Phong distribution

In the Phong reflection model, the intensity of the specular highlight is calculated as:

${\displaystyle k_{\mathrm {spec} }=\|R\|\|V\|\cos ^{n}\beta =({\hat {R}}\cdot {\hat {V}})^{n}}$

Where R is the mirror reflection of the light vector off the surface, and V is the viewpoint vector.

In the Blinn–Phong shading model, the intensity of a specular highlight is calculated as:

${\displaystyle k_{\mathrm {spec} }=\|N\|\|H\|\cos ^{n}\beta =({\hat {N}}\cdot {\hat {H}})^{n}}$

Where N is the smooth surface normal and H is the half-angle direction (the direction vector midway between L, the vector to the light, and V, the viewpoint vector).

The number n is called the Phong exponent, and is a user-chosen value that controls the apparent smoothness of the surface. These equations imply that the distribution of microfacet normals is an approximately Gaussian distribution (for large ${\displaystyle n}$), or approximately Pearson type II distribution, of the corresponding angle.^[1] While this is a useful heuristic and produces believable results, it is not a physically based model.

Another similar formula, but only calculated differently:

${\displaystyle k=({\vec {L}}\cdot {\vec {R}})^{n}=[{\vec {L}}\cdot ({\vec {E}}-2{\vec {N}}({\vec {N}}\cdot {\vec {E}}))]^{n},}$

where R is an eye reflection vector, E is an eye vector (view vector), N is surface normal vector. All vectors are normalized (${\displaystyle \|{\vec {E}}\|=\|{\vec {N}}\|=1}$). L is a light vector. For example, ${\displaystyle {\vec {N}}=\{0;\;1;\;0\};\;\;{\vec {E}}=\{{\frac {\sqrt {3}}{2}};\;{\frac {1}{2}};\;0\};\;\;{\vec {L}}=\{-0.6;\;0.8;\;0\};\;\;n=3}$ then:

${\displaystyle k=[{\vec {L}}\cdot ({\vec {E}}-2{\vec {N}}({\vec {N}}\cdot {\vec {E}}))]^{n}=[{\vec {L}}\cdot ({\vec {E}}-2{\vec {N}}(0\cdot {\frac {\sqrt {3}}{2}}+1\cdot 0.5+0\cdot 0))]^{3}=}$

${\displaystyle =[{\vec {L}}\cdot ({\vec {E}}-{\vec {N}})]^{3}=[{\vec {L}}\cdot (\{{\frac {\sqrt {3}}{2}}-0;\;{\frac {1}{2}}-1;\;0-0\})]^{3}=[-0.6\cdot {\frac {\sqrt {3}}{2}}+0.8\cdot (-0.5)+0\cdot 0]^{3}=(-0.5196-0.4)^{3}=0.9196^{3}=0.7777.}$

Approximate formula is this:

${\displaystyle k=({\vec {N}}\cdot {\vec {H}})^{n}=({\vec {N}}\cdot (({\vec {L}}+{\vec {E}})/2))^{n}=({\vec {N}}\cdot ((\{-0.6+{\frac {\sqrt {3}}{2}};\;0.8+0.5;\;0+0\})/2))^{3}=({\vec {N}}\cdot ((\{0.266;\;1.3;\;0\})/2))^{3}=}$

${\displaystyle =({\vec {N}}\cdot (\{0.133;\;0.65;\;0\}))^{3}=(0\cdot 0.133+1\cdot 0.65+0)^{3}=0.65^{3}=0.274625.}$

If vector H is normalized ${\displaystyle {\frac {{\vec {H}}\{0.133;\;0.65;\;0\}}{\|{\vec {H}}\|}}={\frac {{\vec {H}}\{0.133;\;0.65;\;0\}}{\sqrt {0.133^{2}+0.65^{2}}}}={\frac {{\vec {H}}\{0.133;\;0.65;\;0\}}{0.668}}=\{0.20048;0.979701;0\},}$ then

