9: Lecture Monte Carlo Rendering
Chapters 4 and 5 in Advanced GI
Homework
HW 1out, due Oct 5 Assignments done separately
Might revisit this policy for later assignments
Fall 2004 Kavita Bala Computer Science Cornell University
Kavita Bala, Computer Science, Cornell University
Stochastic Ray Tracing
Parameters?
# starting rays per pixel # random rays for each surface point (branching factor)
Algorithm so far ...
Shoot # viewing rays through each pixel Shoot # indirect rays, sampled over hemisphere
Path tracing shoots only 1 indirect ray
Path Tracing
Branching factor == 1
Terminate recursion using Russian Roulette
Kavita Bala, Computer Science, Cornell University
Kavita Bala, Computer Science, Cornell University
Performance/Error
Want better quality with smaller number of samples
Fewer samples/better performance Stratified sampling Quasi Monte Carlo: well-distributed samples
Stratified Sampling
Samples could be arbitrarily close Split integral in subparts
I=
X1
f (x)
f ( x)dx + K + f ( x)dx
XN
Faster convergence
Importance sampling: next-event estimation
Estimator
I strat 1 = N
0
1
i =1
N
f ( xi ) p ( xi )
Variance:
Kavita Bala, Computer Science, Cornell University
strat sec
Kavita Bala, Computer Science, Cornell University
1
Higher Dimensions
Stratified grid sampling:
Quasi Monte Carlo
Eliminates randomness to find welldistributed samples Samples are determinisitic but appear random
Nd samples
N-rooks sampling:
Kavita Bala, Computer Science, Cornell University Kavita Bala, Computer Science, Cornell University
N samples
Performance/Error
Want better quality with smaller number of samples
Fewer samples/better performance Stratified sampling Quasi Monte Carlo: well-distributed samples
Next Event Estimation
L( x ) = Le + Ldirect + Lindirect
= Le +
x
f r cos +
x
f r cos
Faster convergence
Importance sampling: next-event estimation
So sample direct and indirect with separate MC integration
Kavita Bala, Computer Science, Cornell University
Kavita Bala, Computer Science, Cornell University
How to sample direct illumination
Sampling a single light source Sampling for many lights
L ( x ) =
Direct Illumination
Asource
f ( x, ) L( y ) G( x, y) dA
r
y
G ( x, y ) =
cos(nx , ) cos(n y , )Vis ( x, y )
2 rxy
y ny Vis(x,y)=? nx x rxy nx x
hemisphere integration
Kavita Bala, Computer Science, Cornell University
area integration
Kavita Bala, Computer Science, Cornell University
2
Estimator for direct lighting
Pick a point on the lights surface with pdf
p( y)
PDF for sampling light
Uniform
p( y) = 1 Areasource
For N samples, direct light at point x is:
f r Lsource cos x cos yi rx2yi p ( yi ) Vis ( x, yi )
Pick a point uniformly over lights area
Can stratify samples
1 E ( x) = N
i =1
N
Estimator: Areasource E ( x) = N
fL
i =1 r
N
cos x cos yi rx2yi
source
Vis ( x, yi )
Kavita Bala, Computer Science, Cornell University
Kavita Bala, Computer Science, Cornell University
Different pdfs
Uniform
p( y) = 1 Areasource
Parameters
How to distribute paths within light source?
Uniform Solid angle What about light distribution?
Solid angle sampling
Removes cosine and distance from integrand Better when significant foreshortening
How many paths (shadow-rays)?
Total? Per light source? (~intensity, importance, )
E ( x) =
1 N
i =1
N
f r Lsource
cos x cos yi rx2yi p ( yi )
Vis ( x, yi )
Kavita Bala, Computer Science, Cornell University
Kavita Bala, Computer Science, Cornell University
Formulation over all lights
When M is large, each direct lighting sample is very expensive We would like to importance sample the lights Instead of M integrals
L ( x ) =
i =1 M Asource
Why?
