Matrix Row-Column Sampling for the Many-Light Problem

1
Matrix Row-Column Sampling for the Many-Light
Problem
Miloš Hašan (Cornell University) Fabio Pellacini
(Dartmouth College) Kavita Bala (Cornell
University)
2
Complex Illumination A Challenge
3
Conversion to Many Lights

• Area, indirect, sun/sky

Courtesy Walter et al., Lightcuts, SIGGRAPH 05/06
4
A Matrix Interpretation
Lights (100,000)
Pixels (2,000,000)
5
Problem Statement

• Compute sum of columns
• Note We dont have the matrix data

Lights
S (
)
Pixels
6
Indirect Illumination ? Many Lights

S (
)
100,000 point lights
7
Environment Map ? Many Lights

S (
)
100,000 point lights
8
Sun, Sky, Indirect ? Many Lights

S (
)
100,000 point lights
9
Brute Force Takes Minutes

• Why not sum all columns?
• With 100,000 lights, still several minutes

10 min
13 min
20 min
10
Our Contribution

• Fast, accurate, GPU-based approximation
• Application Preview for lighting design

Brute force
10 min
13 min
20 min
Our result
3.8 sec
13.5 sec
16.9 sec
11
Related Work

• Many lights (CPU-based) Walter et al 05/06, Ward
  94, Paquette et al 98, Wald et al 03,
• Instant radiosity related Keller 97,
  Dachsbacher Stamminger 05/06, Laine et al 07,
• Environment maps Agarwal et al 03, Ostromoukhov
  et al 04,
• Precomputation-based Sloan et al 02/03, Ng et al
  03/04, Ben-Artzi et al 06, Hasan et al 06,
  Ritschel et al 07,
• Other global illumination Ward et al 88, Jensen
  96, Hanrahan et al 91, Christensen 97, Scheel
  01/02, Gautron et al 05, Krivanek et al 06,
  Dachsbacher et al 07,

12
Insight 1 Matrix has structure

• Compute small subset of elements
• Reconstruct

643 lights
900 pixels
A simple scene 30 x 30 image
The matrix
13
Insight 2 Sampling Pattern Matters
Lights
Pixels
Point-to-point visibility Ray-tracing
Point-to-many-points visibility Shadow-mapping
14
Row-Column Duality

• Columns Regular Shadow Mapping

Shadow map at light position
Surface samples
15
Row-Column Duality

• Rows Also Shadow Mapping!

Shadow map at sample position
16
Image as a Weighted Column Sum

• The following is possible

compute very small subset of columns
compute weighted sum

• Use rows to choose a good set of columns!

17
Exploration and Exploitation
?
how to choose columns and weights?
compute rows (explore)
compute columns (exploit)
weighted sum
choose columns and weights
18
Reduced Matrix
Reduced columns
19
Clustering Approach
Choose representative columns
Reduced columns
Choose k clusters
20
Reduced ? Full
Use the same representatives for the full matrix
Representative columns
Weighted sum
21
Visualizing the Reduced Columns
Reduced columns vectors in high-dimensional space
visualize as
22
Clustering Illustration
Columns with various intensities can be clustered
Strong but similar columns
Weak columns can be clustered more easily
23
The Clustering Metric

• Minimize
• where

total cost of all clusters
norms of the reduced columns
squared distance between normalized reduced
columns
cost of a cluster
sum over all pairs in it
24
How to minimize?

• Problem is NP-hard
• Not much previous research
• Should handle large input
• 100,000 points
• 1000 clusters
• We introduce 2 heuristics
• Random sampling
• Divide conquer

25
Clustering by Random Sampling
Very fast (use optimized BLAS) Some clusters
might be too small / large
26
Clustering by Divide Conquer
Splitting small clusters is fast Splitting large
clusters is slow
27
Combined Clustering Algorithm
28
Combined Clustering Algorithm
29
Full Algorithm
Assemble rows into reduced matrix
Cluster reduced columns
Compute rows (GPU)
Choose representatives
Weighted sum
Compute columns (GPU)
30
Results

• We show 5 scenes
• Show reference and 5x difference image
• All scenes have 100,000 lights
• Timings
• NVidia GeForce 8800 GTX
• Light / surface sample creation not included

Temple
Bunny
Kitchen
Trees
Grand Central
31
Results Kitchen
5x diff

• 388k polygons
• Mostly indirect illumination
• Glossy surfaces
• Indirect shadows

Reference 13 min (using all 100k lights)
Our result 13.5 sec (432 rows 864
columns)
32
Results Temple
5x diff

• 2.1m polygons
• Mostly indirect sky illumination
• Indirect shadows

Our result 16.9 sec (300 rows 900 columns)
Reference 20 min (using all 100k lights)
33
Results Trees

• 328k polygons
• Complex incoherent geometry

5x diff
Reference 14 min (using all 100k lights)
Our result 2.9 sec (100 rows 200
columns)
34
Results Bunny

• 869k polygons
• Incoherent geometry
• High-frequency lighting
• Kajiya-Kay hair shader

5x diff
Our result 3.8 sec (100 rows 200
columns)
Reference 10 min (using all 100k lights)
35
Results Grand Central
5x diff

• 1.5m polygons
• Point lights between stone blocks

Our result 24.2 sec (588 rows 1176
columns)
Reference 44 min (using all 100k lights)
36
The Value of Exploration
Our result (432 rows 864 columns)
No exploration (Using 1455 lights)
Equal time comparison
37
The Value of Exploration
Our result
No exploration
Equal time comparison 5x difference from
reference
38
Conclusion

• Fast, high quality approximation for many lights
• GPU-oriented
• Sample rows to explore low-rank structure
• Sample well-chosen columns
• Application Preview for lighting design
• Indirect illumination
• Environment maps
• Arbitrary lights and shaders

39
Future Work

• How many rows columns?
• Pick automatically
• Row / column alternation
• Progressive algorithm
• stop when user likes the image
• Render multiple frames at once?

40
Acknowledgments

• Veronica Sundstedt and Patrick Ledda
• Temple scene
• Bruce Walter, PCG _at_ Cornell
• NSF CAREER 0644175
• Affinito-Stewart Award

41
Thank You
42
Discarded slides
43
Indirect Illumination ? Many Lights

• shoot photons from light sources
• deposit on every bounce

• treat photons as point lights
• cosine-weighted emission

44
Low Rank Assumption

• Worst case lights with very local contribution

45
The Value of Exploration
Our result (432 rows 864 columns)
No exploration (Using 1992 lights)
Equal time comparison
46
The Value of Exploration
Our result
No exploration
Equal time comparison 5x difference image