An algorithm of computing 3D geometric attenuation factor

Linli Sun; Shaojie Tang; Feng Zhao

doi:10.1364/OE.27.002056

1. Introduction

An important unsolved problem in the research field of computer vision is the evaluation of bidirectional reflectance distribution function (BRDF) on rough surfaces. In particular, the effect of geometrical attenuation factor is quite critical, and the problem becomes significantly challenging.

Microfacet method is often used by researchers to study geometrical attenuation factor. Microfacet models assume that the surface consists of a large quantity of small flat facets, which behave as micro mirrors, reflecting light only in the specular direction. The reflection of incident light by a rough surface can be studied by using either a physical or geometrical method. It has been assumed that, the reflecting surface is a perfect electrical conductor, meaning that the incident lights arrive from a near-normal direction, and the multiple reflections and shadowing by surface asperities almost do not occur.

The geometrical attenuation factor illuminates the shadowing and masking effect, while the shadowing or masking occurs as a part of the incident ray or the outgoing ray is blocked by the neighboring topography. So the geometrical attenuation factor, gauged by the proportion of light that is not attenuated, demonstrates the effect of shadowing, masking, and shadowing-masking on rough surfaces.

In the study of analytical solution to geometrical attenuation factor, theory of the analytical 2D shadowing function has been researched by Beckmann [1], Smith [2], and Wagner [3]. The solving process was simplified with some assumptions. Smith proposed the physical optical shadowing function by neglecting the correlation between the height and slope on Gaussian surfaces, and the visibility of non-backfacing point on the microsurface depending on its height but not on its normal. Torrance and Sparrow [4] presented the microfacet distribution function and the geometric attenuation function, which are based on the V-groove cavity assumption. Blinn [5] simplified the function based on Torrance and Sparrow’s theory, and the final expression is widely adopted in computer graphics or rendering.

Bourlier studied the 2D shadowing properties on monostatic and bistatic on random rough surfaces [6–8]. He extended the approaches developed by Smith and Wagner from 2D to 3D on any height and slope uncorrelated or correlated surfaces. Whatever the assumed surface would be, the slope and height probability density function (PDF) must be defined by integrating over the PDF of surface slope. The shadowing function was performed for Gaussian, Laplacian, and exponential slope PDFs in his research. Moreover, for 2D case, Bourlier explained in detail these different shadowing functions (Smith’s, Wagner’s, and Sto and Ricciardi’s) and noted that Smith’s result was the best. In recent years, Eric Heitz developed the microfacet light reflectance theory, defining one multiple-scattering model with the Smith masking-shadowing function [9].

Notation. used throughout this paper

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Eric Heitz [10–13] and Bruce Walter [14] presented the shadowing function $G_{1} (ω_{i}, ω_{m})$ , the masking function $G_{1} (ω_{o}, ω_{m})$ , and the simultaneously occurred shadowing-masking function $G_{1} (ω_{i}, ω_{o}, ω_{m})$ for interpreting the diverse situations on rough surfaces. We notice that, the Smith masking function is integrated over the slope on the microsurface and its form is derived with a ray tracing formulation of the masking probability. However, the Smith’s model didn’t emphasize the exactness on its result. Actually, Smith’s solution can be viewed as an approximation. In practice, we consider light scattering on rough surfaces as a spatial problem, the normal on the microfacet is directional, and so the directional scattering pattern matters. The geometrical attenuation factor should be considered in 3D space. In this work, we validate the 3D geometrical attenuation factor by numerical solution on real Gaussian random surfaces. To the best of our knowledge, we are the first in either graphics or physics to derive and investigate the numerical analysis of geometrical attenuation factor on 3D surfaces. In this paper, wave effects (e.g., light strikes the surface twice or more and the mutual reflection) and the volumetric scattering are neglected.

Our work makes the following contributions:

i). A 3D geometrical attenuation factor on real surfaces is defined.
ii). A numerical solution for calculating the 3D geometrical attenuation factor with four functions is derived.
iii). The shadowing and masking effects on radiation and irradiation are discussed, respectively.
iv). An illumination factor to indicate the demarcation as the shadowing or masking occurs is defined.

2. Geometrical attenuation factor

2.1 Smith’s 2D geometrical attenuation factor

Smith shadowing function, being equal to the ratio of illuminated surface to the entire rough surface, is given in [2, (10)]. The model is used to describe the rough surface as a mean smooth surface, upon which the positive and negative undulations of height generated by a stationary random process are superimposed. The density of surface height deviation from the mean plane in the vertical direction is described by a continuous probability function of zero mean. He chose Gaussian for ease of computation.