Do not need a minimum of M rays/sample Can use only one ray/sample Still need N samples, but 1 ray/sample Ray is distributed over the whole integration domain
Can importance sample the lights
f ( x , ) L
r
source
( y ) G ( x, y ) dAy
Formulation over 1 integration domain
L ( x ) =
Aall lights
Kavita Bala, Computer Science, Cornell University
f r ( x, ) Lsource ( y ) G ( x, y ) dAy
Kavita Bala, Computer Science, Cornell University
3
Anti-aliasing
Can piggy-back on the anti-aliasing of pixel
lights eye pixel 10 rays 10 rays
How to sample the lights?
A discrete pdf pL(ki) picks the light ki A surface point is then picked with pdf p(yi|ki)
Estimator with N samples: 1 N f L G ( x, yi ) E ( x) = r source N i =1 p L (ki ) p ( yi | ki )
Kavita Bala, Computer Science, Cornell University Kavita Bala, Computer Science, Cornell University
Strategies for picking light
Uniform p L (k ) = 1 M Area
Research on many lights
Ward 91 Shirley, Wang, Zimmerman 94 Fernandez, Bala, Greenberg 02 Wald and Slusallek 03
Ak p L (k ) = Ak p L (k ) =
Power
P
Pk
k
Dont take visibility into account
Kavita Bala, Computer Cornell Science, University Kavita Bala, Computer Science, Cornell University
Alternative direct paths
Shoot paths at random over hemisphere; check if they hit light source
paths not used efficiently noise in image might work if light source occupies large portion on hemisphere
Alternative direct paths
1 path / point
16 paths / point
256 paths / point
Kavita Bala, Computer Science, Cornell University
Kavita Bala, Computer Science, Cornell University
4
Direct paths
Different path generators produce different estimators and different error characteristics Direct illumination general algorithm:
Stochastic Ray Tracing
Sample area of light source for direct term Sample hemisphere with random rays for indirect term Optimizations:
Stratified sampling Importance sampling Combine multiple probability density functions into a single PDF
Kavita Bala, Computer Science, Cornell University
compute_radiance (point, direction) est_rad = 0; for (i=0; i<n; i++) p = generate_path; est_rad += energy_transfer(p) / probability(p); est_rad = est_rad / n; return(est_rad);
Kavita Bala, Computer Science, Cornell University
Indirect Illumination
Paths of length > 1 Many different path generators possible Efficiency depends on:
BRDFs along the path Visibility function ...
Indirect paths - surface sampling
Simple generator (path length = 2):
select point on light source select random point on surfaces per path:
2 visibility checks
Kavita Bala, Computer Science, Cornell University
Kavita Bala, Computer Science, Cornell University
Indirect paths - surface sampling
Indirect illumination (path length 2):
L( x ) =
Asource A
Indirect paths - source shooting
Shoot ray from light source, find hit location Connect hit point to receiver
L ( y ) f ( z ,
1 r
1
2 )G ( z, y ) f r ( x, 2 )G ( z , x) dAz dAy
L ( x ) =
1 N
i =1
N
L( yi 1i ) f r ( zi ,1i 2i )G ( zi , yi ) f r ( x,2i )G ( zi , x) p y ( yi ) p z ( z i )
per path:
1 ray intersection 1 visibility check
2 visibility values cause noise
which might be 0
Kavita Bala, Computer Science, Cornell University
Kavita Bala, Computer Science, Cornell University
5
Indirect paths - receiver gathering
Shoot ray from receiver point, find hit location Connect hit point to random point on light source
Indirect paths
per path:
1 ray intersection 1 visibility check
Surface sampling - 2 visibility terms; can be 0 Source shooting - 1 visibility term - 1 ray intersection Receiver gathering - 1 visibility term - 1 ray intersection
Kavita Bala, Computer Science, Cornell University
Kavita Bala, Computer Science, Cornell University
More variants ...
Shoot ray from receiver point, find hit location Shoot ray from hit point, check if on light source
per path:
2 ray intersections Le might be zero
Indirect paths
Same principles apply to paths of length > 2
generate multiple surface points generate multiple bounces from light sources and ...