Matrix Row-Column Sampling for the Many-Light Problem

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Matrix Row-Column Sampling for the Many-Light Problem. Milo š Ha š an (Cornell University) Fabio Pellacini (Dartmouth College) Kavita Bala (Cornell University). Complex Illumination: A Challenge. Conversion to Many Lights. Area, indirect, sun/sky.

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Matrix Row-Column Sampling for the Many-Light Problem

Miloš Hašan (Cornell University)

Fabio Pellacini (Dartmouth College)

Kavita Bala (Cornell University)

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
• Compute sum of columns
• Note: We don’t have the matrix data

Lights

= Σ (

)

Pixels

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
• Application: Preview for lighting design

Brute force:

10 min

13 min

20 min

Our result:

3.8 sec

13.5 sec

16.9 sec

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
• Compute small subset of elements
• Reconstruct

643 lights

900 pixels

A simple scene

30 x 30 image

The matrix

Insight #2: Sampling Pattern Matters

Lights

Pixels

Point-to-point visibility: Ray-tracing

Row-Column Duality

Surface samples

Row-Column Duality

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 to choose columns and weights?

compute rows (explore)

choose columns and weights

compute columns (exploit)

weighted sum

Reduced Matrix

Reduced columns

Clustering Approach

Choose representative columns

Reduced columns

Choose k clusters

Reduced  Full

Use the same representatives for the full matrix

Representative columns

Weighted sum

Visualizing the Reduced Columns

Reduced columns: vectors in high-dimensional space

visualize as …

Clustering Illustration

Columns with various intensities can be clustered

Strong but similar columns

Weak columns can be clustered more easily

The Clustering Metric
• Minimize:
• where:

total cost of all clusters

squared distance between normalized reduced columns

norms of the reduced columns

cost of a cluster

sum over all pairs in it

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

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

Kitchen

Temple

Trees

Bunny

Grand Central

Results: Kitchen

5x diff

• 388k polygons
• Mostly indirect illumination
• Glossy surfaces

Reference: 13 min (using all 100k lights)

Our result: 13.5 sec (432 rows + 864 columns)

Results: Temple

5x diff

• 2.1m polygons
• Mostly indirect & sky illumination

Our result: 16.9 sec (300 rows + 900 columns)

Reference: 20 min (using all 100k lights)

Results: Trees

5x diff

• 328k polygons
• Complex incoherent geometry

Reference: 14 min (using all 100k lights)

Our result: 2.9 sec (100 rows + 200 columns)

Results: Bunny

5x diff

• 869k polygons
• Incoherent geometry
• High-frequency lighting

Our result: 3.8 sec (100 rows + 200 columns)

Reference: 10 min (using all 100k lights)

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)

The Value of Exploration

Our result

(432 rows + 864 columns)

No exploration

(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
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
Indirect Illumination  Many Lights
• shoot photons from light sources
• deposit on every bounce
• treat photons as point lights
• cosine-weighted emission
Low Rank Assumption

Worst case: lights with very local contribution

The Value of Exploration

Our result

(432 rows + 864 columns)

No exploration

(Using 1992 lights)

Equal time comparison

The Value of Exploration

Our result

No exploration

Equal time comparison: 5x difference image