Update 2025-05-06-Microfacet-Theory.md

Author: Assassin-plus

_posts/2025-05-06-Microfacet-Theory.md

@@ -10,17 +10,17 @@ tags: [graphics]     # TAG names should always be lowercase
 ---
 # Microfacet Theory
 
-Many BRDF models are based on a mathematical analysis of the effects of microgeometry on reflectance called microfacet theory. This tool was first developed by researchers in the optics community [124]. It was introduced to computer graphics in 1977 by Blinn [159] and again in 1981 by Cook and Torrance [285]. The theory is based on the modeling of microgeometry as a collection of *microfacets*.
+Many BRDF models are based on a mathematical analysis of the effects of microgeometry on reflectance called microfacet theory. This tool was first developed by researchers in the optics community . It was introduced to computer graphics in 1977 by Blinn  and again in 1981 by Cook and Torrance . The theory is based on the modeling of microgeometry as a collection of *microfacets*.
 
-Each of these tiny facets is flat, with a single microfacet normal $m$. The microfacets individually reflect light according to the micro-BRDF $f_\mu(l, v, m)$, with the combined reflectance across all the microfacets adding up to the overall surface BRDF. The usual choice is for each microfacet to be a perfect Fresnel mirror, resulting in a specular microfacet BRDF for modeling surface reflection. However, other choices are possible. Diffuse micro-BRDFs have been used to create several local subsurface scattering models [574, 657, 709, 1198, 1337]. A diffraction micro-BRDF was used to create a shading model combining geometrical and wave optics effects [763].
+Each of these tiny facets is flat, with a single microfacet normal $m$. The microfacets individually reflect light according to the micro-BRDF $f_\mu(l, v, m)$, with the combined reflectance across all the microfacets adding up to the overall surface BRDF. The usual choice is for each microfacet to be a perfect Fresnel mirror, resulting in a specular microfacet BRDF for modeling surface reflection. However, other choices are possible. Diffuse micro-BRDFs have been used to create several local subsurface scattering models . A diffraction micro-BRDF was used to create a shading model combining geometrical and wave optics effects .
 
 An important property of a microfacet model is the statistical distribution of the microfacet normals $m$. This distribution is defined by the surface¡Çs **normal distribution function**, or NDF. Some references use the term *distribution of normals* to avoid confusion with the Gaussian normal distribution. We will use $$D(m)$$ to refer to the NDF in equations.
 
-The NDF $D(m)$ is the statistical distribution of microfacet surface normals over the microgeometry surface area [708]. Integrating $D(m)$ over the entire sphere of microfacet normals gives the area of the microsurface. More usefully, integrating $D(m)(n \cdot m)$, the projection of $D(m)$ onto the macrosurface plane, gives the area of the macrosurface patch that is equal to 1 by convention, as shown on the left side of Figure 9.31. In other words, the projection $D(m)(n \cdot m)$ is normalized:
+The NDF $D(m)$ is the statistical distribution of microfacet surface normals over the microgeometry surface area . Integrating $D(m)$ over the entire sphere of microfacet normals gives the area of the microsurface. More usefully, integrating $D(m)(n \cdot m)$, the projection of $D(m)$ onto the macrosurface plane, gives the area of the macrosurface patch that is equal to 1 by convention, as shown on the left side of Figure 9.31. In other words, the projection $D(m)(n \cdot m)$ is normalized:
 
 $$ \int_{m\in \Theta} D(m)(n \cdot m) dm = 1$$
 
-The integral is over the entire sphere, represented here by $\Theta$, unlike previous spherical integrals in this chapter that integrated over only the hemisphere centered on $n$, represented by $\Omega$. This notation is used in most graphics publications, though some references [708] use $\Omega$ to denote the complete sphere. In practice, most microstructure models used in graphics are heightfields, which means that $D(m) = 0$ for all directions m outside $\Omega$. However, Equation above is valid for non-heightfield microstructures as well.
+The integral is over the entire sphere, represented here by $\Theta$, unlike previous spherical integrals in this chapter that integrated over only the hemisphere centered on $n$, represented by $\Omega$. This notation is used in most graphics publications, though some references  use $\Omega$ to denote the complete sphere. In practice, most microstructure models used in graphics are heightfields, which means that $D(m) = 0$ for all directions m outside $\Omega$. However, Equation above is valid for non-heightfield microstructures as well.
 
 ![Figure 9.31](/images/fig9.31.png)
 > Figure 9.31. Side view of a microsurface. On the left, we see that integrating $D(m)(n \cdot m)$, the microfacet areas projected onto the macrosurface plane, yields the area (length, in this side view) of the macrosurface, which is 1 by convention. On the right, integrating $D(m)(v \cdot m)$, the microfacet areas projected onto the plane perpendicular to $v$, yields the projection of the macrosurface onto that plane, which is cos $\theta_o$ or $(v \cdot n)$. When the projections of multiple microfacets overlap, the negative projected areas of the backfacing microfacets cancel out the "extra" frontfacing microfacets.
@@ -38,7 +38,7 @@ Take a second look at the right side of Figure 9.31. Although there are many mic
 
 $$ \int_{m\in \Theta} G_1(m, v)D(m)(v \cdot m)^+ dm = v \cdot n$$
 
-as shown in Figure 9.32. The dot product in Equation above is clamped to zero. Backfacing microfacets are not visible, so they are not counted in this case. The product $G_1(m, v)D(m)$ is the **distribution of visible normals** [708].
+as shown in Figure 9.32. The dot product in Equation above is clamped to zero. Backfacing microfacets are not visible, so they are not counted in this case. The product $G_1(m, v)D(m)$ is the **distribution of visible normals** .