and L r is the emerging radiance resulting from irradiation. The condition of being irradiated with a large enough area A i will be commented on later; at this point, it is convenient to define the quantity: f * r (r i ; x r , r r ; A i ) = dL r (r i ; x r , r r ; where the only difference from Eq. (1) is that no condition is imposed on the size of the irradiated area. In this case, the dependence on the collection position on the surface x r is explicitly specified, where both f * r and L r depend on the position with respect to the center of the irradiated area in objects allowing transmission of optical radiation under the surface, in contrast with f r , where the "large enough area" condition is applied precisely to avoid this dependence.
Differentials are used to indicate that the irradiation is incident from a differential element of solid angle ω i in direction of r i , with dE(r i ) = L i (r i ) cos θ i dω i , where L i and θ i are the  incident radiance and the incidence angle, respectively. In this and in the following equations, Leibniz notation is used, for which quotients of differentials can be interpreted as derivatives (non-standard analysis). To our knowledge, this interpretation does not incur any contradictions in analogous expressions with radiometric quantities. Nicodemus et al. [5] related the BRDF to a more fundamental distribution function, the Bidirectional Scattering -Surface Reflectance Distribution Function (BSSRDF, for which the symbol f ssr is used in this article; the subscript "ssr" stands for "Scattering -Surface Reflectance"): where L ssr is the emerging radiance produced by the incident radiant flux Φ i . Although both dL r and dL ssr are radiances at a specific position x r , the notation used in Eq. (3) differs from that used in Eqs. (1) and (2) because dL r and dL ssr refer to differentials of different orders. For dL ssr , both radiance and BSSRDF depend explicitly on the positions of the irradiation (x i ) and the collection (x r ), and not only on the directions. This is relevant for addressing the spatial dimension of the scattering, as will be done in this article. It must be noted that the BSSRDF is a particular example of the even more general scattering function called S in Ref. [6], which also includes spectral and polarization variables. Note that, while the BSSRDF given in Ref. [5] refers to flat surfaces, which is the most feasible approach for metrologists, in this article, it is considered in a more generalized way for any kind of surface, such as in computer graphics [7], where the BSSRDF is interpreted in a way similar to the scattering function [6,8].
The reason why a scattering quantity that depends on x i and x r must be defined is the existence of subsurface scattering. The scattering caused by an object is often due to two optical phenomena: reflection at the air-matter interface (here referred to as insurface reflection), and scattering within the bulk (referred to as subsurface scattering). These two phenomena generate different lateral light propagation distances, which can be defined as the distance between the position where the incident radiant flux irradiates the object and the furthest position where a proportion of it may exit. Insurface reflection produces a very short lateral propagation due to inter-reflections within the concavities in the micro-rough structure of the surface, and approaches zero for flat surfaces. However, subsurface scattering can produce considerable lateral propagation, and the irradiation at x i gives rise to non-negligible radiance at non-irradiated positions x r . Very low scattering and absorption coefficients yield a transparent aspect, whereas, at the other extreme, very high scattering coefficients or absorption coefficients provide an opaque aspect. Translucent objects are those with appropriate intermediate scattering and absorption coefficients and thicknesses for permitting light to pass through but not allowing the objects on the opposite side to be clearly visible.
According to Nicodemus et al. [5] [Eqs. (5)-(9) therein], the BRDF at a position x r can be calculated by integrating the BSSRDF via the following reasoning.
Let dL r (r i ; x r , r r ; A i ) be an element of radiance at position x r for given irradiation and collection directions (r i , r r ), produced by irradiation of an area A i . The area A i can be considered a collection of differential elements of area dA i , each with an element of incident radiant flux, dΦ i (x i , r i ) (see Fig. 1, Table 1). According to the definition in Eq. (3), dΦ i (x i , r i ) contributes to dL r (r i ; x r , r r ; A i ) with a proportion given by: Thus, dL r (r i ; x r , r r ; A i ) is expressed as the integral of these elementary radiances, i.e.: where a uniform irradiation has been assumed; in this way, the irradiance dE becomes independent of the point of incidence. By rearranging terms, Eq. (5) can be written as: According to Nicodemus et al. [5], this equation allows the relation between the BRDF and the BSSRDF to be established under the condition with a (as stated by the authors) "uniform irradiance over a large enough area (A i ) of a uniform and isotropic surface". A "large enough area" A i is required to make the BRDF invariant around x r . Because of the dependence of the BSSRDF on x i and x r , this condition is required even in the absence of local variations of the scattering properties of the surface, or under the assumption that any spatial inhomogeneity of the reflectance is spatially averaged by the detector. Therefore, due to this "large enough area" condition, the BRDF definition excludes situations faced with smaller irradiation areas; this leaves the following question unanswered: How can the radiance generated at an object's surface be described when the irradiated area is not "large enough"?
