Matrix Row-Column Sampling
for the Many-Light Problem
Miloš Hašan (Cornell University)
Fabio Pellacini (Dartmouth College)
Kavita Bala (Cornell University)
Complex Illumination: A Challenge
Conversion to Many Lights
• Area, indirect, sun/sky
Courtesy Walter et al., Lightcuts, SIGGRAPH 05/06
A Matrix Interpretation
Lights (100,000)
Pixels
(2,000,000)
Problem Statement
=Σ(
Lights
Pixels
• Compute sum of columns
• Note: We don’t have the matrix data
)
Indirect Illumination  Many Lights
=
Σ (
)
100,000 point lights
Environment Map  Many Lights
=
Σ (
)
100,000 point lights
Sun, Sky, Indirect  Many Lights
=
Σ (
)
100,000 point lights
Brute Force Takes Minutes
• Why not sum all columns?
– With 100,000 lights, still several minutes
10 min
13 min
20 min
Our Contribution
• Fast, accurate, GPU-based approximation
Brute
force:
10 min
13 min
20 min
3.8 sec
13.5 sec
16.9 sec
Our
result:
• Application: Preview for lighting design
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, …
Insight #1: Matrix has structure
900 pixels
• Compute small subset of elements
• Reconstruct
643 lights
A simple scene
30 x 30 image
The matrix
Insight #2: Sampling Pattern Matters
Pixels
Lights
Point-to-many-points
Point-to-point visibility:
visibility:Ray-tracing
Shadow-mapping
Row-Column Duality
• Columns: Regular Shadow Mapping
Surface
samples
Shadow map
at light position
Row-Column Duality
• Rows: Also Shadow Mapping!
Shadow map
at sample position
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!
Exploration and Exploitation
?
how tocolumns
choose compute columns
compute rows choose
columns
and
(explore)
and weights
(exploit)
weights?
weighted
sum
Reduced Matrix
Reduced
columns
Clustering Approach
Reduced
columns
Choose k clusters
Choose
representative
columns
Reduced  Full
Representative
columns
Use the same
representatives for
the full matrix
Weighted
sum
Visualizing the Reduced Columns
Reduced columns:
vectors in highdimensional space
visualize as …
radius = norm
Clustering Illustration
Columns with
various intensities
can be clustered
Strong but
similar columns
Weak columns
can be clustered
more easily
The Clustering Metric
• Minimize:
total cost of all clusters
• where:
cost of a
cluster
sum over all
pairs in it
norms of the squared distance
reduced between normalized
reduced columns
columns
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
Clustering by Random Sampling
Very fast (use optimized BLAS)
Some clusters might be too small / large
Clustering by Divide & Conquer
Splitting small clusters is fast
Splitting large clusters is slow
Combined Clustering Algorithm
Combined Clustering Algorithm
Full Algorithm
Assemble rows into
reduced matrix
Cluster reduced
columns
Compute rows
(GPU)
Choose
representatives
Weighted sum
Compute columns
(GPU)
Results
• We show 5 scenes:
Kitchen
Temple
Trees
Bunny
Grand Central
• Show reference and 5x difference image
• All scenes have 100,000+ lights
• Timings
– NVidia GeForce 8800 GTX
– Light / surface sample creation not included
Results: Kitchen
•
•
•
•
5x diff
388k polygons
Mostly indirect illumination
Glossy surfaces
Indirect shadows
Our result: 13.5 sec
Reference: 13 min
(432 rows + 864 columns)
(using all 100k lights)
Results: Temple
• 2.1m polygons
• Mostly indirect & sky illumination
• Indirect shadows
5x diff
Our result: 16.9 sec
Reference: 20 min
(300 rows + 900 columns)
(using all 100k lights)
Results: Trees
• 328k polygons
• Complex incoherent geometry
5x diff
Our result: 2.9 sec
Reference: 14 min
(100 rows + 200 columns)
(using all 100k lights)
Results: Bunny
•
•
•
•
869k polygons
Incoherent geometry
High-frequency lighting
Kajiya-Kay hair shader
5x diff
Our result: 3.8 sec
Reference: 10 min
(100 rows + 200 columns)
(using all 100k lights)
Results: Grand Central
5x diff
• 1.5m polygons
• Point lights between
stone blocks
Our result: 24.2 sec
Reference: 44 min
(588 rows + 1176 columns)
(using all 100k lights)
The Value of Exploration
Our result
No exploration
(432 rows + 864 columns)
(Using 1455 lights)
Equal time comparison
The Value of Exploration
Our result
No exploration
Equal time comparison:
5x difference from reference
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
Future Work
• How many rows + columns?
– Pick automatically
• Row / column alternation
• Progressive algorithm:
– stop when user likes the image
• Render multiple frames at once?
Acknowledgments
• Veronica Sundstedt and Patrick Ledda
– Temple scene
• Bruce Walter, PCG @ Cornell
• NSF CAREER 0644175
• Affinito-Stewart Award
Thank You
Discarded slides
Indirect Illumination  Many Lights
• shoot photons from light sources
• treat photons as point lights
• deposit on every bounce
• cosine-weighted emission
Low Rank Assumption
• Worst case: lights with very local contribution
…
The Value of Exploration
Our result
No exploration
(432 rows + 864 columns)
(Using 1992 lights)
Equal time comparison
The Value of Exploration
Our result
No exploration
Equal time comparison:
5x difference image