The Rendering Equation

--- breaks: False --- # 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 $y$. - $\epsilon(x,y)$ -- 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 **B**idirectional **R**eflectance **D**istribution **F**unction (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 $\lambda$. $$L(x,y) = g(x,y)\left[ \epsilon(x,y) + \int_S \rho(x,y,z) L(y, z) \mathrm{d}z\right],$$ Operator form: $$L = g\epsilon + gML,$$ $$M: L\mapsto (x,y) \mapsto \int_S \rho(x, y, \cdot) L(y,\cdot).$$ Can be rewritten as: $$(1 - gM)L = g\epsilon.$$ If $\operatorname{spr}(gM) < 1$, the inverse operator is given by (converging) Neumann series: $$(1 - gM)^{-1} = \sum_{k\geq 0} g^kM^k,$$ and the solution: $$\begin{split}L &= (\sum_{k\geq 0} g^kM^k)g\epsilon = g\epsilon + \sum_{k\geq 1} g^kM^kg\epsilon\\ &= g\epsilon + gMg\epsilon + gMgMg\epsilon + \cdots \end{split}$$ Note that in this series expansion: - $g\epsilon$ term stands for the direct light, - $gMg\epsilon$ is the once-scattered light, - $(gM)^k g\epsilon$ is the $k$-times scattered light. Kajiya claims that in cases of interest the $\operatorname{spr}(gM)$ 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](@newkozlukov/rJR8GHeEH) - [Torrance&Sparrow](/eHJK60yMSuyfyH1mmIndNg) - [Blinn-Phong](/N2NFMir6SvuF3rgeYBfdfA) - [Dichromatic Reflection Model (Shafer)](/prmEE7wXScWGwDurE-CxrQ) - Also see the [rest of the collection](/@newkozlukov/SkjMyrlEr/) ## 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](http://intro-to-dxr.cwyman.org/) - [TU Wien Rendering/Ray Tracing Course on Youtube](https://www.youtube.com/playlist?list=PLujxSBD-JXgnGmsn7gEyN28P1DnRZG7qi) - 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