The shadowing problem is described below:

Since shadows are cast on the surface, it is given by the ratio of illuminated to total area subtending the surface, which for a stationary surface equals the probability that an arbitrary point on the surface is illuminated. Then, a great simplicity is gained by neglecting the correlation between height and slope, after the conditional probability function on the surface is integrated. Smith got the expression of shadowing function as follows,

G (θ_{i}) = \frac{1 - \frac{1}{2} e r f c (μ / \sqrt{2} σ)}{Λ (μ) + 1}

where

θ_{i}

is the angle of incidence,

μ = \cot θ_{i}

and erfc represents the error function complement. The shadowing is independent of the surface height autocorrelation function. Smith’s 2D geometrical attenuation factor depends on the surface slope PDF of root mean square (rms)

σ

.

2.2 3D geometrical attenuation factor

For a 3D continuous surface $z (x, y)$ , we discretize it as $z (x, y)$ , $x = (0, 1, 2, \dots X - 1)$ , $y = (0, 1, 2, \dots, Y - 1)$ . We take X and Y discrete points along x and y axes, respectively. Sampling intervals are set as $τ_{x}$ and $τ_{y}$ respectively, with $τ_{x}$ being equal to $τ_{y}$ . So $z (x, y)$ can be expressed in term of matrix as follows:

[z (x, y)] = [\begin{matrix} z (0, 0) & z (0, 1) & \dots & z (0, Y - 1) \\ z (1, 0) & z (1, 1) & \dots & z (1, Y - 1) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ z (X - 1, 0) & z (X - 1, 1) & \dots & z (X - 1, Y - 1) \end{matrix}]

where each element

z (x, y)

in the matrix represents the corresponding microfacet on the 3D surface. We evaluate the geometrical attenuation factor on each microfacet to find out whether the microfacet is lighted or not.

Whether the microfacet $z (x, y)$ is illuminated or not depends on its normal orientation $ω_{m}$ , which is not backfacing, shadowing function is used to illustrate the incident effect. Although we must consider that the incidence ray towards the local microfacet is not blocked by other microfacets, local shadowing function $G_{1}^{l o c a l} (ω_{i}, ω_{m})$ is raised. Notice that it is binary.

However, we do care the masking function because it is related with the outgoing ray. For the microfacets, the masking function is used to illustrate outgoing ray: whether the local microfacet is visible or not. So it shows that it is a binary discard of backfacing microfacets. Considering the blockage of neighboring microfacets, the local masking function is put forward as $G_{1}^{l o c a l} (ω_{m}, ω_{o})$ .

For the specular microfacet-based shadowing and masking function given above, the local shadowing-masking function is defined by:

G_{1}^{l o c a l} (ω_{i}, ω_{m}, ω_{o}) = G_{1}^{l o c a l} (ω_{i}, ω_{m}) G_{1}^{l o c a l} (ω_{m}, ω_{o})

On the basis of the above analysis, the local shadowing-masking function

G_{1}^{l o c a l} (ω_{i}, ω_{m}, ω_{o})

is binary, which is further explained as follows:

if $G_{1}^{l o c a l} (ω_{i}, ω_{m}, ω_{o}) = 0$ , the microfacet $z (x, y)$ is not lighted. That is to say, the incident ray cannot reach the microfacet, or the reflected ray cannot go out, and the blockage of light occurs on the microsurface.

if $G_{1}^{l o c a l} (ω_{i}, ω_{m}, ω_{o}) = 1$ , the normal of microfacet is not backfacing $(ω_{m} \cdot ω_{o}) > 0$ , incident ray can reach the microfacet and the reflected ray can go out, so the microfacet $z (x, y)$ is illuminated and can be observed finally.

The self-shadowing or self-masking effects by adjacent microfacets on the microsurface are also taking into account. The self-shadowing function $G_{1}^{d i s t} (ω_{i})$ , self-masking function $G_{1}^{d i s t} (ω_{o})$ and self-shadowing-masking function are presented in section 3.2.

