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,
Thus
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 (
):
Then
How are radiosity and outgoing radiance related for a Lambertian surface?
