The Rendering Equation

Key nodes:

• Kajiya's "the rendering equation" which introduces the integral equation for radiance, suggests MCMC solution, introduces importance sampling.
• Tzu Mao's 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.
• Some basics on the pipeline and how shaders work: see "Intro to DXR" in the links.
• …

Kajiya'86

• $L(x,y)$ – radiance passed from
  $y$ to
  $x$.
• $g(x,y)$ – geometry term, depends on distance, visibility, and, apparently, the media;
  $g(x,y)=0$ unless
  $x$ is visible from
  $y$.
• $ϵ(x,y)$ – emmitance term; light emmited (by a light source) from
  $y$ to
  $x$.
• $\rho (x,y,z)$ – scattering term; density of distribution of light scattered from
  $z$ to
  $x$ through
  $y$; usually realized by some Bidirectional Reflectance Distribution Function (BRDF) that depends on the vector
  ${\omega }_{\text{LS}}\propto z-y$ pointing towards lightsource and vector
  ${\omega }_{\text{V}}\propto x-y$ towards viewer:
  $\rho (x,y,z)=f({\omega }_{\text{LS}},{\omega }_{\text{V}}).$
• $\rho (x,y,z)L(y,z)$ – radiance scattered from
  $z$ to
  $x$ through
  $y$.
• Far as I understand, equations should be refined to account for wavelengths, which are ignored in the paper; in equation on the wiki,
  $L$ depends on wavelength
  $\lambda$.

$$L(x,y)=g(x,y)[ϵ(x,y)+{\int }_S\rho (x,y,z)L(y,z)\mathrm{d}z],$$

Operator form:

$$L=gϵ+gML,$$

$$M:L\mapsto (x,y)\mapsto {\int }_S\rho (x,y,\cdot )L(y,\cdot ).$$

Can be rewritten as:

$$(1-gM)L=gϵ.$$

If

$\mathrm{spr}(gM)<1$, the inverse operator is given by (converging) Neumann series:

$$(1-gM{)}^{-1}=\sum _{k\ge 0}{g}^k{M}^k,$$

and the solution:

$$\begin{array}{rl}L& =(\sum _{k\ge 0}{g}^k{M}^k)gϵ=gϵ+\sum _{k\ge 1}{g}^k{M}^{k}gϵ\\ & =gϵ+gMgϵ+gMgMgϵ+\cdots \end{array}$$

Note that in this series expansion:

• $gϵ$ term stands for the direct light,
• $gMgϵ$ is the once-scattered light,
• $(gM{)}^{k}gϵ$ is the
  $k$-times scattered light.

Kajiya claims that in cases of interest the

$\mathrm{spr}(gM)$ is indeed

$<1$ and series is convergent.

Materials and approximations

• 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

  $\rho (x,y,z)=\cos (z-y,n(y))$ in Kajiya's model, with
  $n(y)$ the surface normal at
  $y$.

• Whitted

  $M$ is the sum of two "delta functions" (deltas not in
  ${\mathbb{R}}^3$ but in angles, AFAIU) – one for reflection, one for refraction – and the diffuse term
  ${\int }_S\cos (\cdot -y,n)L(y,\cdot ).$

• Phong

• Torrance&Sparrow

• Blinn-Phong

• Dichromatic Reflection Model (Shafer)

• Also see the rest of the collection

Links

Seminal works

• Whitted'80: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.107.3997&rep=rep1&type=pdf
• Cook'84: https://artis.inrialpes.fr/Enseignement/TRSA/CookDistributed84.pdf
• Kajiya'86: http://www.cse.chalmers.se/edu/year/2011/course/TDA361/2007/rend_eq.pdf

Shaders, raytracing

• Siggraph 2018 course on DirectX Raytracing
• TU Wien Rendering/Ray Tracing Course on Youtube
• https://users.cg.tuwien.ac.at/zsolnai/gfx/rendering-course/
• https://rgl.epfl.ch/publications/Dupuy2018Adaptive

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.

• https://www.dneg.com/black-holes/
• https://www.dneg.com/show/interstellar/
• Gravitational renderer (1-page report): https://authors.library.caltech.edu/71785/1/a21-james.pdf
• Gravitational lensing by spinning blackholes: https://arxiv.org/abs/1502.03808
• DNEG youtube channel: https://www.youtube.com/channel/UCdVCZDUJkSsWDfQkS4dHvYA/
• Corridor and Blender Guru's breakdown of interstellar: https://youtu.be/D7Cv7x6jjYQ?t=749

Interstellar's wormhole

• https://www.dneg.com/interstellar-wormhole/
• https://aapt.scitation.org/doi/pdf/10.1119/1.4916949?class=pdf
• Kip Thorne's '87 paper on wormholes for teaching general relativity: http://www.cmp.caltech.edu/refael/league/thorne-morris.pdf
• SE on accretion disk in interstellar: https://physics.stackexchange.com/questions/151301/shape-of-accretion-disk-surrounding-the-black-hole-in-interstellar-film

Real-time raytracing

• unity/realtime raytracing https://youtu.be/Gz9weuemhDA?list=PLUPhVMQuDB_aTQ8Ir8tfL1tLr_TVOnvL5&t=5562