Ray Tracing¶

Whitted-Style Ray Tracing¶

Why Ray Tracing?¶

Rasterization couldn't handle global effects well
Rasterization is fast, but quality is relatively low
Ray tracing is accurate, but is very slow

Basic Ray-Tracing Algorithm¶

Ray-Surface Intersection¶

Ray Equation¶

\[ \mathbf{r} = \mathbf{o} + t\mathbf{d}\quad 0\leq t < \infty \]

$\mathbf{r}$ : point along ray
t : "time"
$\mathbf{o}$ : origin
$\mathbf{d}$ : (normalized) direction

Plane Equation¶

\[ \mathbf{p}: (\mathbf{p} - \mathbf{p}')\cdot \mathbf{N}=0 \]

$\mathbf{p}$ : all points on plane
$\mathbf{p}'$ : one point on plane
$\mathbf{N}$ : normal vector

Ray Intersection With Plane¶

\[ \mathbf{p}=\mathbf{r}(t)\\ t = \frac{(\mathbf{p}'-\mathbf{o})\cdot \mathbf{N}}{\mathbf{d}\cdot\mathbf{N}} \]

Check: $0\leq t < \infty$

Möller–Trumbore Algorithm¶

A faster approach, giving barycentric coordinate directly $$ \mathbf{O} + t\mathbf{D} = (1-b_1-b_2)\mathbf{P}_0 + b_1\mathbf{P}_1 + b_2\mathbf{P}_2 $$

\[ \begin{bmatrix} t\\b_1\\b_2 \end{bmatrix} = \frac{1}{\mathbf{S}_1\cdot\mathbf{E}_1} \begin{bmatrix} \mathbf{S}_2\cdot \mathbf{E}_2\\ \mathbf{S}_1\cdot \mathbf{S}\\ \mathbf{S}_2\cdot \mathbf{D} \end{bmatrix} \]

where

$\mathbf{E}_1 = \mathbf{P}_1 - \mathbf{P}_0$ , $\mathbf{E}_2 = \mathbf{P}_2 - \mathbf{P}_0$
$\mathbf{S} = \mathbf{O} - \mathbf{P}_0$
$\mathbf{S}_1 = \mathbf{D}\times \mathbf{E}_2$ , $\mathbf{S}_2 = \mathbf{S}\times \mathbf{E}_1$

Cost = (1 div, 27 mul, 17 add)

Bounding Volumes¶

Axis-Aligned Bounding Box(AABB)¶

$t_{\text{enter}}=\max\{t_{\text{min}}\}$, $t_{\text{exit}}=\min\{t_{\text{max}}\}$ $$ t=\frac{\mathbf{p}_x' - \mathbf{o}_x}{\mathbf{d}_x} $$

Using AABBs to accelerate ray tracing¶

Preprocess - Build Acceleration Grid¶

Find bounding box
Create grid
Store each object in overlapping cells

Step through grid in ray traversal order

For each grid cell, test intersection with all objects stored at that cell

Grid Resolution¶

Heuristic:

#cells = C * #objs
$C\approx 27$ in 3D

Spatial Partitions¶

KD-Tree Pre-Processing¶

Internal nodes store

split axis: x-, y-, or z-axis
split position: coordinate of split plane along axis
children: pointers to child nodes
No objects are stored in internal nodes

Leaf nodes store

list of objects

Object Partitions & Bounding Volume Hierarchy (BVH)¶

Building BVH¶

Find bounding box
Recursively split set of objects in two subsets
Recompute the bounding box of the subsets
Stop when necessary
Store objects in each leaf node

bvh

Internal nodes store

Bounding box
Children: pointers to child nodes

Leaf nodes store

Bounding box
List of objects

Nodes represent subset of primitives in scene

All objects in subtree

BVH Traversal¶

C++

Intersect(Ray ray, BVH node) {
  if (ray misses node.bbox) return;

  if (node is a leaf node) {
    test intersection with all objs;
    return closest iontersection;
  }

  hit1 = Intersect(ray, node.child1);
  hit2 = Intersect(ray, node.child2);

  return the closest of hit1, hit2;
}

Spatial vs Object Partitions¶

Spatial partition (e.g. KD-tree)

Partition space into non-overlapping regions
An object can be contained in multiple regions

Object partition (e.g. BVH)