The radiance of the surface is produced under two different conditions. On the one hand, the radiance at x r = x i is caused by both insurface reflection and subsurface scattering, while on the other hand, the radiance at x r ≠ x i is caused only by subsurface scattering. Therefore, a discontinuity in the BSSRDF is expected at x r = x i . This is a very special and important condition that needs to be closely examined.
This article gives insight into the condition in which the radiance is evaluated within the irradiated area and, consequently, is produced by both the insurface reflection and the subsurface scattering. By explicitly considering both contributions, two other scattering quantities are defined: one accounting exclusively for the insurface reflection and the other only for the subsurface scattering. In this regard, these quantities might be considered more fundamental than the BRDF and the BSSRDF, although they are coincident with these two functions apart from the above-mentioned special condition and for materials with negligible subsurface scattering.

Discontinuity and dependence on the irradiated area
As introduced above, the radiant flux dΦ i (x i , r i ) that irradiates each elementary area dA i contributes to dL r (r i ; x r , r r ; A i ) with a proportion given by Eq. (4). In the following derivation, the discontinuity of dL ssr (x i , r i ; x r , r r ) at x i = x r is explicitly taken into account.
The radiance dL ssr is expressed as the sum of the radiance produced by the insurface reflection (dL is ) and the radiance produced by the subsurface scattering (dL ss ): where there is no spatial dependence on x r or on x i for dL is because it describes the radiance that is exclusively due to the scattering at the air-matter interface and is therefore non-zero only for Since dL is is zero when x r ≠ x i , Eq. (4) can be written as: where the definition of the function f ss is similar to that of the BSSRDF but excludes dL is : The BSSRDF can be obtained directly from Eq. (8) as: where f is is defined by: which is similar to f * r but excludes dL ss .
The differential dA i at the denominator in the second term of Eq. (10) describes the fact that the variation of the radiance with respect to the radiant flux increases for smaller irradiation areas, and that this variation is not completely defined unless the second term of the equation is negligible with respect to the first.
On the other hand, f * r is obtained from Eq. (8) via integration over A i to obtain dL r (r i ; x r , r r ; A i ). In analogy with Eq. (5), where a uniform irradiation has again been assumed. The integration over A i involves the condition Finally, f * r can be expressed as: Whereas the first condition of the equation is related to the BRDF, the interpretation of the second one (x r ∉ A i ) is not obvious. At the back of a planar sample with parallel interfaces, the second condition is related to the scattering quantity usually referred to as the Bidirectional Transmittance Distribution Function (BTDF) [9][10][11]. In such a case, it can be expressed as: where the subscript "r" for "reflectance" in f * r and L r was replaced with "t" for "transmittance". As in the case of f * r , this quantity is invariant with respect to an area A i beyond a certain limit area. In this limit area, there is a significant possibility that entering photons will exit through the collection area. Very favorable situations are present when the thickness of the object is much smaller than √ A i , or when it is almost transparent, provided that no significant light guiding takes place within the sample via successive total internal reflections. By analogy with the BRDF, this quantity should be called BTDF (f t ) only if this "large enough area" condition is fulfilled. When the radiance L t must be characterized for any size of the irradiated area, f ss (or f ssr , which is equivalent in these conditions) is the appropriate quantity.

Fundamental scattering quantities for reflectance and transmittance
The analysis in the previous section demonstrates the convenience of defining new fundamental quantities to explicitly emphasize the spatial distribution dependence of the irradiation and to completely determine the reflectance and the transmittance of objects without the restriction imposed by such dependence.
The function f is (r i ; r r ), which is referred to here as the Bidirectional Insurface Scattering function (BIS), is defined, in analogy to f * r , by Eq. (11) and has units sr −1 . In this equation, as commented on above, L is refers to the radiance produced exclusively by insurface reflection. In contrast with the function f * r defined in Eq. (2), this function is independent of the irradiated area A i because the radiance involved excludes by definition any radiance produced by subsurface scattering, which is proportional to the total radiant flux on the surface (E × A i ).
The function f ss (x i , r i ; x r , r r ), which is referred to here as the Bidirectional Bipositional Subsurface Scattering function (BBSS), is defined, in analogy with the BSSRDF, by Eq. (9) and has units m −2 · sr −1 . In this equation, L ss refers to the radiance produced exclusively by the subsurface scattering. Unlike the BSSRDF, it excludes by definition any radiance produced by the insurface reflection.
