Path Notation (2)

An accurate method must handle all four types : L (D|S|G)* E

Local Illumination Model : L (D|S|G) E

First, let’s introduce some notation for paths. Each path is terminated by the eye
and a light.
E - the eye.
L - the light.
Each bounce involves an interaction with a surface. We characterize the interaction
as either reflection or transmission. There are different types of reflection and
transmission functions. At a high-level, we characterize them as
D - diffuse reflection or transmission
G - glossy reflection or transmission
S - specular reflection or refraction
Diffuse implies that light is equally likely to be scattered in any direction. Specular
implies that there is a single direction; that is, given an incoming direction there is
a unique outgoing direction. Finally, glossy is somewhere in between.
Particular ray-tracing techniques may be characterized by the paths that they
consider.
Appel Ray casting: E(D | G)L
Whitted Recursive ray tracing: E[S*](D | G)L
Kajiya Path Tracing: E[(D | G | S) + (D | G)]L
Goral Radiosity: ED*L
The set of traced paths are specified using regular expressions, as was first proposed
by Shirley. Since all paths must involve a light L, the eye E, and at least one
surface, all paths have length at least equal to 3.
A nice thing about this notation is that it is clear when certain types of paths
are not traced, and hence when certain types of light transport is not considered
by the algorithm. For example, Appel’s algorithm only traces paths of length 3,
ignoring longer paths; thus, only direct lighting is considered. Whitted’s algorithm
traces paths of any length, but all paths begin with a sequence of 0 or more mirror
reflection and refraction steps. Thus, Whitted’s technique ignores paths such as
the following EDSDSL or E(D | G)* L. Distributed ray tracing and path tracing
includes multiple bounces involving non-specular scattering such as E(D | G)* L.
However, even these methods ignore paths of the form E(D | G)S* L; that is, multi-ple
specular bounces from the light source as in a caustic. Obviously, any technique
that ignores whole classes of paths will not correctly compute the solution to the
rendering equation.


Radiosity : L D* E

Whitted Recursive Ray tracing: E S* (D|G) L   

Combining Radiosity and Ray Tracing

 


First Pass - formalized by Rushmeier and Torrance

Diffuse Transmission

Specular Transmission

Specular Reflection

With these extensions, we can now account for:

Once this pass is complete, we then perform the 2nd pass to compute specular - specular and diffuse - specular

Specular - specular is given by ray tracing

For diffuse - specular, we would need to send out many rays from the point through the hemisphere around the point, weight the rays by the bidirectional specular reflectivity, then sum them together.