In our work, we use four functions in evaluating the geometrical attenuation factor $G_{2} (ω_{i}, ω_{o})$ : local shadowing function $G_{1}^{l o c a l} (ω_{i}, ω_{m})$ , local masking function $G_{1}^{l o c a l} (ω_{m}, ω_{o})$ , self-shadowing function $G_{1}^{d i s t} (ω_{i})$ , and self-masking function $G_{1}^{d i s t} (ω_{o})$ . The binary value represents whether the microfacet $z (x, y)$ is illuminated or not. So $G_{2} (ω_{i}, ω_{o})$ can be expressed in a matrix as follows:

[(G_{2} (ω_{i}, ω_{o})) (x, y)] = [\begin{matrix} G_{2} (0, 0) & G_{2} (0, 1) & \dots & G_{2} (0, Y - 1) \\ G_{2} (1, 0) & G_{2} (1, 1) & \dots & G_{2} (1, Y - 1) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ G_{2} (X - 1, 0) & G_{2} (X - 1, 1) & \dots & G_{2} (X - 1, Y - 1) \end{matrix}]

where the number of zeros is

N_{s}

, that is to say, the local point is not lighted, maybe it is shadowed or masked for the given incident ray. And the geometrical attenuation factor on 3D surface is

G_{2} (ω_{i}, ω_{o}) = 1 - \frac{N_{s}}{X Y}

which is physically based since

G_{2} (ω_{i}, ω_{o}) = 1

when

N_{s} = 0

,and

ω_{i} = ω_{g}

, denoting the surface is entirely illuminated, while

G_{2} (ω_{i}, ω_{o}) = 0

as the incidence angle goes to 90^◦,

ω_{i} = (0, 1, 0)

, the tile of the incidence light

θ_{i} =

90^◦, and

N_{s} = XY

, denoting the entire surface is invisible.

We have presented a method for calculating the 3D geometrical attenuation factor. Our method makes no assumptions about microsurface, and so the numerical solution can be used on arbitrary surface.

3. Algorithms of computing 3D geometric attenuation factor

3.1 Generation of a Gaussian random surface

By assuming that the surface is a perfect electrical conductor, only single scattering is modeled, and wave effects and light striking the surface twice or more are ignored. The formulation is based on geometrical optics and applies when $(σ / λ) ≫ 1$ , where $λ$ is the wavelength of the radiation, and $σ$ is rms height of the rough surface. The ray theory is valid only when the surface roughness is larger than the wavelength of the radiation.

For convenience of comparison with the Smith’s result, we generate a random rough Gaussian surface $z (x, y)$ , which is essentially a random stationary process [15–20]. The simplified algorithm is in the following:

Algorithm 1. Generation of a Gaussian random surface $z (x, y)$

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In this paper, the generation of a random rough surface by defining truncation length Lx = Ly = 12.8, correlation length lx = ly = 0.5, discrete interval $τ_{x} = τ_{y}$ = 0.05. A set of isotropic random rough surfaces $σ$ = 0.1, 0.2, 0.3,∙∙∙,1.0 is generated. As depicted in Fig. 1, the rsm height of a random rough surface can significantly affect its shape on the surface.

Fig. 1 Gaussian random rough surface. We generated the surface by using Algorithm 1, and by defining truncation length Lx = Ly = 12.8, correlation length lx = ly = 0.5, discrete interval $τ_{x} = τ_{y}$ = 0.05 and $σ$ = 0.1.

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3.2 Algorithms

The coordinate system used to describe directions is shown in Fig. 1 which demonstrates the Gaussian random rough surface. The microfacets are assumed with ideal specular reflection while the microsfacet acts as a collection of tiny flat mirrors. We use four functions to denote the geometrical attenuation factor above mentioned, and the following sections will discuss them in detail.

3.2.1. Local shadowing function

Figure 2(a) shows the shadowing and masking effects. The incident ray casts shadow on the rough surface, and shadowing function $G_{1}^{l o c a l} (ω_{i}, ω_{m})$ illustrates the shadowing of the point with a local normal $ω_{m}$ : it evaluates to 0 if the point is shadowed and to 1 if it is visible. So the shadowing function has binary values:

G_{1}^{l o c a l} (ω_{i}, ω_{m}) = χ^{+} (i \cdot m) χ^{+} (m \cdot n)

The microfacet with a back-facing normal is invisible, and the shadowing function concerns about the direction between incident light and the normal on the local point of the microsurface.

Fig. 2 (a) Illustration of the shadowing effect: the incident ray can reach the surface, and (b) Illustration of the masking effect: the outgoing ray is blocked by adjacent microsurfaces.

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The analysis on local shadowing function is shown in Algorithm 2.

