Lambertian reflectance

Lambertian reflectance is the property that defines an ideal "matte" or diffusely reflecting surface. The apparent brightness of a Lambertian surface to an observer is the same regardless of the observer's angle of view.[1] More technically, the surface's luminance is isotropic, and the luminous intensity obeys Lambert's cosine law. Lambertian reflectance is named after Johann Heinrich Lambert, who introduced the concept of perfect diffusion in his 1760 book Photometria.

Examples

Unfinished wood exhibits roughly Lambertian reflectance, but wood finished with a glossy coat of polyurethane does not, since the glossy coating creates specular highlights. Freshly fallen snow and charcoal are approximately Lambertian surfaces of high and low reflectance respectively. Though not all rough surfaces are Lambertian, this is often a good approximation, and is frequently used when the characteristics of the surface are unknown.[2]

Spectralon is a material which is designed to exhibit an almost perfect Lambertian reflectance.[1]

Use in computer graphics

In computer graphics, Lambertian reflection is often used as a model for diffuse reflection. This technique causes all closed polygons (such as a triangle within a 3D mesh) to reflect light equally in all directions when rendered. In effect, a small flat region rotated around its normal vector will not change the way it reflects light. However, the region will change the way it reflects light if it is tilted away from its initial normal vector because the area is illuminated by a smaller fraction of the incident radiation.[3]

The reflection is calculated by taking the dot product of the surface's normal vector, ${\displaystyle \mathbf {N} }$, and a normalized light-direction vector, ${\displaystyle \mathbf {L} }$, pointing from the surface to the light source. This number is then multiplied by the color of the surface and the intensity of the light hitting the surface:

${\displaystyle I_{D}=\mathbf {L} \cdot \mathbf {N} CI_{L}}$,

where ${\displaystyle I_{D}}$ is the intensity of the diffusely reflected light (surface brightness), ${\displaystyle C}$ is the color and ${\displaystyle I_{L}}$ is the intensity of the incoming light. Because

${\displaystyle \mathbf {L} \cdot \mathbf {N} =|N||L|\cos {\alpha }=\cos {\alpha }}$,

where ${\displaystyle \alpha }$ is the angle between the directions of the two vectors, the intensity will be the highest if the normal vector points in the same direction as the light vector (${\displaystyle \cos {(0)}=1}$, the surface will be perpendicular to the direction of the light), and the lowest if the normal vector is perpendicular to the light vector (${\displaystyle \cos {(\pi /2)}=0}$, the surface runs parallel with the direction of the light).

Lambertian reflection from polished surfaces are typically accompanied by specular reflection (gloss), where the surface luminance is highest when the observer is situated at the perfect reflection direction (i.e. where the direction of the reflected light is a reflection of the direction of the incident light in the surface), and falls off sharply. This is simulated in computer graphics with various specular reflection models such as Phong, Cook-Torrance. etc.

Other waves

While Lambertian reflectance usually refers to the reflection of light by an object, it can be used to refer to the reflection of any wave. For example, in ultrasound imaging, "rough" tissues are said to exhibit Lambertian reflectance.

See also

• List of common shading algorithms
• Gamma correction

References

1. Ikeuchi, Katsushi (2014). "Lambertian Reflectance". Encyclopedia of Computer Vision. Springer. pp. 441–443. doi:10.1007/978-0-387-31439-6_534. ISBN 978-0-387-30771-8.
2. Lu, Renfu (2016). Light Scattering Technology for Food Property, Quality and Safety Assessment. CRC Press. p. 26. ISBN 9781482263350.
3. Angel, Edward (2003). Interactive Computer Graphics: A Top-Down Approach Using OpenGL (third ed.). Addison-Wesley. ISBN 978-0-321-31252-5.