Partition set of objects into disjoint subsets
Bounding boxes for each set may overlap in space

Basic Radiometry¶

Measurement system and units for illumination

Accurately measure the spatial properties of light

Radiant flux, intensity, irradiance, radiance

Perform lighting calculations in a physically correct manner

Radiant Energy and Flux (Power)¶

\[ Q\ [\text{J}=\text{Joule}] \]

\[ \Phi \equiv \frac{\text{d}Q}{\text{d}t}\ [\text{W}=\text{Watt}]\ [\text{lm}=\text{lumen}] \]

Flux - #photons flowing through a sensor in unit time¶

phy

Solid Angle¶

Definition: Ratio of subtended area on sphere to radius squared. $$ \Omega = \frac{A}{r^2} $$

\[ \text{d}A = (r\text{d}\theta)(r\sin\text{d}\phi) = r^2\sin\theta\text{d}\theta\text{d}\phi\\ \text{d}\omega = \frac{\text{d}A}{r^2} = \sin\theta \text{d}\theta\text{d}\phi \]

Radiant Intensity¶

Definition: The radiant (luminous) intensity is the power per unit solid angle emitted by a point light source. $$ I(\omega)\equiv \frac{\text{d}\Phi}{\text{d}\omega}\ \left[ \frac{\text{W}}{\text{sr}} \right] \left[ \frac{\text{lm}}{\text{sr}} = \text{cd} = \text{candela} \right] $$

Irradiance¶

Definition: The irradiance is the power per projected unit area incident on a surface point. $$ E(\mathbf{x}) \equiv \frac{\text{d}\Phi(\mathbf{x})}{\text{d}A}\\ \left[ \frac{\text{W}}{\text{m}^2} \right] \left[ \frac{\text{lm}}{\text{m}^2} = \text{lux}\right] $$

Lambert's Cosine Law¶

Radiance¶

Definition: The radiance (luminance) is the power per unit solid angle per projected unit area. $$ L(\text{p},\omega) \equiv \frac{\text{d}^2\Phi(\text{p},\omega)}{\text{d}\omega\text{d}A\cos\theta} $$ Recall

Irradiance: power per projected unit area
Intensity: power per solid angle

So

Radiance: Irradiance per solid angle
Radiance: Intensity per projected unit area

Irradiance vs. Radiance¶

Irradiance: total power received by area $\text{d}A$

Radiance: power received by area $\text{d}A$ from "direction" $\text{d}\omega$

Bidirectional Reflectance Distribution Function(BRDF)¶

BRDF¶

The BRDF represents how much light is reflected into each outgoing direction $\omega_r$ from each incoming direction $$ f_r(\omega_i\to \omega_r) = \frac{\text{d}L_r(\omega_r)}{\text{d}E_i(\omega_i)} = \frac{\text{d}L_r(\omega_r)}{L_i(\omega_i)\cos\theta_i\text{d}\omega_i} \begin{bmatrix} 1\newline \text{sr} \end{bmatrix} $$

The Reflection Equation¶

$$ L_r(\text{p},\omega_r) = \int_{H^2} f_r(\text{p},\omega_i\to\omega_r)L_i(\text{p},\omega_i)\cos\theta_i\text{d}\omega_i $$ Adding an Emission term to make it general: $$ L_o(\text{p},\omega_o) = L_e(\text{p},\omega_o) + \int_{\Omega^+}L_i(\text{p},\omega_i) f_r(\text{p},\omega_i, \omega_o)(n\cdot \omega_i)\text{d}\omega_i $$ Note: now, we assume that all directions are pointing outwards!

Understanding the Rendering Equation¶

Linear Operator Equation¶

\[ l(u) = e(u) + \int l(v)\underbrace{K(u,v)\text{d}v}_{\text{Kernel of equation}}\\ L=E+KL \]

\[ L=E+KE+K^2E+K^3E+\cdots \]

Path Tracing¶

Problem¶

Where should the ray be reflected for glossy materials?
No reflections between diffuse materials?