Algorithm 2. Local shadowing function $G_{1}^{l o c a l} (ω_{i}, ω_{m})$

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3.2.2. Local masking function

The masking function $G_{1}^{l o c a l} (ω_{m}, ω_{o})$ illustrates whether the outgoing light is masked by adjacent microfacets or not. The masking effect is shown in Fig. 2(b). If the microfacet is masked it is 0 while if visible it is 1. There is a precondition on masking functions: the microfacet is visible. We concern the direction between local normal and the outgoing reflected light. The explanation is shown in Fig. 3. Because the reflected light is not masked, the following conditions must be satisfied:

Fig. 3 Illustration on masking function. The outgoing ray o can be seen depending on two conditions: i). the incident ray can illuminate the microfacet, ii). the outgoing ray o is not blocked by adjacent microfacets.

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i). The reflected light lies on the upper half plane,
ii). The angle between the local normal and the reflected light is less than 90^◦.
- The masking function is defined as: $G_{1}^{l o c a l} (ω_{m}, ω_{o}) = χ^{+} (n \cdot o) χ^{+} (m \cdot n) χ^{+} (m \cdot o)$

Algorithm 3. Local masking function $G_{1}^{l o c a l} (ω_{m}, ω_{o})$

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3.2.3. Parallel incident light plane

The calculation about self-shadowing and self-masking functions involves constructing the parallel incident light source. Now we construct a matrix M0 to simulate the incident parallel light source as coaxial light source. The height of the top surface M0 has the same height, the height value is the maximum of the generated Gaussian surface, and it can be moved along the y axis. The initial state is that the projection of M0 onto the xoy plane is coincident with the bottom surface M1. As shown in Fig. 4, M0 simulates the movement of the light source. M0 and M1 are matrixes with the same size. As shown in Fig. 6, every point on M0 emits light and penetrates the Gaussian surface Z, and arrives at the corresponding point of M1.As shown in Fig. 4.

Fig. 4 Top surface M0 and the bottom surface M1. The top surface simulates a coaxial light source, and so the parallel incidence light emitting out from it can be moved along y axis. The M1 is the orthographic projection of surface Z on xoy plane.

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3.2.3. Self-shadowing function

In Fig. 5(a), L is a ray emitted from the simulated incident source. Light L will intersect surface Z and it may have several intersection points. Figure 5(b) shows three intersection points P1, P2 and P3 of incident L. Actually reflectance occurs at point P1, where it is the first bounce of incidence on Z. P0 is the highest point on the surface Z, parallel incident light casts a shadow between P0P0’ and Z.

Fig. 5 Illustration on self-shadowing. (a) L intersects Z. L represents an incident ray coming from the parallel light source M0, assuming it intersects surface Z and reached at the bottom surface M1 at the same index of matrix of M0 and M1. (b) Reflection occurs at points P0 and P1 in the first bounce. Assuming an incident ray intersects the surface Z with several intersections, apparently the reflectance occurs at P1.

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Our self-shadowing algorithm aims to figure out the intersections between incident light and the surface. We count the number of intersections, if there is one intersection and the incident line on M0 arrives at the corresponding point of M1, and while the intersection point satisfies the reflection condition, no self-shadowing will happen. If there are two or more intersections, self-shadowing occurs. So P0’, P0, P2 and P3 on the surface Z are all in shadow, as shown in Fig. 5(b). Then these points in shadow on the surface Z are invisible and need to be marked.

Algorithm 4. Self-shadowing function $G_{1}^{d i s t} (ω_{i})$

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3.2.4. Self-masking function

Our self-masking algorithm aims to find out the reflected lights which are blocked by microfacets on the surface Z. Actually, some points do not satisfy the reflection condition, so they are marked by masking function as mentioned earlier. If the reflectance ray intersects the surface Z, it forms self-masking. As shown in Fig. 6, the surface Z blocked the reflected light on P3. So, we mark P3 in self-masking algorithm. P3 is invisible.

Fig. 6 Illustration on self-masking. At point P1, the incidence ray L and the reflectance ray R go without shadowing and masking. At point P2,the reflectance ray is blocked by the surface Z at point P3, the self-shadowing occurs.

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Algorithm 5. Self-masking function $G_{1}^{d i s t} (ω_{o})$

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Now, we combine the four functions together to evaluate the 3D geometrical attenuation factor.

