What is diffuse reflection?

Diffuse reflection is a fundamental process in illuminating the 3D objects in a scene. It plays a significant role in these objects' perceived depth and texture. 

Diffuse or Lambertian reflection is a common form of light reflection used in computer graphics. It describes the way light interacts with the surface of objects. When light hits an object, it gets scattered in multiple directions rather than reflecting in a single direction. It provides the basic colors and shadings to the objects.

Lambert's cosine law and diffuse reflection

Diffuse reflection in computer graphics is modeled using Lambert’s cosine law.

The intensity of the diffusely reflected light is proportional to the cosine of the angle between the surface normal and the direction of the light source.

Where,

• $I$ is the intensity of the light observed.

• $I_0$ is the maximum intensity of the light.

• $\theta$ is the angle between the direction of the light and the surface normal.

The mathematical perspective

The color $c$ of the surface is proportional to the cosine of the angle between the surface normal and the direction of the light.

The cosine of the angle between the surface normal ($\mathbf{n}$) and the direction of the light source ($\mathbf{l}$) can be calculated as given below.

Where,

• Normal ($n$): The normal is a vector that is perpendicular to the surface at the point of interest. 

• Light vector ($l$): The light vector is a directional vector from a point towards the light source. 

• Angle between $l$ and $n$ ($\theta$): It is the angle between the light vector and the normal vector.

$n$ and $l$ are often normalized ($|\mathbf{n} | = |\mathbf{l}| = 1$) which simplifies the formula as given below.

The above equation shows that as the angle between $\mathbf{n}$ and $\mathbf{l}$ decreases (making $\mathbf{n} \cdot \mathbf{l}$ larger), the surface color becomes brighter and vice versa.

The equation can be rewritten as given below.

The exact amount of the diffuse reflection ($I_{diff}$) can then be determined by multiplying the cosine of the angle ($\cos\theta$) by the intensity of the light ($I_l$) and the diffuse reflection coefficient ($k_d$) of the surface.

Where,

• $I_l$ is the light source intensity

• $k_d = m_dC$ is the surface reflectance coefficient $[0 \; 1]$ and the color

• $\theta$ is the angle between the normal ($\mathbf{n}$) and the light ($\mathbf{l}$)

Since the value of the cosine of the angle is known to be $n.l$, the above equation can be rewritten as given below.

Note:

• The diffuse reflection coefficient ($k_d$) lies between $0$ and $1$.

• It represents the property of the material that determines how much it diffuses the incident light.

Handling backface lighting

To avoid calculating negative values when the surface is facing away from the light source ($\mathbf{n} \cdot \mathbf{l} < 0$), we use the $max$ function to ensure that the negative diffuse reflections are bounded to zero.

• The $\max(0, \mathbf{n} \cdot \mathbf{l})$ component means that if $\mathbf{n} \cdot \mathbf{l} < 0$, it is treated as $0$ which indicates that no light is reflected in such cases.

The above equation avoids inaccurate negative reflections and prevents incorrect rendering of the backface lighting.

Examples

• Same spheres lit differently from different lighting angles (independent of the viewing angle)

• Same spheres but with different diffuse reflection coefficient ($k_d$)

When the value of $k_d$ changes, the reflectance property of the surface changes.

• A higher $k_d$ means that the surface will reflect more light, making it appear brighter.

• A lower $k_d$ means that the surface will reflect less light, making it appear darker.

The variations of $k_d$ can be used to represent a wide range of materials in computer graphics, from dull and matte surfaces (low $k_d$) to bright and reflective surfaces (high $k_d$).

Read more

• What is ambient reflection?

• What is specular reflection?

• What are the types of light sources?

• What is Lambert's cosine law?

• What is bilinear interpolation?

• What are barycentric coordinates?