A Simple Monte Carlo Solution¶

Compute the radiance at $p$ towards the camera

\[ L_o(\text{p},\omega_o) = \int_{\Omega^+}L_i(\text{p},\omega_i) f_r(\text{p},\omega_i, \omega_o)(n\cdot \omega_i)\text{d}\omega_i \]

Monte Carlo Integration: $\displaystyle \int f(x)\text{d}x = \frac{1}{N} \sum_{i=1}^N\frac{f(X_i)}{p(X_i)}\quad X_i\sim p(x)$

"$f(x)$": $L_i(\text{p},\omega_i) f_r(\text{p},\omega_i, \omega_o)(n\cdot \omega_i)$

PDF: $p(\omega_i)=1/2\pi$ (assume uniformly sampling the hemisphere)

\[ \begin{aligned} L_o(\text{p},\omega_o) &= \int_{\Omega^+}L_i(\text{p},\omega_i) f_r(\text{p},\omega_i, \omega_o)(n\cdot \omega_i)\text{d}\omega_i\\ &\approx \frac{1}{N}\sum_{i=1}^N\frac{L_i(\text{p},\omega_i) f_r(\text{p},\omega_i, \omega_o)(n\cdot \omega_i)}{p(\omega_i)} \end{aligned} \]

Introducing Global Illumination¶

What if a ray hits an object? The direct illumination at the object!

Sampling the Light¶

$\displaystyle \text{d}\omega = \frac{\text{d}\cos \theta'}{\|x'-x\|^2}$

\[ \begin{aligned} L_o(x,\omega_o) &= \int_{\Omega^+}L_i(x,\omega_i) f_r(x,\omega_i, \omega_o)(n\cdot \omega_i)\text{d}\omega_i\\ &= \int_AL_i(x,\omega_i) f_r(x,\omega_i, \omega_o)\frac{\cos \theta\cos \theta'}{\|x'-x\|^2}\text{d}A \end{aligned} \]

Now we consider the radiance coming from two parts:

light source (direct, no need to have RR)
other reflectors (indirect, RR)

C

shade(p, wo)
  // Contribution from the light source.
  Uniformly sample the light at x’ (pdf_light = 1 / A)
  Shoot a ray from p to x'
  if the ray is not blocked in the middle
    L_dir = L_i * f_r * cos θ * cos θ' / |x' - p| ^2 / pdf_light

  // Contribution from other reflectors.
  L_indir = 0.0
  Test Russian Roulette with probability P_RR
  Uniformly sample the hemisphere toward wi (pdf_hemi = 1 / 2pi)
  Trace a ray r(p, wi)
  if ray r hit a non-emitting object at q
    L_indir = shade(q, -wi) * f_r * cos θ / pdf_hemi / P_RR

  return L_dir + L_indir

Core Code¶

C++

Vector3f Scene::castRay(const Ray &ray, int depth) const {
  if (depth > maxDepth) {
    return Vector3f(0.0f);
  }

  Intersection inter = intersect(ray);
  if (!inter.happened) {
    return Vector3f(0.0f);
  }

  if (inter.m->hasEmission()) {
    return inter.m->getEmission();
  }

  Vector3f hitPoint = inter.coords;
  Vector3f N = inter.normal;

  Vector3f L_dir(0.0f);

  Intersection lightInter;
  float pdf_light = 0.0f;
  sampleLight(lightInter, pdf_light);

  Vector3f x = lightInter.coords;
  Vector3f ws = (x - hitPoint).normalized();
  Vector3f NN = lightInter.normal;

  Ray shadowRay(hitPoint, ws);

  Intersection shadowInter = intersect(shadowRay);
  if (shadowInter.distance - (x - hitPoint).norm() > -0.005f) {
    L_dir = lightInter.emit * inter.m->eval(ray.direction, ws, N) * dotProduct(ws, N) * dotProduct(-ws, NN) / dotProduct(x - hitPoint, x - hitPoint) / pdf_light;
  }

  Vector3f L_indir(0.0f);

  if (get_random_float() < RussianRoulette) {
    Vector3f wi = inter.m->sample(ray.direction, N).normalized();
    Ray reflectionRay(hitPoint, wi);
    Intersection reflectionInter = intersect(reflectionRay);

    if (reflectionInter.happened && !reflectionInter.m->hasEmission()) {
      float pdf_bsdf = inter.m->pdf(ray.direction, wi, N);
      if (pdf_bsdf > 0) {
        L_indir = castRay(reflectionRay, depth + 1) * inter.m->eval(ray.direction, wi, N) * dotProduct(wi, N) / pdf_bsdf / RussianRoulette;
      }
    }
  }

  return L_dir + L_indir;
}