Algorithm 6. Geometrical attenuation factor $G_{2} (ω_{i}, ω_{o})$

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The combining algorithm uses four sub-functions to calculate points which do not satisfy illumination condition. These points, either shadowed or marked, are marked with NaN, and we can calculate the 3D geometrical attenuation factor $G_{2} (ω_{i}, ω_{o})$ at last. These four functions include: local shadowing, local masking, self-shadowing and self-masking functions.

In our algorithm, the subscript i means that we get the geometrical attenuation factor with a given incident direction. The top surface M0 simulates the variation of the tilt of the incidence, while $ω_{i}$ is from (0, 0, 1) to (0, 1, 0). The tilt of incident is from 0^◦to 90^◦. The observation orientation is set as (0, 0, 1).

4. Data and simulation

For given parameters, truncation length Lx = Ly = 12.8, correlation length lx = ly = 0.5, discrete interval $τ_{x} = τ_{y} = 0.05$ , we generated various Gaussian random surfaces via the different values of rsm height $σ$ from 0.1 to 1.0.. And for a given rms value $σ$ , different surface Z we obtained, though. So in order to ensure the accuracy of the data, we conducted the repeated experiments.

For the given parameters in one experiment, such as truncation length, correlation length, discrete interval and the rsm height, we generated the surface Z for 20 times, and computed our proposed algorithm to get the geometrical attenuation factors and get the average of the results as the final data.

During the simulation, the Gaussian random surface is generated with ten different $σ$ values. We mark the points that are invisible on the microsurface. In fact, we use NaNs to mark them out on the surface Z with MATLAB.

5. Results and discussions

5.1 Computing result of 3D geometrical attenuation factor

We follow the algorithm step by step to calculate the four functions, and the evaluation of 3D geometrical attenuation factors with different $σ$ values has been shown in Fig. 7. As the tilt of incident is zero, the geometrical attenuation factor is 1; while as the tilt of incident is 90^◦, the factor is zero. The curve is concentrated in the center as the surface roughness becomes larger. For a generated Gaussian random surface with a small $σ$ value, e.g., $σ$ = 0.1, the shadow is very weak. As the incident angle is very large, e.g., the incident is 85.8^◦, the average illumination factor equals 1. This means that the whole surface is still illuminated, and no shadow occurs. But the occurred shadowing is coming strong when the incident angle is larger than 85.8^◦, while the geometrical attenuation factor decreases rapidly to 0, as the incident increases to 90^◦. The comparation between our 3D geometrical attenuation factor and the Smith’s for $σ$ = 0.1, 0.3, 0.5 is shown in Fig. 8.

Fig. 7 Our proposed algorithm is applied to ten rough surfaces with $σ$ = 0.1 to 1.0. The numerical solutions for geometrical attenuation factor curves are continuous, coinciding with the law of physics. As the incidence angle is 0^◦, the geometrical attenuation factor is 1. The surface is illuminated, the incidence angle is 90^◦, while the surface is invisible, and the factor is 0. From the result, demarcation point, between the lighted and not lighted surfaces, is given.

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Fig. 8 Our 3D geometrical attenuation factor and the Smith’s. $σ$ = 0.1, 0.3, 0.5 respectively.

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5.2 Illumination factor

As for the surface Z with a larger $σ$ , shadowing is even observed. For an example with $σ$ = 0.5 as shown in Fig. 7, the tilt of incident is larger than 21.4^◦, and the shadow becomes more obvious. When $σ$ = 0.9 and the tilt of incident is larger than 15.6^◦, the geometrical attenuation factor decreases rapidly. We name the demarcation point as an illumination factor, which is between the lighted and shadow occurred surfaces as shown in Table 1, compared with Smith model.

Table 1. Illumination factor

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The data in Table 1 illustrates the illumination factor we get by the approach of the Smith method, are larger than our algorithms start with $σ$ is larger than 0.2. The illumination factor is important for the demarcation of the incident ray, denoting that the surface is between illuminated and shadow, occurs for a given observation direction. The surface is entirely illuminated when the incident angle is less than the critical point, while shadow occurs when the incident angle is bigger than the critical point. The illumination factor is put forward to illustrate the illumination on the surface of an object.

The Cook-Torrance BRDF of the surface is given as follows:

f (ω_{i}, ω_{m}) = \frac{F (ω_{o}) G_{2} (ω_{i}, ω_{o}) D (ω_{m})}{4 | ω_{i} \cdot ω_{g} | | ω_{o} \cdot ω_{g} |}

where

F (ω_{o})

is the Fresnel factor which is 1 such as metal in this paper.

