Monte Carlo Path Tracing - graphics.stanford.edu

CS348B Lecture 14 Pat Hanrahan, Spring 2009

Monte Carlo Path Tracing

Path tracing starting from the eye

Path tracing starting from the lights

Which direction is best?

Bidirectional ray tracing

Random walks and Markov chains

Irradiance caching

Photon mapping

The Rendering Equation

Solving the Rendering Equation

Rendering Equation

Solution

Successive Approximation

Light Path

Solving the Rendering Equation

One path

Solution is the integral over all paths

Solve using Monte Carlo Integration

Question: How to generate a random path?

Path Tracing from the Eye

Path Tracing: From Camera

Step 1. Choose a camera ray r given the (x,y,u,v,t) sample

weight = 1;

Step 2. Find ray-surface intersection

Step 3.

if hit light

return weight * Le(r);

weight *= reflectance(r)

Choose new ray r’ ~ BRDF(O|I)

Go to Step 2.

Path Tracing

10 paths / pixel

M. Fajardo Arnold Path Tracer

Works well for diffuse surfaces

And large hemispherical light sources

How Many Bounces?

Avoid path that carry little energy

Terminate when the weight is low

Photons with similar power is a good thing

Think of importance sampling

Integrand is f(x)/p(x) which is constant

Russian Roulette

Terminate photon with probability p

Adjust weight of the result by 1/(1-p)

Intuition:

Reflecting from a surface with R=.5

100 incoming photons with power 2 W

1. Reflect 100 photons with power 1 W

2 Reflect 50 photons with power 2 W

Path Tracing: Include Direct Lighting

Step 1. Choose a camera ray r given the

(x,y,u,v,t) sample

weight = 1;

Step 3.

L += weight * Lr(light sources)

weight *= reflectance(r)

Choose new ray r’ ~ BRDF pdf(r)

Go to Step 2.

Penumbra: Trees vs. Paths

4 eye rays per pixel

16 shadow rays per eye ray

64 eye rays per pixel

1 shadow ray per eye ray

Variance Decreases with N

10 rays per pixel 100 rays per pixel

From Jensen, Realistic Image Synthesis Using Photon Maps

Fixed Sampling (Not Random Enough)

10 paths / pixel

Light Ray Tracing

Classic Ray Tracing

Forward (from eye): E S* (D|G) L

From Heckbert

Photon Paths

From Heckbert

Caustics

Radiosity

Early Example [Arvo, 1986]

“Backward“ ray tracing

Path Tracing: From Lights

Step 1. Choose a light ray.

Choose a ray from the light source

distribution function

x ~ p(x)

d ~ p(d|x)

r = (x, d)

weight = ;

Step 1. Choose a light ray

Step 3. Reflect or transmit

u = Uniform()

if u < reflectance(x)

Choose new direction d ~ BRDF(O|I)

goto Step 2

else u < reflectance(x)+transmittance(x)

Choose new direction d ~ BTDF(O|I)

goto Step 2

else // absorption=1–reflectance-transmittance

terminate on surface; deposit energy

Step 1. Choose a light ray

Step 3. Interaction

u = Uniform() / (reflectance(x)+transmittance(x)

if u < reflectance(x)

Choose new direction d ~ BRDF(O|I)

weight *= reflectance(x)

Choose new direction d ~ BTDF(O|I)

weight *= transmittance(x)

goto Step 2

Bidirectional Path Tracing

Symmetric Light Path

Bidirectional Ray Tracing

Path Pyramid

From Veach and Guibas

Comparison

Bidirectional path tracing

25 rays per pixel

Path tracing

56 rays per pixel

From Veach and Guibas

Same amount of time

Which Direction?

Solve a linear system

Solve for a single xi?

Solve the reverse equation

Source

Estimator

More efficient than solving for all the unknowns

[von Neumann and Ulam]

Discrete Random Walk

Discrete Random Process

Creation Termination

Transition

States

Assign probabilities to each process

Discrete Random Process

Transition

States

Equilibrium number of particles in each state

Equilibrium Distribution of States

Total probability of being in states P

Solve this equation

Discrete Random Walk

1. Generate random particles from sources.

2. Undertake a discrete random walk.

3. Count how many terminate in state i [von Neumann and Ulam; Forsythe and Leibler; 1950s]

Transition

States

Monte Carlo Algorithm

Define a random variable on the space of paths

Probability:

Estimator:

Expectation:

Monte Carlo Algorithm

Define a random variable on the space of paths

Probability:

Estimator:

Estimator

Count the number of particles terminating in state j

Equilibrium Distribution of States

Total probability of being in states P

Note that this is the solution of the equation

Thus, the discrete random walk is an unbiased

estimate of the equilibrium number of particles

in each state

Monte Carlo Path Tracing - graphics.stanford.edu

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