The explicit dependence of the scattering quantities f * r , f ssr and f * t on the irradiation area and on these proposed fundamental scattering quantities is given in Eq. (13), Eq. (10) and Eq. (14), respectively, where the "large enough area" condition is not applied.
Note that, for objects for which the optical radiation transmitted at the air-matter interface is negligible with respect to the reflected optical radiation, the following is obtained: regardless of the size of the irradiation area, and that for any object, the following identity can be used except for x r = x i :

Relations between the proposed fundamental scattering quantities and f * r
The radiance produced by insurface reflection (L is ) and the radiance produced by subsurface scattering (L ss ) are not directly evaluable by measurement when x r ∈ A i ; as a consequence, neither f is nor f ss is directly measurable except in the case described by Eq. (15). However, they can be related to f * r as explained in the following. Over a range of |x i − x r | length values much smaller than the absorption mean free path, the optical radiation flux crossing the air-matter interface at position x i and then undergoing multiple scattering within the bulk can go back across the air-matter interface at any position within this range with a very similar probability. Therefore, assuming √ A i to be much smaller than the absorption mean free path, f ss can be regarded as independent of the spatial dimensions x i and x r [f ss (r i ; r r ) = f ss (x i , r i ; x r , r r )]; for x r ∈ A i , Eq. (13) can be written as: By taking the derivative with respect to A i , the following is obtained: The dependence of f is on f * r can be expressed from Eqs. (17) and (18) by: for an A i small enough to make f ss independent of spatial variables.
Equations (18) and (19) relate f ss and f is , respectively, to f * r . Equation (18) expresses the fact that the function f ss is the derivative of f * r under the specific condition x r = x i . Since, at this condition, the radiance L ss is not always directly measurable because both the insurface reflection and the subsurface scattering contribute to the reflected radiant flux, it can be more practical to define f ss as: For measurements, Eq. (18) and Eq. (19) can be approximated as: andf wheref ss andf is are the measurements of f ss and f is , respectively, and A i,1 and A i,2 are irradiated areas of different sizes, small enough to make Eq. (17) valid. According to Eq. (10), f ssr (usually obtained as L r /Φ i ) equals f ss when it is much larger than f is /dA i , that is, when the condition dA i ≫ f is /f ss is fulfilled. This condition, which involves an infinitesimal element, may be confusing, but makes sense in a BSSRDF measurement, where ∆A i instead of dA i would denote the finite-size irradiated area. The condition implies that L ss ≫ L is and that most of the total radiance is produced by the subsurface scattering term. On the other hand, according to Eq. (17), f * r (usually obtained as L r /E) equals f is for small enough values of A i , i.e., when From these considerations, a very simple formula can be given for the measurement of f is and f ss at the condition Here, the increment ∆A i is used instead of the infinitesimal dA i because finite intervals are involved in the measurements.
Notice that Eqs. (23) and (24) require at least rough previous knowledge of the order of magnitude of the values of f is and f ss to select the adequate irradiated area. Otherwise, an iterative procedure can be used.
One may question why a value of ∆A i that is larger than A i needs to be used for measuring f ss , since dA i is included in A i (Fig. 1, Table 1). However, it should be noted that f is is not related to f ss , and that both measurements described in Eqs. (23) and (24) are independent. The difference in the nomenclature of both irradiated areas simply allows these equations to be related to Eqs. (10) and (17).