D (ω_{m})

represents the distribution of visible normals.

Note that, as the surface is without shadow, the BRDF is relevant with the incident and the outgoing directions and the distribution of visible normal, when the incident angle is less than the evaluated illumination factor. However, when the incident angle is larger than the demarcation angle, the geometrical attenuation factor must be taken into account. The illumination factor makes the decision whether to introduce the geometrical attenuation factor into the BRDF or not. So the Smith model overestimates the geometrical attenuation factor with $σ$ value is larger than 0.2.

Our method considers the local shadowing and local masking effects, and the self-shadowing and self-masking effects. The generated Gaussian random surface is calculated directly with our four corresponding algorithms, and the result shows that the curves are continuous and more physically reasonable. From the procedure of our solution, we consider not only the correlation between height and slope, but also the self-shadowing and self-masking effects. It is a better prediction of BRDF and it will make the BRDF model closer to reality and more accurate.

5.3 Discussion about grazing angles

The value of the geometrical attenuation factor is between 0 and 1. When the tile of the incident is 0^◦ the factor is 1 and the tile of the incident is 90^◦ the factor is 0. When conducting the geometrical attenuation factor into the BRDF model, the geometrical attenuation factor begins to pull down the BRDF model when the tilt of the incident increases. That is to say, shadowing or masking occurs.

The research work of Becca E. Ewing showed a wide variety of materials were used to test three methods for fitting the BRDF with emphasis on improving the accuracy near grazing angles [21]. Models are the traditional Cook-Torrance, a modified Cook-Torrance and the new model which neglected the geometrical attenuation factor term. And the adopted the geometrical attenuation factor is the Blinn model based on the V-grooved assumption. The result has shown that the general trend over all materials is that Model 3, by neglecting the geometric attenuation factor and using the Butler’s S approximation, produces a better fit of the BRDF data at near grazing angles [22].

Some researchers ignore data with incident or outgoing angle larger than 80^◦ as measurements close to extreme grazing angles are in general unreliable [23]. The $| ω_{i} \cdot ω_{g} | | ω_{o} \cdot ω_{g} |$ term on the denominator in Eq. (8) we called the cosine factor, leads the BRDF value goes to infinity when incident or outgoing angle converges to grazing angles. Neumann made some appropriate modification of the Cook-Torrance model for improving the behavior at grazing angles [24], we call it as the Neumann-Cook-Torrance model in this paper.

Now we adopt the traditional Cook-Torrance and the Neumann-Cook-Torrance model, the geometrical attenuation term is Smith and ours respectively. As shown in Fig. 9.

Fig. 9 The comparison between the Cook-Torrance BRDF model (black) and the Neumann- Cook-Torrance BRDF model (blue). The geometrical attenuation factor is Smith (dotted markers) and ours (solid curves).

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Figure 9 had shown the comparison between the traditional Cook-Torrance BRDF model and the Neumann-Cook-Torrance BRDF model. The Neumann-Cook-Torrance BRDF model eliminated the undesired behavior of the traditional Cook-Torrance BRDF model near the grazing angle on some extent. The Smith geometrical attenuation factor used in the two models made the BRDF value would be unacceptable high at grazing angles and the energy balance could not be preserved. Our geometrical attenuation factor is computed from a brute force computation algorithm, so the BRDF model using our geometrical attenuation factor is energy preserved.

Figure 10 has shown the comparison between Smith geometrical attenuation factor and ours. The value of the geometrical attenuation factor decreases when the surface becomes rougher. When incident angle is close to the grazing angle, the value of our geometrical attenuation factor (solid line) is close to 0. Our result is much smaller than Smith’s on some extent. And the result is more accurate by comparing with Smith’s geometrical attenuation factor.

Fig. 10 Smith geometrical attenuation factor (dotted markers) and Ours(solid line) are shown for $θ_{i}$ = 30^◦(black), $θ_{i}$ = 45^◦ (bottle green), $θ_{i}$ = 60^◦ (blue), $θ_{i}$ = 75^◦ (rose red) and $θ_{i}$ = 85^◦ (grass green) . $σ$ is from 0.1 to1.0 respectively.

