Path Tracing

Parallel Algorithms

David Mayerich

Scalable Tissue Imaging and Modeling (STIM) Laboratory

Department of Electrical and Computer Engineering

Cullen College of Engineering

University of Houston

David Mayerich

STIM Laboratory, University of Houston

Global Illumination

David Mayerich

STIM Laboratory, University of Houston

\(\int \cdots d\theta\)

\(\mathbf{p}\)

\(e=(1,1,1)\)
\(d=(0,0,0)\)

\(\mathbf{v}\)

light from \(\phi\)

emission
from \(\phi\)

\(e=(0,0,0)\)
\(d=(1,1,1)\)

hemisphere

material

collected from \(\theta\)

L(\phi) = E(\phi) + \int_\Omega d(\phi, \theta) C(\theta) d \theta

\(-\frac{\pi}{2}\)

\(\frac{\pi}{2}\)

\(L(\theta)=\)

\(\theta\)

\(\theta=\frac{\pi}{2}\)

\(\theta=-\frac{\pi}{2}\)

Material Properties

David Mayerich

STIM Laboratory, University of Houston

\(\mathbf{p}\)

\(\mathbf{v}\)

\(e=(0,0,1)\)
\(d=(0,0,0)\)

\(e=(0,0,0)\)
\(d=(1,0,0)\)

\(e=(1,1,1)\)
\(d=(0,0,0)\)

\(e=(0,0,0)\)
\(d=(1,1,1)\)

\(e=(0,1,0)\)
\(d=(0,0,1)\)

Rendering Equation

\(L\) is the wavelength \(\lambda\) light from direction \(\phi\)
\(E\) is the wavelength \(\lambda\) light emitted from the object from direction \(\phi\)
\(R\) is the wavelength \(\lambda\) light reflected by the object at from direction \(\phi\)

David Mayerich

STIM Laboratory, University of Houston

L(\phi, \lambda) = E(\phi,\lambda) + R(\phi, \lambda)

R(\phi, \lambda) = \int_\Omega d(\phi, \theta, \lambda) C(\theta, \lambda) d \theta

\(d\) defines how the material transmits light \(\lambda\) from direction \(\theta\) to \(\phi\)
\(C\) is the light of wavelength \(\lambda\) coming from angle \(\theta\)

L(\phi, \lambda) = E(\phi,\lambda) + \int_\Omega d(\phi, \theta, \lambda) C(\theta, \lambda) d \theta

Bidirectional Reflectance Distribution

The material function \(d(\phi, \theta, \lambda)\in\mathbb{R}\) is known as the BRDF
bidirectional reflectance distribution function
Defines how light is reflected off of a surface
A BRDF is always positive: \(d(\phi, \theta, \lambda)\geq 0\)
Works the same in both directions: \(d(\phi, \theta, \lambda) = d(\theta, \phi, \lambda)\)
It doesn't add energy: \(\int_\Omega d(\phi, \theta) \cos(|\theta-\phi|)d\theta \leq 1\)

David Mayerich

STIM Laboratory, University of Houston

L(\phi, \lambda) = E(\phi,\lambda) + \int_\Omega d(\phi, \theta, \lambda) C(\theta, \lambda) d \theta

\(\theta\)

\(\phi\)

\(\mathbf{n}\)

Path Tracing Algorithm

calculate a ray \((\mathbf{p}_0, \mathbf{v}_0)\)
initialize the ray color \(\mathbf{c}\leftarrow(0, 0, 0)\)
initialize the current throughput \(\mathbf{r}=(1,1,1)\) to ensure energy is conserved
while throughput is high: \(|\mathbf{r}| \geq R_{min}\)
- find the closest intersection \(\mathbf{p}_b = \mathbf{p}_{b-1}+t\mathbf{v}_{b-1}\) with object \(s[i]\)
  - if you hit the background, set \(\mathbf{r}=(0,0,0)\) which will exit the loop
- add the emissive light from the object based on throughput: \(\mathbf{c} =\mathbf{c} + \mathbf{r}\circ\mathbf{s}[i]_e\)
- create a new ray \((\mathbf{p}_b, \mathbf{v}_b)\) with a random direction \(\mathbf{v}_b\) based on the BRDF
- reduce the throughput based on the BRDF: \(\mathbf{r} = \mathbf{r} \circ \mathbf{d}(\mathbf{v}_{b-1}, \mathbf{v}_{b})\)
assign \(\mathbf{c}\) to the pixel

David Mayerich

STIM Laboratory, University of Houston

Provided Code

a tira::Camera class that provides ray directions based on a camera position

David Mayerich

STIM Laboratory, University of Houston

tira::Camera c;
// initialize camera with UI (provided)
glm::vec3 v = c.Ray(-0.2, 0.3)

an ImGui user interface for controlling the camera position with a GUI
a template for two functions that you can expand to create your path tracer

\(-0.5,-0.5\)

\(0.5,0.5\)

\(-0.2,0.3\)

Basic Algorithms

some sort of struct or class containing properties of the sphere, at the very least:
- position \(\mathbf{p}\) and radius \(a\)
- emissive color \(\mathbf{e}=(e_r, e_g, e_b)\)
- a basic diffuse BRDF color: \(\mathbf{d}=(d_r, d_g, d_b)\)
code to calculate the intersection of a ray and a sphere
a way to generate random numbers so that you can create new random rays after an intersection
the path tracing loop described in the previous slide
there are great websites describing all of these, but most importantly: You Must Understand Your Own Code!

David Mayerich

STIM Laboratory, University of Houston

Recommended Resources

Ray Tracing in One Weekend (Shirley, Black, and Hollasch):
https://raytracing.github.io/books/RayTracingInOneWeekend.html
note: this covers recursive ray tracing, not path tracing
Scratchpixel has an online book on 3D rendering:
https://www.scratchapixel.com/
note: you can probably skip to the "Global Illumination and Path Tracing" chapter
Intersection tests between objects:
https://www.realtimerendering.com/intersections.html

David Mayerich

STIM Laboratory, University of Houston

Programming Assignment 2

Create a scene with at least 10 spheres:
3+ spheres must emit light at different wavelengths (colors)
Your results will be very noisy in the first pass, so average multiple images

David Mayerich

STIM Laboratory, University of Houston

10X

100X

10,000X

See the README.md file for implementation details (I'll update it as they come up)
Start small (render one purely emissive sphere at the origin

What To Submit:

a ZIP of your repository: make sure it compiles from scratch with CMake
include a PDF of your flame graph and a description of your hot path

Template Demo

You can pull a repository with a template for the path tracer here:
https://codeberg.org/stimlab/par-raytrace

David Mayerich

STIM Laboratory, University of Houston