Illustration by means of a measurement
Experimental results are shown in Fig. 2 to illustrate the conclusions from this article. The measurements were carried out by means of the goniospectrophotometer described in [12,13], with a camera for spatial resolution in the detection path. The radiance of the reflected optical radiation, the size of the irradiated area and the irradiance on the sample were measured, allowing both f * r and f ssr to be obtained from the same measuring procedure. The irradiated area was uniform and always much larger than the collection area, which is given by the field-of-view area of the pixels of the camera (∆A r = 2 × 10 −3 mm 2 ). The data presented in Fig. 2(a) consist of BSSRDF measurements of a translucent sample made of a polycarbonate with organic scattering particles. The data was obtained at a fixed polar angle of incidence of 15 o , a fixed polar angle of collection of 10 o , fixed azimuth angles of incidence and collection of 0 o with respect to the incidence plane, and a fixed wavelength of 600 nm. The figure represents the BSSRDF as a function of a, which is the distance from the center of the irradiation spot (x i ) to the collection position (x r ) (Fig. 1, Table 1). Two sets of measurements are shown for two different irradiated areas (∆A i,1 and ∆A i,2 ), which are circular areas of 2.1 mm 2 and 8.4 mm 2 , respectively. Note that the a-axis is on a logarithmic scale to better show the results at the irradiated area. The vertical lines in the plot show the positions of the edges of the irradiation spots. It must be noted that, although the very small collection area allows better insight into the spatial distribution, only one integrated f ssr value should be reported within the irradiation spot, which corresponds to the elementary irradiated area ∆A i for assessing the BSSRDF. This f ssr value at x r = x i can be given as the ratio between the average radiance within the irradiated area and the total radiant flux. The data presented in Fig. 2(b) consists of the BSSDRF data in Fig. 2(a) multiplied by the size of the irradiated area. Therefore, the importance of this additional figure lies in its absolute value. The data represents measurements of f * r and can only be interpreted as BRDF (f r ) measurements for "large enough" irradiated areas, and at irradiated positions (x r ∈ A i ), on the left of the vertical lines, but not when x r ∉ A i . Note that, to better relate the experimental results to the previous theoretical analysis, the irradiated areas are denoted as ∆A i for BSSRDF measurements and as A i for f * r measurements, although they represent the same area. Several observations support the proposed formalism: 1. In Fig. 2(a), f ssr is independent of ∆A i outside the irradiation spot (right side of the vertical line) but not within the irradiation spot (left side), where the smaller spot produces a much higher result.
2. In Fig. 2(b), the f * r measurements show a clear dependence on A i . The larger the irradiated area is, the larger the f * r value is, which corresponds to Eq. (13).
3. The two figures show that neither f * r nor f ssr are completely independent of the irradiated areas.
4. The variation of f ssr from condition x r = x i to condition x r ≠ x i (within and outside the irradiated area) is larger for the smaller irradiated area [ Fig. 2(a)]. That would mean that this abrupt variation tends to disappear for larger areas, as predicted by Eq. (10) (f is /∆A i → 0 for very large values of ∆A i ).
5. By using the data shown in Fig. 2(b), an estimated value at x r = x i of f ss ≈ 413 m −2 · sr −1 was calculated in accordance with the approximation in Eq. (21). This value is coherent with the f ssr value at the edge of the larger irradiation spot (336 m −2 · sr −1 ), as shown in Fig. 2(a). This is expected because, as predicted by Eq. (10), f ss approaches f ssr for large irradiated areas.
6. By again using the data shown in Fig. 2(b), two estimated values of f is were calculated from Eq. (22), one for each irradiated area. For this calculation, the f * r values at the center of the spot were used. Very similar values were obtained regardless of whether the large or the small irradiated area (0.0011 sr −1 and 0.0012 sr −1 , respectively) was used for the calculation. These values are in agreement with the difference between the f * r values for small irradiated areas at the center of the spot (0.0021 sr −1 ) and outside and close to the edge (0.0009 sr −1 ) as shown in Fig. 2(b). Equation (13) predicts that f is approaches f * r for small irradiated areas; therefore, it is expected that, for smaller irradiated areas, the value of f * r at the center of the spots would approach f is (around 0.001 sr −1 , as calculated above) and that its value outside the spots would approach zero. These trends can even be observed in the two curves for different areas shown in Fig. 2(b). 7. The previous estimations provide that f is /f ss ≈ 2.5 mm 2 ; according to Eqs. (23) and (24), this means that the larger area ∆A i,2 is more convenient for measuring f ss (8.4 mm 2 is more than three times larger than 2.5 mm 2 ) than the smaller area A i,1 is for measuring f is (2.1 mm 2 , which is similar to 2.5 mm 2 ).

Discussion
The BRDF is well accepted as a fundamental scattering quantity for the determination of reflectance. However, for objects allowing the optical radiation to be transmitted under the surface, the BRDF needs to be defined with "large enough" irradiated areas because it is invariant for larger irradiated areas only at this condition. Consequently, the BRDF does not completely describe the reflectance of these objects for any irradiation condition. For the "large enough" condition, the radiance caused by the subsurface scattering prevails over the radiance from insurface reflection. For the "small enough" condition, the insurface reflection would prevail over the subsurface scattering contribution. For arbitrary intermediate sizes, both contributions are important. The BSSRDF can be regarded exclusively as a fundamental scattering quantity with the exception of the condition x r = x i . For this condition, the radiance is produced both by the insurface reflection (proportional to the irradiance) and by the subsurface scattering (proportional to the radiant flux).
