GAMES101 Lecture 13 - Ray Tracing 1 (Whitted-Style Ray Tracing)

GAMES101_Lecture_13.pdf

I. Why Ray Tracing?

• Rasterization couldn't handle global effects well

  • (Soft) shadows

  • Especially when the light sources bounce more than once

    • Glossy reflection

    • Indirect Illumination

• Rasterization is fast, but quality is relatively low

• Ray tracing is accurate, but is very slow

  • Rasterization: real-time

  • Ray tracing: offline

  • ~10K CPU core hours to render one frame in production

II. Basic Ray-Tracing Algorithms

Three assumptions:

• Light travels in straight lines (physically wrong)

• Light rays do not "collide" with each other if they cross (physically wrong)

• Light rays travel from the light sources to the eye (but the physics is invariant under path reversal - reciprocity)

Ray Casting

• Pinhole Camera Model

• Point Light Source

• Start an eye ray at the eye and goes through pixel.

  • Collide with the first object, and then

  • (Local) Perform shading calculation to compute color of pixel

• Directly project to the light source after the eye ray has collide with an object.

  • Then the light bounces only once.

Recursive (Whitted-Style) Ray Tracing

"An improved illumination model for shaded display." - T. Whitted, CACM 1980

Simulates the recursive bouncing of light rays.

• Taking both Reflection and Refraction into consideration

• Different rays:

  

  • Primary Ray: Starts at the eye

  • Secondary Ray: Generated upon collision of primary ray

  • Shadow Ray: Directly connected from the intersection to the light source.

Ray-Surface Intersection

• Ray Equation: A ray is defined by its origin $\mathbf{\text{o}}$$\textbf{o}$ and a direction vector $\mathbf{\text{d}}$$\textbf{d}$:

  $$\mathbf{\text{r}}(t)=\mathbf{\text{o}}+t\mathbf{\text{d}}0\le t\le \mathrm{\infty }$$

• Intersections:

  • Implicit Surface: Solve for real, positive roots.

    $$\mathbf{\text{p}}:f(\mathbf{\text{p}})=0f(\mathbf{\text{o}}+t\mathbf{\text{d}})=0$$

  • Triangle Mesh

    • Naïve Procedure:

      1. Ray-plane intersection

         $$(\mathbf{\text{p}}-{\mathbf{\text{p}}}^{\prime })\cdot \mathbf{\text{N}}=(\mathbf{\text{o}}+t\mathbf{\text{d}}-{\mathbf{\text{p}}}^{\prime })\cdot \mathbf{\text{N}}=0$$

         Solving the equation gives

         $$t=\frac{({\mathbf{\text{p}}}^{\prime }-\mathbf{\text{o}})\cdot N}{\mathbf{\text{d}}\cdot \mathbf{\text{N}}}.$$

         Remember to check whether $0\le t\le \mathrm{\infty }$$0 \leq t \le \infty$.

      2. Test if the point is inside the triangle

          

    • Möller Trumbore Algorithm: A faster approach by examining the barycentric coordinate.

      $$\mathbf{\text{o}}+t\mathbf{\text{d}}=(1-b_1-b_2){\mathbf{\text{p}}}_0+b_1{\mathbf{\text{p}}}_1+b_2{\mathbf{\text{p}}}_2$$

      Solving the equation gives:

      $$\left[\begin{array}{c}t\\ b_1\\ b_2\end{array}\right]=\frac{1}{{\mathbf{\text{S}}}_1\cdot {\mathbf{\text{E}}}_1}\left[\begin{array}{c}{\mathbf{\text{S}}}_2\cdot {\mathbf{\text{E}}}_2\\ {\mathbf{\text{S}}}_1\cdot \mathbf{\text{S}}\\ {\mathbf{\text{S}}}_2\cdot \mathbf{\text{D}}\end{array}\right]$$

      where

    $${\mathbf{\text{E}}}_1={\mathbf{\text{p}}}_1-{\mathbf{\text{p}}}_0$$
    $${\mathbf{\text{E}}}_2={\mathbf{\text{p}}}_2-{\mathbf{\text{p}}}_0$$
    $$\mathbf{\text{S}}=\mathbf{\text{o}}-{\mathbf{\text{p}}}_0$$
    $${\mathbf{\text{S}}}_1=\mathbf{\text{d}}\times {\mathbf{\text{E}}}_2$$
    $${\mathbf{\text{S}}}_2=\mathbf{\text{S}}\times {\mathbf{\text{E}}}_1$$

    See Cramer's rule.

  • Inside/outside test: Fire a ray from a given point. Even number of intersections if the point is outside.

Accelerating Ray-Surface Intersection

Performance Challenges

Simple ray-scene intersection

• Exhaustively test ray-intersection with every triangle

• Find the closest hit (i.e. minimum $t$$t$)

Bounding Volumes

Build a larger volume for testing the potential intersection quickly.

• Feature:

  • Object is fully contained in the volume.

  • If the ray doesn't hit the volume, then it won't hit the object

  • Test BVol first, then test the object if BVol is hit.

• Can be in any shape as long as it speeds up the test.

Ray-Intersection With Box

• Axis-Aligned Bounding Box (AABB)

  • For the 3D box, $t_{\text{enter}}=max\{t_{\text{min}}\}$$t_\text{enter} = \max{\{t_\text{min}\}}$, $t_{\text{exit}}=min\{t_{\text{max}}\}$$t_{\text{exit}} = \min{\{t_{\text{max}}\}}$.

  • If $t_{\text{enter}}<=t_{\text{exit}}$$t_{\text{enter}} <= t_{\text{exit}}$ and $t_{\text{exit}}>0$$t_\text{exit} > 0$, then there is an intersection.

    • Equality for BBoxes that contains an axis-aligned plane only

• Why Axis-Aligned?

  • General Plane:

    $$t=\frac{({\mathbf{\text{p}}}^{\prime }-\mathbf{\text{o}})\cdot \mathbf{\text{N}}}{\mathbf{\text{d}}\cdot \mathbf{\text{N}}}$$

    3 subtractions, 6 multiplications, 1 division

  • Slabs perpendicular to $x$$x$ axis:

    $$t=\frac{{\mathbf{\text{p}}}_x^{\prime }-{\mathbf{\text{o}}}_x}{{\mathbf{\text{d}}}_x}$$

    1 subtraction, 1 division