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6. Conclusion

This paper presents a 3D geometrical attenuation factor formulation for calculating the effects of shadowing, masking, self-shadowing and self-masking. The analytical Smith’s method neglected the correlation between height and slope, and the self-shadowing or self-masking effects on microsurfaces. So, their result leads to an overestimation of the geometrical attenuation factor. Our proposed numerical analysis of 3D geometrical attenuation factor can precisely evaluate the BRDF model.

Funding

Education Department of Shaanxi Province (2013JK1023); Key Lab of Computer Networks and Information Integration, Southeastern University, Ministry of Education, China (K93-9-2017-03); Natural Science Foundation of Shaanxi Province (2016JM8034); Key Research and Development Projects of Shaanxi Province (2018ZDXM-GY-091); Key Laboratory of High Performance Complex Manufacturing, Central South University (Kfkt2016-12).

Acknowledgements

The authors would like to sincerely thank all reviewers for their helpful comments and suggestions.

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Symbol	Meaning
$(θ, φ)$	spherical coordinates: $ω = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$
$ω_{i} = (x_{i}, y_{i}, z_{i})$	incident direction
$ω_{o} = (x_{o}, y_{o}, z_{o})$	outgoing direction
$ω_{m} = (x_{m}, y_{m}, z_{m})$	microfacet normal
$ω_{g} = (0, 0, 1)$	geometrical normal
$χ^{+} (a)$	Heaviside function, 1 if a>0, and 0 if a≤0
$G_{1}^{l o c a l} (ω_{i}, ω_{m})$	shadowing function at microfacet normal $ω_{m}$ ,binary value
$G_{1}^{l o c a l} (ω_{m}, ω_{o})$	masking function at microfacet normal $ω_{m}$ , binary value
$G_{1}^{l o c a l} (ω_{i}, ω_{m}, ω_{o})$	local shadowing-masking function at microfacet normal $ω_{m}$ , binary value
$G_{1}^{d i s t} (ω_{i})$	self-shadowing function, in[0,1]
$G_{1}^{d i s t} (ω_{o})$	self-masking function, in[0,1]
$G_{1}^{d i s t} (ω_{i}, ω_{o})$	self-shadowing-masking function, in[0,1]
$G_{2} (ω_{i}, ω_{o})$	geometrical attenuation factor
$Λ (x)$	the Smith $Λ$ function

Step	Meaning
1	$σ$ ←rms
2	Lx, Ly←truncation length
3	lx,ly←correlation length
4	$τ_{x}$ , $τ_{y}$ ←discrete interval
5	$X = L x / 2 / τ_{x}$ , $Y = L y / 2 / τ_{y}$
6	_{$α_{m n}, β_{m n}, m = 0 ~ 2 X, n = 0 ~ 2 Y$}←generate the independent standard Gaussian random number matrix
7	combine $α_{m n}$ with $β_{m n}$ into one complex random number $γ_{m 0} = α_{m 0} / 2$ , $γ_{0 n} = α_{0 n} / 2$ , $γ_{m n} = α_{m n} + j β_{m n}$
8	define the Gaussian spectral density function $W (k_{x}, k_{y}) = \frac{l_{x} l_{y} σ^{2}}{4 π} \exp (- \frac{k_{x}^{2} l_{x}^{2}}{4} - \frac{k_{y}^{2} l_{y}^{2}}{4})$
9	calculate expansion coefficient, $χ_{m n} = W (\frac{2 π m}{L x}, \frac{2 π n}{L y})$ $χ_{(2 X - m) (2 Y - n)} = χ_{m (2 Y - n)} = χ_{m (2 X - m)} = χ_{m n}$ , $b_{m n} = π \sqrt{\frac{χ_{m n}}{L_{x} L_{y}} γ_{m n}}$
10	calculate the height using DFT transform: $\begin{array}{l} z (x, y) = \sum_{m = 0}^{2 X} \sum_{n = 0}^{2 Y} b_{m n} \exp (j (\frac{2 π}{X} m x + \frac{2 π}{Y} n y)) \end{array}$

Step	Meaning
1	$ω_{i}$ ←incident direction
2	$ω_{m}$ ←microfacet normal
3	_{$χ^{+} (i \cdot m)$} ←incident ray can be reflected on the microfacet
4	_{$χ^{+} (m \cdot n)$} ←the normal is not backfacing on the microfacet
5	calculate the local masking function on every point on the microfacet: $G_{1}^{l o c a l} (ω_{i}, ω_{m}) = χ^{+} (i \cdot m) χ^{+} (m \cdot n)$
6	if $G_{1}^{l o c a l} (ω_{i}, ω_{m})$ = 0, then replace the value of $z (x, y)$ with NaN
7	N₁←count the number of NaNs

