Cg Programming/Unity/Specular Highlights at Silhouettes

This tutorial covers the Fresnel factor for specular highlights.

It is one of several tutorials about lighting that go beyond the Phong reflection model. However, it is based on lighting with the Phong reflection model as described in (for per-vertex lighting) and  (for per-pixel lighting). If you haven't read those tutorials yet, you should read them first.

Many materials (e.g. matte paper) show strong specular reflection when light grazes the surface; i.e., when backlight is reflected from the opposite direction to the viewer as in the photo. The Fresnel factor explains this strong reflection for some materials. Of course, there are also other reasons for bright silhouettes, e.g. translucent hair or fabrics (see ).

Interestingly, the effect is often hardly visible because it is most likely when the background of the silhouette is very bright. In this case, however, a bright silhouette will just blend into the background and thus become hardly noticeable.



Schlick's Approximation of the Fresnel Factor
The Fresnel factor $$F_\lambda$$ describes the specular reflectance of nonconducting materials for unpolarized light of wavelength $$\lambda$$. Schlick's approximation is:

$$F_\lambda = f_\lambda + (1 - f_\lambda)(1 - \mathbf{H}\cdot\mathbf{V})^5$$

where V is the normalized direction to the viewer and H is the normalized halfway vector: H = (V + L) / |V + L| with L being the normalized direction to the light source. $$f_\lambda$$ is the reflectance for H·V = 1, i.e. when the direction to the light source, the direction to the viewer, and the halfway vector are all identical. On the other hand, $$F_\lambda$$ becomes 1 for H·V = 0, i.e. when the halfway vector is orthogonal to the direction to the viewer, which means that the direction to the light source is opposite to the direction to the viewer (i.e. the case of a grazing light reflection). In fact, $$F_\lambda$$ is independent of the wavelength in this case and the material behaves just like a perfect mirror.

Using the built-in Cg function  we can rewrite Schlick's approximation as:

$$F_\lambda = f_\lambda + (1 - f_\lambda)(1 - \mathbf{H}\cdot\mathbf{V})^5$$ $$ = f_\lambda \left(1 - (1 - \mathbf{H}\cdot\mathbf{V})^5\right) + (1 - \mathbf{H}\cdot\mathbf{V})^5$$ $$ = \text{lerp}\left(f_\lambda, 1, (1 - \mathbf{H}\cdot\mathbf{V})^5\right)$$

which might be slightly more efficient, at least on some GPUs. We will take the dependency on the wavelength into account by allowing for different values of $$f_\lambda$$ for each color component; i.e. we consider it an RGB vector. In fact, we identify it with the constant material color $$k_\text{specular}$$ from. In other words, the Fresnel factor adds a dependency of the material color $$k_\text{specular}$$ on the angle between the direction to the viewer and the halfway vector. Thus, we replace the constant material color $$k_\text{specular}$$ with Schlick's approximation (using $$f_\lambda = k_\text{specular}$$) in any calculation of the specular reflection.

For example, our equation for the specular term in the Phong reflection model was (see ):

$$I_\text{specular} = I_\text{incoming}\,k_\text{specular} \max(0, \mathbf{R}\cdot \mathbf{V})^{n_\text{shininess}}$$

Replacing $$k_\text{specular}$$ by Schlick's approximation for the Fresnel factor with $$f_\lambda = k_\text{specular}$$ yields:

$$I_\text{specular} = I_\text{incoming}\,\text{lerp}\left(k_\text{specular}, 1, (1 - \mathbf{H}\cdot\mathbf{V})^5\right) \max(0, \mathbf{R}\cdot \mathbf{V})^{n_\text{shininess}}$$

Implementation
The implementation is based on the shader code from. It just computes the halfway vector and includes the approximation of the Fresnel factor:

Complete Shader Code
Putting the code snippet from above in the complete shader from results in this shader:

Artistic Control
A useful modification of the implementation above is to replace the power  by a user-specified shader property. This would give CG artists the option to exaggerate or attenuate the effect of the Fresnel factor depending on their artistic needs.

Consequences for Semitransparent Surfaces
Apart from influencing specular highlights, a Fresnel factor should also influence the opacity $$\alpha$$ of semitransparent surfaces. In fact, the Fresnel factor describes how a surface becomes more reflective for grazing light rays, which implies that less light is absorbed, refracted, or transmitted, i.e. the transparency $$T$$ decreases and therefore the opacity $$\alpha = 1 - T$$ increases. To this end, a Fresnel factor could be computed with the surface normal vector N instead of the halfway vector H and the opacity of a semitransparent surface could increase from a user-specified value $$\alpha_0$$ (for viewing in the direction of the surface normal) to 1 (independently of the wavelength) with

$$\alpha_\text{Fresnel} = \alpha_0 + (1 - \alpha_0)(1 - \mathbf{N}\cdot\mathbf{V})^5$$.

In the opacity was considered to result from an attenuation of light as it passes through a layer of semitransparent material. This opacity should be combined with the opacity due to increased reflectivity in the following way: the total opacity $$\alpha_\text{total}$$ is 1 minus the total transparency $$T_\text{total}$$ which is the product of the transparency due to attenuation $$T_\text{attenuation}$$ (which is 1 minus $$\alpha_\text{attenuation}$$) and the transparency due to the Fresnel factor $$T_\text{Fresnel}$$ (which is 1 minus $$\alpha_\text{Fresnel}$$), i.e.:

$$\alpha_\text{total} = 1 - T_\text{total}\,$$ $$\,= 1 - T_\text{attenuation} T_\text{Fresnel}$$ $$\,= 1 - (1 - \alpha_\text{attenuation})(1 - \alpha_\text{Fresnel})$$

$$\alpha_\text{Fresnel}$$ is the opacity as computed above while $$\alpha_\text{attenuation}$$ is the opacity as computed in. For the view direction parallel to the surface normal vector, $$\alpha_\text{total}$$ and $$\alpha_0$$ could be specified by the user. Then the equation fixes $$\alpha_\text{attenuation}$$ for the normal direction and, in fact, it fixes all constants and therefore $$\alpha_\text{total}$$ can be computed for all view directions. Note that neither the diffuse reflection nor the specular reflection should be multiplied with the opacity $$\alpha_\text{total}$$ since the specular reflection is already multiplied with the Fresnel factor and the diffuse reflection should only be multiplied with the opacity due to attenuation $$\alpha_\text{attenuation}$$.

Summary
Congratulations, you finished one of the somewhat advanced tutorials! We have seen:
 * What the Fresnel factor is.
 * What Schlick's approximation to the Fresnel factor is.
 * How to implement Schlick's approximation for specular highlights.
 * How to add more artistic control to the implementation.
 * How to use the Fresnel factor for semitransparent surfaces.