Recall from 2D than the angle subtended by an object, O, from a point, x, is the length of the projection of O onto a unit circle centred at x. It is measured in "unit angles", or radians (rad). The full circle subtends radians.
In 3D, the angle subtended by an object, O, from a point, x, is the area of the projection of O onto a unit sphere centred at x. It is measured in "unit solid angles", or steradians (sr). The full sphere subtends steradians.
Radiant power ( ) is the rate at which light energy is transmitted, and is measured as energy (photons) per unit time, in Watts.
Flux is the radiant power crossing a particular surface.
Flux density ( ) is the radiant power per unit area of the surface, measured in Watts per unit area. At a point, , surrounded by an infinitesimal surface, , the flux density is
Incident flux density is the flux density of particles arriving at a surface from all directions. It is also called irradiance (E).
Exitant flux density is the flux density of particles leaving a surface in all directions. It is also called radiosity (B).
Consider a point light source. Its radiant intensity (I) is the power radiated per unit solid angle, and is measured in Watts per steradian.
In a particular direction, , surrounded by an infinitesimal solid angle, , the radiant intensity is
Radiance (L) is the flux density at a point, , in a direction, . Sometimes, the direction is given by two angles: elevation (from the surface normal, ) and azimuth "phi" (from a given direction on the surface).
The flux density is measured with respect to a surface perpendicular to the direction. If the surface at is not perpendicular to direction , it is projected onto a surface that is.
The figure below shows the radiance at as the flux density across the surface in the direction :
Since the ratio of the areas, and , is (where is the angle between and , the surface normal at ), radiance is typically written as
Radiosity is the exitant flux density from a point: . How does radiosity relate to radiance?
Let be the set of directions above the surface at . From the radiance equation above,
Similarly, for incoming energy, the irradiance is
Intuitively, the cosine term occurs because the radiance in direction is taken with respect to a surface, , perpendicular to the direction, not with respect to the surface itself:
[Sillion & Puech 2.1.2]
Given two points, and , the radiance shot from toward is equal to that received by from the direction of . Radiance is not attenuated with distance.
As a consequence, all we need in order to render an image is the radiance, in the direction of the eye, leaving each point of the scene.
[Sillion & Puech 2.1.4]
The bidirectional reflectance distribution function at a point, , is the ratio of the radiance going out in one direction to the flux density coming in from another direction.
Let and be the incoming and outgoing directions, respectively.
The incoming directional flux density, , is the flux density of energy arriving from an infinitesimal solid angle, , around . From the radiance equation,
The infinitesimal flux density, , is taken with respect the the infinitesimal solid angle, . The BRDF is then
Note that the units of the BRDF are per steradian. The reflected radiance due to radiance incoming from a particular direction is
A Lambertian surface, or "ideal diffuse surface", has a constant BRDF:
The radiance of such a surface is independent of direction. The total outgoing radiance in a particular direction, , is the integral of reflected radiance from all incoming directions, :
How are radiosity and irradiance related for a Lambertian surface?
Inuitively, the quantity is the fraction of irradiance, , that is reflected (as radiosity, ). This quantity is known as the albedo ( ):
How are radiosity and outgoing radiance related for a Lambertian surface?