Matrix Row-Column Sampling for the Many-Light Problem

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

Courtesy Walter et al., Lightcuts, SIGGRAPH 05/06

• Area, indirect, sun/sky

A Matrix Interpretation

Pixels(2,000,000)

Lights (100,000)

• Compute sum of columns

• Note: We don’t have the matrix data

Problem Statement

= Σ ( )P

ixel

s

Lights

Indirect Illumination Many Lights

=

100,000 point lights

Σ ( )

Environment Map Many Lights

=

100,000 point lights

Σ ( )

Sun, Sky, Indirect Many Lights

=

100,000 point lights

Σ ( )

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

10 min 13 min 20 min

3.8 sec 13.5 sec 16.9 sec

Brute force:

Our result:

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

A simple scene

30 x 30 image The matrix

643 lights

900

pixe

ls

Insight #2: Sampling Pattern Matters

Lights

Pix

els

Point-to-point visibility: Ray-tracingPoint-to-many-points visibility: Shadow-mapping

Row-Column Duality

Shadow map at light position

Surface samples

• Columns: Regular Shadow Mapping

Row-Column Duality

Shadow map at sample position

• Rows: Also Shadow Mapping!

Image as a Weighted Column Sum

compute very small subset of columns

compute weighted sum

• The following is possible:

• Use rows to choose a good set of columns!

Exploration and Exploitation

compute rows (explore)

compute columns (exploit)

weighted sum

?

choose columns and weights

how to choose columns and

weights?

Reduced Matrix

Reduced columns

Clustering Approach

Choose k clustersChoose

representative columns

Reduced columns

Reduced Full

Use the same representatives for

the full matrix

Weighted sum

Representative columns

Visualizing the Reduced Columns

Reduced columns: vectors in high-

dimensional space

visualize as …

radius = norm

Clustering Illustration

Strong but similar columns

Weak columns can be clustered more easily

Columns with various intensities can be clustered

• Minimize:

• where:

The Clustering Metric

total cost of all clusters

cost of a cluster

sum over all pairs in it

norms of the reduced columns

squared distance between normalized

reduced 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

Compute rows (GPU)

Weighted sum

Assemble rows into reduced matrix

Cluster reduced columns

Choose representatives

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

Temple BunnyKitchen Trees Grand Central

Results: Kitchen• 388k polygons• Mostly indirect illumination• Glossy surfaces• Indirect shadows

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

Reference: 13 min (using all 100k lights)

5x diff

Results: Temple

• 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)

5x diff

Results: Trees• 328k polygons• Complex incoherent geometry

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

Reference: 14 min (using all 100k lights)

5x diff

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

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

Reference: 10 min (using all 100k lights)

5x diff

Results: Grand Central

• 1.5m polygons• Point lights between

stone blocks

Our result: 24.2 sec (588 rows + 1176 columns)

Reference: 44 min (using all 100k lights)

5x diff

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– 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

• 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