Step	Meaning
1	$ω_{o}$ ←outgoing direction
2	$ω_{m}$ ←microfacet normal
3	_{$χ^{+} (n \cdot o)$} ←outgoing direction is on upper half plane
4	_{$χ^{+} (m \cdot n)$}←the normal is not backfacing on the microfacet
5	_{$χ^{+} [\cos (π / 2 - \cos^{- 1} (m \cdot o))]$}←the angle between the local normal and the reflected light is less than 90^◦
6	calculate the local masking function of every point on the surface: $G_{1}^{l o c a l} (ω_{m}, ω_{o}) = χ^{+} (n \cdot o) χ^{+} (n \cdot m) χ^{+} [\cos (π / 2 - \cos^{- 1} (m \cdot o))]$
7	if $G_{1}^{l o c a l} (ω_{m}, ω_{o})$ = 0, then replace the value of $z (x, y)$ with NaN
8	N₂←count the number of NaNs

Step	Meaning
1	$ω_{i}$ ←incident direction
2	for each incident ray, calculate the intersection P_k, k = 1, 2, ∙∙∙,v of incident ray and surface Z
3	_{$N_{(m, n)} = \sum_{k = 2}^{v} \sum_{m = 0}^{2 X} \sum_{n = 0}^{2 Y} P_{k} (x, y)$} ←count the number of intersection except the first intersection
4	mark the intersections on $z (x, y)$ except the first intersection point, and then replace the points’ value on $z (x, y)$ with NaN
5	N₃←count the number of NaNs
6	$G_{1}^{d i s t} (ω_{i}) = \sum_{m = 0}^{2 X} \sum_{n = 0}^{2 Y} N_{3 (m, n)} / [(2 X + 1) (2 Y + 1)]$

An algorithm of computing 3D geometric attenuation factor

Abstract

1. Introduction

2. Geometrical attenuation factor

2.1 Smith’s 2D geometrical attenuation factor

2.2 3D geometrical attenuation factor

3. Algorithms of computing 3D geometric attenuation factor

3.1 Generation of a Gaussian random surface

3.2 Algorithms

3.2.1. Local shadowing function

3.2.2. Local masking function

3.2.3. Parallel incident light plane

3.2.3. Self-shadowing function

3.2.4. Self-masking function

4. Data and simulation

5. Results and discussions

5.1 Computing result of 3D geometrical attenuation factor

5.2 Illumination factor

5.3 Discussion about grazing angles

6. Conclusion

Funding

Acknowledgements

References

Cited By

Figures (10)

Tables (8)

Equations (8)

Optics Express

Step	Meaning
1	$ω_{o}$ ←outgoing direction
2	for each outgoing ray, calculate the intersection P_k, k = 1, 2, ∙∙∙,v of outgoing ray and surface $z (x, y)$
3	_{$N_{(m, n)} = \sum_{k = 1}^{v} \sum_{m = 0}^{2 X} \sum_{n = 0}^{2 Y} P_{k} (x, y)$} ←count the number of intersection except the first intersection
4	mark the intersections on $z (x, y)$ and then replace the points’ value with NaN
5	N₄←count the number of NaNs
6	$G_{1}^{d i s t} (ω_{o}) = \sum_{m = 0}^{2 X} \sum_{n = 0}^{2 Y} N_{4 (m, n)} / [(2 X + 1) (2 Y + 1)]$

Step	Meaning
1	N₁←the number of NaNs of shadowing function
2	N₂←the number of NaNs of masking function
3	N₃←the number of NaNs of self-shadowing function
4	N₄←the number of NaNs of self-masking function
5	eliminate the duplicate NaN data of $z (x, y)$ on the surface, the final number of NaNs are $N_{k}^{'}$ .k = 1, 2, 3, 4.
6	$G_{2} (ω_{i}, ω_{o}) = 1 - \frac{\sum_{k = 1}^{4} N_{k}^{'}}{(2 X + 1) (2 Y + 1)}$

$σ$	illumination factor (degree)
$σ$	our	Smith’s
0.1	85.8	73.8
0.2	48.3	61.7
0.3	31.4	53.3
0.4	25.1	42.4
0.5	21.4	41.2
0.6	23.2	31.8
0.7	16.1	27.7
0.8	17.5	23.8
0.9	15.6	20.5
1.0	15.2	19.7