Path Tracing

Path Tracing



CS 319

Advanced Topics in Computer Graphics

John C. Hart

Ray Tracing

• Whitted, CACM 80

• LDS*E

• Rays cast from eye into scene objects lights

Why? Because most rays cast from

light wouldn’t reach eye

• Shadow rays cast at each intersection

Why? Because most rays wouldn’t

reach the light source

• Workload badly distributed

Why? Because # of rays grows

exponentially, and their result

becomes less influential

Tough Cases

• Caustics

– Light focuses through a specular

surface onto a diffuse surface

– LSDE

– Which direction should secondary

rays be cast to detect caustic?

• Bleeding

– Color of diffuse surface reflected in

another diffuse surface

– LDDE

– Which direction should secondary

rays be cast to detect bleeding?

Path Tracing

• Kajiya, SIGGRAPH 86

• Diffuse reflection spawns infinite rays

• Pick one ray at random objects lights

• Cuts a path through the dense ray tree

• Still cast an extra shadow ray toward

light source at each step in path

• Trace 40 paths per pixel

Approximate

Integration

• Can’t solve integral symbolically

• Replace continuous integral with

discrete summation

 f(x)dx  Sf(xi)Dxi

• Discretization requires sampling

• Potential Sampling Problems

• Aliasing

– Jitter samples

• Biasing

– Ensure samples properly distributed

• Inaccuracy

– Ensure samples representative

Importance Sampling

• Areas of constant illumination need

few samples

– walls, floors, table tops

• Areas of interesting illumination need

many samples

– edges, shadow boundaries, caustic

boundaries

– cast rays in high variance directions

of reflectivity

• Important that bright areas receive at

least one sample

– cast a ray directly to light source

– cast a ray in highlight direction of

reflectivity

Stratified Sampling

• Add sample until variance falls below

threshold

• Sequential Uniform Sampling

– Lee, Redner and Uselton

SIGGRAPH 85

– Subdivide largest area first

– Breadth first subdivision

• Hierarchical Integration

– Weight samples by area represented

– Variance becomes O(1/n2) instead

of O(1/n)

Examples









Jensen, Stanford

Probability Distributions

p(z) Uniform pdf

from a to b



• Continuous real random variable z 1/(b-a)

• Probability density function (pdf) p(z) 0 z

– Prob. of choosing a (P[z=a]) is p(a) a b



– Prob. of a  z  b is ab p(z)dz

p(z) = 2z

– Pdf is always positive: p(z) 0 z

– Pdf sums to 100%:  p(z)dz = 1 2



• Expected value: E[f(z)] =  f(z)p(z)dz 1

– Linear, regardless of corr. of f and g

E[f(z)+g(z)] = E[f(z)] + E[g(z)] 0 z

0 1

• Variance: V[z] = E[(z – E[z])2] Prob. z < 50% is 25%

– Easier to compute as: E[z2] – E[z]2

L(w)

Monte Carlo Integration

• Expected value of f w

Uniform samples waste too

– Given random variable z much time measuring low light

L(w)

– With pdf p

E[f(z)] =  f(z)p(z)dz

• Implemented as w

Non-uniform samples skewed

1 N

 f (z ) p(z )dz  N  f (z i )

i 1

toward brighter directions, but

bias result to appear too bright



• So to integrate some function g Dividing by pdf gives



– Pick samples z with pdf p, then

multiple samplesin brighter

areas less weight.

1 N g (z i ) Ideal pdf is a normalized

 g (z )dz  N  p(z i )

i 1

version of the integrand.

Monte Carlo Path Tracing

• Reflectance equation

Lr(wr) = Le(wr) + Wfr(wi,wr)Li(wi) cosqi dwi

• Path tracing

– Repeated recursive evaluation of

f r (wi , wr ) Li (wi ) cos q i Dwi

Lr (wr )  Le (wr ) 

p(wi )

• Assume diffuse surface fr(wi,wr) = r/p

• Setting p(wi) = (1/p) cos qi yields

Lr(wr) = Le(wr) + rLi(wi) Dwi

• How can we pick wi with prob. (1/p) cos qi ?

– Use Nusselt analog of random disk points

Sampling Strategies

• Sampling incident radiance

L(x, x)   f r (x, x, x)G(x, x) L(x, x)dx



– Bias samples toward direction of

maximal incident radiance

• Sampling the BRDF



L(wr )   f r (wi , wr ) L(wi ) cos q i dwi



– Bias samples toward directions of

maximal reflectance