Given these considerations, insurface and subsurface scattering quantities need to be given separately, as they are quantities that are more fundamental for reflectance than the BRDF and the BSSRDF.
For transmittance, the condition x r = x i is always excluded. However, the BTDF, as well as the BRDF, needs to be defined with a large enough irradiation area to be invariant for larger irradiation areas and makes the Bidirectional Bipositional Subsurface Scattering function [BBSS, Eq. (9)] a more fundamental quantity for transmittance.
The proposed fundamental scattering quantities can be related to the radiance at the surface of an object in a more practical way as: where the Dirac delta function δ(x i − x r ) replaces the discontinuous function in Eq. (10). The Dirac delta function is adequate for integration over A i , resulting in the value of the integrated function f is (r i ; r r ) as long as x r belongs to A i , and zero otherwise. Note that δ(x i − x r ) has the inverse dimension of its argument by definition. In this case, it has the dimension of inverse squared length because f is is relative to a surface and x i and x r are regarded as two-dimensional variables. Equation (25) describes the variation of the radiance of an object with the incident radiant flux using the two fundamental scattering quantities defined in this article.
In this work, we conclude that the BRDF and the BSSRDF, as defined in Eqs. (1) and (3), can be regarded as fundamental scattering quantities under most of the conditions, but the following points must be considered: 1. The "large enough area" condition in the definition of the BRDF favors the inclusion of the subsurface scattering contribution, whereas a "small enough area" condition would make the insurface reflection contribution prevail. Under this latter condition, f * r coincides with the Bidirectional Insurface Scattering function, BIS.
2. The BSSRDF coincides with the Bidirectional Bipositional Subsurface Scattering function, BBSS, under the condition x r ≠ x i , which includes only the subsurface scattering; however, when x r = x i , the insurface reflection is also included, meaning that the BSSRDF is no longer a fundamental scattering quantity. In such a case, Eqs. (9) and (18) are more adequate definitions for the subsurface scattering quantity.
To obtain representative measurements of samples, the size of the evaluated area must be selected such that the result is spatially invariant. In this article, it is assumed that the diameters of both irradiated and collection areas contain many times the wavelength of the optical radiation and the correlation length of any internal structure. This is fulfilled for the shown experimental results, where the material does not have a surface structure and the surface roughness is very small (the amplitude parameter R a is around 5 nm).
In addition, uniform irradiation has been assumed, meaning that radiance and irradiance are well specified. In some practical cases, uniform irradiation on the scale required for irradiance and radiance measurements is not possible, and these quantities must be determined as the average values of radiant fluxes. In such cases, the measurement equation for the BRDF and the BTDF is obtained from Eq. (2) as: where the subscript "r" (or "t") is replaced by the more general "s" for "scattering" and θ s is the scattering angle with respect to the surface normal. A i is the irradiated area and A s is the area including the full scattered radiant flux. When the object is non-translucent and there is no lateral scattering, both areas are coincident and Eq. (26) is usually simplified as [9][10][11]: However, when the object is translucent, this simplification could be wrong, under conditions in which a significant part of the scattered radiant flux emerges from an area larger than A i . In these cases, such a measurement approach is not recommended; here the measurement of the BBSS is a better option, since it does not require a uniform irradiation but a known value of the radiant flux.

Conclusions
In this work, relating radiance to irradiation conditions at the surface of objects has been studied theoretically. The Bidirectional Reflectance Distribution Function (BRDF) is only invariant for "large enough" irradiated areas, and cannot be used to predict radiance for smaller irradiated areas. The Bidirectional Scattering -Surface Reflectance Distribution Function (BSSRDF) can include both subsurface and insurface scattering contributions when the radiance is evaluated at an irradiated position, which makes it dependent on the irradiation area. Therefore, the BRDF and the BSSRDF cannot be regarded as fundamental scattering quantities for determining reflectance and transmittance under every irradiation condition. We have proposed the definition of two different scattering quantities, one which exclusively describes insurface reflection and the other which exclusively describes subsurface scattering; these quantities are completely independent of the irradiation conditions and can therefore be regarded as more fundamental than the BRDF and the BSSRDF. Therefore, the proposed quantities should allow the reflectance and transmittance of an object to be calculated under any given irradiation condition. The conclusions drawn from the formalism presented have been supported by measurements. In addition, relating these quantities to the Bidirectional Transmittance Distribution Function (BTDF) has been examined and some practical measurement issues discussed. New procedures to provide traceability to measurements of BIS and BBSS need to be developed, and that will be part of future work.