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Last Time

• Raytracing and PBRT Structure• Radiometric quantities

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Today

• Improving Efficiency with Monte Carlo Integration

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Monte Carlo Efficiency

• We can get an estimate faster (do less work per sample)• Or we can get an estimate with lower variance as a function

of N• Either improves the efficiency of an estimator F:

FTFVF 1

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Ways to Improve Efficiency (PBR Chap. 15)

• Less work per sample:– Russian Roulette– Splitting

• Careful sample placement:– Stratified Sampling– Low-discrepancy sequences– Best-Candidate samplers

• Introducing Bias• Importance Sampling

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Russian Roulette (PBR 15.1)

• Say the integrand, f(x), is expensive to compute– It may require tracing rays and evaluating reflectance functions, or it

may even require an infinite amount of work

• For each sample, choose some value q• Sample • If <q, then use a constant c instead of f(x)• Otherwise, evaluate the integrand, but weight it• Why does it work …

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Roulette Math

• Never decreases variance• But can reduce time without increasing variance if only

samples with low F are terminated

FEqcqqcFqFE

q

cqqcF

F

1)1(

1 otherwise if

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Choosing q and c

• Try to base q on the anticipated value of F• Integrating the direct contribution of lights requires a

shadow ray test• If the light is far away or the test ray will hit a low-

contribution part of the light, then q should be low• The contribution of reflection rays goes down as the ray-tree

gets deeper• Base q on the depth of the ray tree (the number of reflections

so far)• Choose c=0 in these cases

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Splitting (PBR 15.1.1)

• Say you need to compute a multi-dimensional integral– Such as the integral over the area seen by a pixel and the directions

to an area light source

• Say you expect the integral to vary more slowly in one dimension than the other– Incoming illumination is going to vary more rapidly, due to

occluders, than over positions within the pixel

• Choose a value for the slow varying component, and many values for the fast varying component– One ray through the pixel to find the surface point, then many rays

to the light

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Effects of Splitting

• Reduces time with little increase in variance• The example on the previous slide is extremely common• Extreme contrived case:

• Less contrived case:

5

0

5

0

2dxdyx

5

0

5

0

3 ydxdyx

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Stratified Sampling (PBR Sect 7.3 and 15.2.1)

• Consider uniformly sampling inside a square• Truly at random (choose random x, random y) will give

clumps of points– Uniformly distributed talks about probability, but not the intuitive

definition of uniform (like evenly)

• Instead, break domain into little regions, or strata, and put one sample in each piece– Choose uniformly at random in each strata – jittered sampling

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Stratified Example

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Effect on Images

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Stratification Comments

• Stratification reduces variance• But what if number of samples required is not a product of

two reasonable numbers – N=NxNy?• What do you do in multiple dimensions?

– The curse of dimensionality gets to you

• It isn’t always great

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Latin Hypercube Sampling• Say you want 5 samples in the square• Use a 5x5 grid, and place samples in squares s.t. only one

sample per row and only one sample per column– Can be done by permuting of rows or columns of identity

• Performance degrades for increasing numbers of samples– Can get large empty areas

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Stratifying Multiple Dimensions

• Do not attempt to put a sample in every possible multi-dimensional strata

• For each dimension, stratify independently• For multi-dimensional samples, permute strata, then choose

1st value from every sequence, 2nd value, etc

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Stratification can Fail

• Can get unlucky, typically by clumping in one dimension

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Low-Discrepancy Samplers

• Deterministic sequences that look random in the more natural sense – noisy– Also guarantees on arrangement

• Generation beyond scope of class

• Quasi Monte Carlo: Instead of random samples, use low discrepancy sequences

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Best Candidate Sampling• Poisson distribution: Uniform distribution with condition

that no two points are closer than a minimum distance– Excellent distribution to use, but hard to generate

• Best-Candidate patterns approximate Poisson distributions

• New methods along these lines are always being invented

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Biased Samplers

• A sampler is biased if the expected value is not the correct one– Bias is the difference between estimate and desired value

• Biased estimators for uniform[0,1], with =-0.5/N+1:

FFE

N

N

ii

XXX

XN

,,,max21

11

21

1

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Bias can be Good

• Even with low sample counts, bias can be good– Variance can be lower, e.g. O(N-2)

• Most image reconstruction filters give biased estimates– Reduces variance and hence apparent noise in image

• Standard photon map estimator (later) is biased– But result is less noisy

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Importance Sampling

• The function p is called the importance function• A wise choice of p, as close as possible to f, can

dramatically reduce variance• A poor choice can dramatically increase variance• Choosing importance functions is a well established art in

physically based rendering

y

N

i i

i dyyfxpxf

NE )(

)()(1

1

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Important Function Generalities

• If you are integrating something like f(x)g(x), it can be helpful to choose p=f or p=g

• Multiple Importance Sampling lets you combine results from many importance samplers– Generate some samples according to p– Generate some according to q– Form a weighted sum – details in PBR – good when no one

importance function handles all cases

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Next Time

• Cameras and Film