Key nodes:
diffrt.pdf
and PhD thesis. Shows that under mild assumptions the integral operator introduced by Kajiya has piecewise differentiable kernel with discontinuities at edges. Consequence: differentiability in the weak sense, edges contribute as delta-functions, triangles are modeled as products of three Heaviside functions, Tzu Mao suggests combining usual samples of triangles' interior with sampling edge points. Only silhouette edges are sampled, as far as I understand.Operator form:
Can be rewritten as:
If , the inverse operator is given by (converging) Neumann series:
and the solution:
Note that in this series expansion:
Kajiya claims that in cases of interest the is indeed and series is convergent.
Lambertian/ideal diffuse
Light is scattered uniformly in all directions and intensity depends only on the cosangle betwen light direction and surface normal – not on direction of scattered ray.
This is in Kajiya's model, with the surface normal at .
Whitted
is the sum of two "delta functions" (deltas not in but in angles, AFAIU) – one for reflection, one for refraction – and the diffuse term
Also see the rest of the collection
Seminal works
Shaders, raytracing
Gravity-based ray (beam) tracing, AKA Interstellar's Gargantua blackhole
Kip Thorne and Double Negative team implemented a custom renderer that traces beams of rays (paths and shapes) back in time, along geodesics of general-relativistic spacetime. The beam is initiated as shooting from the eye into a pixel-sized disc on the screen. That's some cool twist between VFX and differential geometry.
Interstellar's wormhole
Real-time raytracing