${\displaystyle k=({\vec {N}}\cdot {\vec {H}})^{n}=(0\cdot 0.2+1\cdot 0.9797+0\cdot 0)^{3}=0.979701^{3}=0.940332.}$

Gaussian distribution

A slightly better model of microfacet distribution can be created using a Gaussian distribution. The usual function calculates specular highlight intensity as:

${\displaystyle k_{\mathrm {spec} }=e^{-\left({\frac {\angle (N,H)}{m}}\right)^{2}}}$

where m is a constant between 0 and 1 that controls the apparent smoothness of the surface.^[2]

Beckmann distribution

A physically based model of microfacet distribution is the Beckmann distribution:^[3]

${\displaystyle k_{\mathrm {spec} }={\frac {\exp {\left(-\tan ^{2}(\alpha )/m^{2}\right)}}{\pi m^{2}\cos ^{4}(\alpha )}},~\alpha =\arccos(N\cdot H)}$

where m is the rms slope of the surface microfacets (the roughness of the material).^[4] Compared to the empirical models above, this function "gives the absolute magnitude of the reflectance without introducing arbitrary constants; the disadvantage is that it requires more computation".^[5] However, this model can be simplified since ${\displaystyle \tan ^{2}(\alpha )/m^{2}={\frac {1-\cos ^{2}(\alpha )}{\cos ^{2}(\alpha )m^{2}}}}$. Also note that the product of ${\displaystyle \cos(\alpha )}$ and a surface distribution function is normalized over the half-sphere which is obeyed by this function.

Heidrich–Seidel anisotropic distribution

The Heidrich–Seidel.^[6] distribution is a simple anisotropic distribution, based on the Phong model. It can be used to model surfaces that have small parallel grooves or fibers, such as brushed metal, satin, and hair.

Ward anisotropic distribution

The Ward anisotropic distribution uses two user-controllable parameters α_x and α_y to control the anisotropy. If the two parameters are equal, then an isotropic highlight results. The specular term in the distribution is:

${\displaystyle k_{\mathrm {spec} }={\frac {\rho _{s}}{\sqrt {(N\cdot L)(N\cdot V)}}}{\frac {N\cdot L}{4\pi \alpha _{x}\alpha _{y}}}\exp \left[-2{\frac {\left({\frac {H\cdot X}{\alpha _{x}}}\right)^{2}+\left({\frac {H\cdot Y}{\alpha _{y}}}\right)^{2}}{1+(H\cdot N)}}\right]}$

The specular term is zero if N·L < 0 or N·V < 0. All vectors are unit vectors. The vector V is the viewing direction, L is the direction from the surface point to the light, H is the half-angle direction between V and L, N is the surface normal, and X and Y are two orthogonal vectors in the normal plane which specify the anisotropic directions.

Cook–Torrance model

The Cook–Torrance model^[5] uses a specular term of the form

${\displaystyle k_{\mathrm {spec} }={\frac {DFG}{4(V\cdot N)(N\cdot L)}}}$.

Here D is the Beckmann distribution factor as above and F is the Fresnel term. For performance reasons, in real-time 3D graphics Schlick's approximation is often used to approximate the Fresnel term.

G is the geometric attenuation term, describing selfshadowing due to the microfacets, and is of the form

${\displaystyle G=\min {\left(1,{\frac {2(H\cdot N)(V\cdot N)}{V\cdot H}},{\frac {2(H\cdot N)(L\cdot N)}{V\cdot H}}\right)}}$.

In these formulas V is the vector to the camera or eye, H is the half-angle vector, L is the vector to the light source and N is the normal vector, and α is the angle between H and N.

Using multiple distributions

If desired, different distributions (usually, using the same distribution function with different values of m or n) can be combined using a weighted average. This is useful for modelling, for example, surfaces that have small smooth and rough patches rather than uniform roughness.