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

Author: Assassin-plus

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

@@ -72,14 +72,28 @@ If the microsurface is a heightfield, which is typically the case for microsurfa
 
 $$ G_2(l, v, m) = \lambda(\phi)G_1(v, m)G_1(l, m) + (1 - \lambda(\phi))\min(G_1(v, m), G_1(l, m)) \tag{9.28}$$
 
-where $\lambda(\psi)$ is some function that increases from 0 to 1 as the angle $\psi$ increases. Ashikhmin et al.  suggested a Gaussian with a standard deviation of $15^\circ$(? 0.26 radians):
+where $\lambda(\psi)$ is some function that increases from 0 to 1 as the angle $\psi$ increases. Ashikhmin et al.  suggested a Gaussian with a standard deviation of $15^\circ$( 0.26 radians):
 
 $$\lambda(\psi)=1-e^{-7.3\psi^2} \tag{9.29}$$
 
 A different $\lambda$ function was proposed by van Ginneken et al.:
 
 $$\lambda(\psi)=\frac{4.41\psi}{4.41\psi+1} \tag{9.30}$$
 
+Regardless of the relative alignment of the light and view directions, there is another reason that masking and shadowing at a given surface point are correlated. Both are related to the point's height relative to the rest of the surface. The probability of masking increases for lower points, and so does the probability of shadowing. If the Smith masking function is used, this correlation can be precisely accounted for by the *Smith height-correlated masking-shadowing function*:
+
+$$G_2(l,v,m)=\frac{\chi^+(m \cdot v)\chi^+(m \cdot l)}{1+\Lambda(v)+\Lambda(l)} \tag{9.31}$$
+
+Heitz also describes a form of Smith $G_2$ that combines direction and height correlation:
+
+$$G_2(l,v,m)=\frac{\chi^+(m \cdot v)\chi^+(m \cdot l)}{1+\max(\Lambda(v),\Lambda(l))+\lambda(v,l)\min(\Lambda(v),\Lambda(l))} \tag{9.32}$$
+
+where the function $\lambda(v, l)$ could be an empirical function such as the ones in Equations 9.29 and 9.30, or one derived specifically for a given NDF.
+
+Out of these alternatives, Heitz  recommends the height-correlated form of the Smith function (Equation 9.31) since it has a similar cost to the uncorrelated form and better accuracy. This form is the most widely used in practice , though some practitioners use the separable form (Equation 9.27) .
+
+The general microfacet BRDF (Equation 9.26) is not used directly for rendering. It is used to derive a closed-form solution (exact or approximate) given a specific choice of micro-BRDF $f_\mu$. The first example of this type of derivation will be shown in the next section
+
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