2023-6-20 21:46:42

In Phong model, there are several material parameters:

i s i_s is​ which is the specular intensity;

i d i_d id​ which is the diffuse intensity;

i a i_a ia​ which is the ambient intensity;

k s k_s ks​ which is the specular reflection constant, the ratio of specular reflection of incoming light;

k d k_d kd​ which is the diffuse reflection constant, the ratio of diffuse reflection of incoming light;

k a k_a ka​ which is the ambient reflection constant, the ratio of reflection of ambient of all the points in the rendered scene;

α \alpha α is the roughness constant.

In the scene, we have those vectors in the figure:

in which,

L m ^ \hat{L_m} Lm​^​ is the direction vector from the point on the surface to the light source;

N ^ \hat{N} N^ is the surface normal;

R m ^ \hat{R_m} Rm​^​ is the specular reflection vector;

V ^ \hat{V} V^ is the viewer direction vector.

The subscript m m m presents the m t h m_{th} mth​ light source. Therefore, for each of light sources, the illumination of each point on the surface I P I_P IP​ can be calculated from:
I P = k a i a + ∑ m ∈ l i g h t s ( k d ( L m ^ ⋅ N ^ ) i m , d + k s ( R m ^ ⋅ V ^ ) α i m , s ) \begin {align*} & I_P = k_ai_a + \sum_{m\in lights}(k_d(\hat{L_m} \cdot \hat{N})i_{m,d}+k_s(\hat{R_m} \cdot \hat{V})^\alpha i_{m,s}) \end {align*} ​IP​=ka​ia​+m∈lights∑​(kd​(Lm​^​⋅N^)im,d​+ks​(Rm​^​⋅V^)αim,s​)​
The specular highlight of each point on the surface is actually decided by the viewing direction and specular reflection direction. The specular direction vector R m ^ \hat{R_m} Rm​^​ is calculated as the reflection of L m ^ \hat{L_m} Lm​^​ on the surface characterized by the surface normal N ^ \hat{N} N^ :

R m ^ = 2 ( L m ^ ⋅ N ^ ) N ^ − L m ^ \begin {align*} & \hat{R_m} = 2(\hat{L_m} \cdot \hat{N})\hat{N}-\hat{L_m} \end {align*} ​Rm​^​=2(Lm​^​⋅N^)N^−Lm​^​​

Remind that in the Phong reflection model, the specular the surface will disappear when the angle between R m ^ \hat{R_m} Rm​^​ and V ^ \hat{V} V^ is larger than 9 0 ∘ 90^\circ 90∘ since the dot product of them is negative, which is usually set as zero.

which results in the effect of cutoff of specular edge of light reflection, as shown in the figure.

Therefore, a new vector named halfway was introduced to eliminate the negative dot product. The halfway vector is defined as:

H m ^ = L m ^ + V ^ ∥ L m ^ + V ^ ∥ \begin {align*} & \hat{H_m} = \frac{\hat{L_m}+\hat{V}}{\left \| \hat{L_m}+\hat{V} \right \| } \end {align*} ​Hm​^​= ​Lm​^​+V^ ​Lm​^​+V^​​
Now just replace the specular term of Phong model:
I B P = k a i a + ∑ m ∈ l i g h t s ( k d ( L m ^ ⋅ N ^ ) i m , d + k s ( N ^ ⋅ H m ^ ) α ′ i m , s ) \begin {align*} & I_{BP} = k_ai_a + \sum_{m\in lights}(k_d(\hat{L_m} \cdot \hat{N})i_{m,d}+k_s(\hat{N} \cdot \hat{H_m})^{\alpha'} i_{m,s}) \end {align*} ​IBP​=ka​ia​+m∈lights∑​(kd​(Lm​^​⋅N^)im,d​+ks​(N^⋅Hm​^​)α′im,s​)​
Because the halfway vector and the surface normal is likely to be smaller than the angle between R m ^ \hat{R_m} Rm​^​ and V ^ \hat{V} V^ used in Phong’s model (unless the surface is viewed from a very steep angle for which it is likely to be larger), the exponent α ′ \alpha' α′ can be set α ′ > α \alpha' > \alpha α′>α to close the Phong model. The rendering result of Blinn-Phong can be seen in the figure: