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MonteCarlo Ray Tracing

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Title: MonteCarlo Ray Tracing

1
Monte-Carlo Ray Tracing
Image by Henrik
2
Antialiasing integration

So far, Antialiasing as signal processing
Now, Antialiasing as integration
Complementary yet not always the same
in particular for jittered sampling

3
Why integration?

Simple version compute pixel coverage
More advanced Filtering (convolution)is an
integralpixel s filter color
And integration is useful in tons of places in
graphics

4
Global illumination

So far, we've seen only direct lighting (red
here)
We also want indirect lighting
Full integral on hemisphere (multiplied by BRDF)
In practice, send tons of random rays

5
What else can we integrate?

Pixel antialiasing
Light sources Soft shadows
Lens Depth of field
Time Motion blur
BRDF glossy reflection
Hemisphere indirect lighting

6
Domains of integration

Pixel, lens (Euclidean 2D domain)
Time (1D)
Hemisphere
Work needed to ensure uniform probability
Light source
Same thing make sure that the probabilities and
the measures are right.

7
Example Light source

Integrate over surface or over angle
Be careful to get probabilities and integration
measure right!
More in 6.839

Sampling the source uniformly
Sampling the hemisphere uniformly
source
hemisphere
8
A little bit of eye candy for motivation

Glossy material rendering
Random reflection rays around mirror direction
1 sample per pixel

9
A little bit of eye candy for motivation

Glossy material rendering
Random reflection rays around mirror direction
256 sample per pixel

10
Error/noise expressed as variance

We use random rays
Run the algorithm again, ? get different image
What is the noise/variance/standard deviation?

11
Goal today

Prove that Monte-Carlo integration works
Notion of expected value
Discuss its convergence speed
Error notion of variance
For all this, we need to study what happens to
expected value and variance when we had tons of
random samples
Apply to ray tracing

12
Questions?

Image by Henrik

13
Integration

compute integral of weird arbitrary function
Continuous problem ? we need to discretize

14
Integration

You know trapezoid

15
Integration

Now come Monte Carlo use random samples and
compute average
We dont keep track of spacing between samples
But we kind of hope it will be on average 1/n

16
Monte-Carlo computation of p

Take a square
Take a random point (x,y) in the square
Test if it is inside the ¼ disc (x2y2 lt 1)
The probability is p /4

Integral of the function that is one inside the
circle, zero outside
y
x
17
Monte-Carlo computation of p

The probability is p /4
Count the inside ratio n inside / total
trials
p ? n 4
The error depends on the number or trials

Demo
long N_SAMPLES(long)1e9 long nSuccess0
long nTrial0 int pow1 for (long
i0 iltN_SAMPLES i) double
xMath.random() double yMath.random() if
(xxyylt1) nSuccess / if
(nTrialpow0) double myPi4.0(double)nSuc
cess/(double)i System.out.println("n"nTrial
" pi"myPi) pow2 / doubl
e myPi4.0(double)nSuccess/(double)N_SAMPLES S
ystem.out.println(myPi)
18
Why not use Simpson integration?

Yeah, to compute p, Monte Carlo is not very
efficient
But convergence is independent of dimension
Better to integrate high-dimensional functions
For d dimensions, Simpson requires Nd domains

19
Questions?

Image from the ARNOLD Renderer by Marcos Fajardo

20
Continuous random variables

Real-valued random variable x
Probability density function (PDF) p(x)
Probability of a value between x and xdx is p(x)
dx

21
Expected value

Expected value is linear
Ef1(x) a f2(x) Ef1(x) a Ef2(x)

22
Monte Carlo integration

Function f(x) of x 2 a b
We want to compute
Consider a random variable x
If x has uniform distribution, IEf(x)
By definition of the expected value

23
Sum of Random Variables

Use N independent identically-distributed (IID)
variables xi
Share same probability (uniform here)
Define
By linearity of the expectationEFN Ef(x)

Monte Carlo estimator
24
Variance

Measure of deviation from expected value
Expected value of square difference (MSE)
Standard deviation s square root of variance
(notion of error, RMS)

25
Dumbest Monte-Carlo integration

Compute 0.5 by flipping a coin
1 flip 0 or 1gt average error 0.5
2 flips 0, 0.5, 0.5 or 1 gtaverage error0. 25
4 flips 0 (1),0.25 (4), 0.5 (6), 0.75(4),
1(1) gt average error 0.1875
Does not converge very fast
Doubling the number of samples does not double
accuracy

26
Study of variance

Recall s2xy s2x s2y 2 Covx,y
We have independent variables Covxi, xj0 if i
? j
s2ax a2 s2x
i.e. stddev s (error) decreases by

27
Example

We know it should be 1.0
In practicewith uniform samples

error
s2
- s2
N
28
Monte-Carlo Recap

Expected value is the integrand
Accurate on average
Variance decrease in 1/N
Error decreases in
Good news
Math are mostly over for today
OK, its bad news if you like math (and you
should)

29
Advantages of MC Integration

Few restrictions on the integrand
Doesnt need to be continuous, smooth, ...
Only need to be able to evaluate at a point
Extends to high-dimensional problems
Same convergence
Conceptually straightforward
Efficient for solving at just a few points

30
Disadvantages of MC

Noisy
Slow convergence
Good implementation is hard
Debugging code
Debugging maths
Choosing appropriate techniques

31
Questions?

Images by Veach and Guibas

Naïve sampling strategy
Optimal sampling strategy
32
Smarter sampling

Sample a non-uniform probability
Called importance sampling
how to get probabilities right?

33
Example Glossy rendering

Integrate over hemisphere
BRDF times cosine times incoming light

Slide courtesy of Jason Lawrence
34
Sampling a BRDF
Slide courtesy of Jason Lawrence
35
Sampling a BRDF
25 Samples/Pixel
Slide courtesy of Jason Lawrence
36
Sampling a BRDF
75 Samples/Pixel
Slide courtesy of Jason Lawrence
37
Questions?
1200 Samples/Pixel
Traditional importance function
Better importance by Lawrence et al.
38
Stratified sampling

With uniform sampling, we can get unlucky
E.g. all samples in a corner
To prevent it, subdivide domain W into
non-overlapping regions Wi
Each region is called a stratum
Take one random samples per Wi

39
Example

Borrowed from Henrik Wann Jensen

Unstratified
Stratified
40
Stratified sampling - bottomline

Cheap and effective
Mostly for low-dimensional domains
Typical example jittering for antialiasing
Signal processing perspective better than
uniform because less aliasing (spatial patterns)
Monte-Carlo perspective better than random
because lower variance (error for a given pixel)

41
Questions?

Image from the ARNOLD Renderer by Marcos Fajardo

42
Global illumination

e.g. indirect lighting bouncing off the walls

43
The Rendering Equation
x
?'
Kajiya 1986
?
x'
L(x',?') E(x',?') ??x'(?,?')L(x,?)G(x,x')V(x,x
') dA
Incominglight
Geometricterm
visibility
emission
BRDF
44
Ray Casting

Cast a ray from the eye through each pixel

45
Ray Tracing

Cast a ray from the eye through each pixel
Trace secondary rays (light, reflection,
refraction)

46
Monte-Carlo Ray Tracing

Cast a ray from the eye through each pixel
Cast random rays from the visible point
Accumulate radiance contribution

47
Monte-Carlo Ray Tracing

Cast a ray from the eye through each pixel
Cast random rays from the visible point
Recurse

48
Monte-Carlo

Cast a ray from the eye through each pixel
Cast random rays from the visible point
Recurse

49
Monte-Carlo

Systematically sample primary light

50
Results
51
Monte Carlo Path Tracing

Trace only one secondary ray per recursion
But send many primary rays per pixel
(performs antialiasing as well)

52
Results
Think about it we compute an infinite-dimension
al integral with 10 samples!!!

10 paths/pixel

53
Results glossy

10 paths/pixel

54
Results glossy

100 paths/pixel

55
Importance of sampling the light
Without explicit light sampling
With explicit light sampling
1 path per pixel
4 path per pixel
56
Why use random numbers?