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Progressive ExpectationMaximization for Hierarchical Volumetric Photon Mapping Wenzel Jakob 1,2 Christian Regg 1,3 Wojciech Jarosz 1 1 Disney Research, Zrich 2 Cornell University 3 ETH Zrich Saturday, August 4, 12 Motivation Volumetric

  1. Progressive Expectation–Maximization for Hierarchical Volumetric Photon Mapping Wenzel Jakob 1,2 Christian Regg 1,3 Wojciech Jarosz 1 1 Disney Research, Zürich 2 Cornell University 3 ETH Zürich Saturday, August 4, 12

  2. Motivation Volumetric photon mapping 1. Trace photons 2. Radiance estimate Issues • high-frequency illumination requires many photons • time spent on photons that contribute very little • prone to temporal fl ickering Saturday, August 4, 12

  3. Motivation Per-pixel Beam radiance estimate : 917K photons render time Saturday, August 4, 12

  4. Motivation Beam radiance estimate : 917K photons Our method: 4K Gaussians Per-pixel Per-pixel Render time: 281 s Render time: 125 s render time render time Our approach: • represent radiance using a Gaussian mixture model ( GMM ) • fi t using progressive expectation maximization ( EM ) • render with multiple levels of detail Saturday, August 4, 12

  5. Motivation Beam radiance estimate : 4M photons Our method: 16K Gaussians Render time: 727s Render time: 457 s Our approach: • represent radiance using a Gaussian mixture model ( GMM ) • fi t using progressive expectation maximization ( EM ) • render with multiple levels of detail Saturday, August 4, 12

  6. Related work • Di ff usion based photon mapping [Schjøth et al. 08] • Photon relaxation [Spencer et al. 09] • Hierarchical photon mapping [Spencer et al. 09] Saturday, August 4, 12

  7. Density estimation Given photons approximately determine their density Nonparametric: • Count the number of photons within a small region Saturday, August 4, 12

  8. Density estimation Given photons approximately determine their density Nonparametric: • Count the number of photons within a small region Parametric: • Find suitable parameters for a known distribution Saturday, August 4, 12

  9. Gaussian mixture models • Photon density modeled as a weighted sum of Gaussians: Saturday, August 4, 12

  10. Gaussian mixture models • Photon density modeled as a weighted sum of Gaussians: [Papas et al.] 256 Gaussians 1024 Gaussians 4096 Gaussians 16384 Gaussians Target density Saturday, August 4, 12

  11. Gaussian mixture models • Photon density modeled as a weighted sum of Gaussians: Unknown parameters : 2. Means 2. Means 3. Covariance matrices 3. Covariance matrices 1. Weights 1. Weights Saturday, August 4, 12

  12. Maximum likelihood estimation Approach: fi nd the “most likely” parameters, i.e. Mixture model Estimated parameters Photon locations Expectation maximization Saturday, August 4, 12

  13. Expectation maximization • Two components: E establish soft assignment between E-Step : photons and Gaussians M maximize the expected likelihood M-Step : • Finds a locally optimal solution good starting guess needed! • Slow and scales poorly — (where : photon count) Saturday, August 4, 12

  14. Expectation maximization Accelerated EM by [Verbeek et al. 06] Saturday, August 4, 12

  15. Accelerated EM Stored cell statistics : • photon count • mean position • average outer product Saturday, August 4, 12

  16. Progressive EM Stored cell statistics : • photon count • mean position • average outer product Our modi fi cations: • better cell re fi nement Saturday, August 4, 12

  17. Progressive EM Stored cell statistics : • photon count • mean position • average outer product Our modi fi cations: • better cell re fi nement • progressive photons shooting passes Saturday, August 4, 12

  18. Progressive EM Stored cell statistics : • photon count • mean position • average outer product Our modi fi cations: • better cell re fi nement • progressive photons shooting passes • reduced complexity Saturday, August 4, 12

  19. Pipeline overview Build Build Progressive EM Progressive EM yes Shoot Shoot E Hierarchy Hierarchy photons photons converged? Shoot more Re fi ne M photons partition Initial Initial Render Render guess guess no Saturday, August 4, 12

  20. Rendering ... Saturday, August 4, 12

  21. Level of detail hierarchy Agglomerative construction: • Repeatedly merge nearby Gaussians based on their Kullback-Leibler divergence 5 6 7 8 1 2 3 4 Saturday, August 4, 12

  22. Example Rendering hierarchy: Criterion 1: bounding box intersected? Criterion 2: solid angle large enough? Criterion 3: attenuation low enough? T r ��� Saturday, August 4, 12

  23. BRE : 1M Photons 23+192 = 215 s 23 Saturday, August 4, 12

  24. Our method : 4K Gaussians 35+24 = 59 s (3.6 × ) ( fi t to 1M photons) 24 Saturday, August 4, 12

  25. BRE : 18M Photons 507+609 = 1116 s 25 Saturday, August 4, 12

  26. Our method : 64K Gaussians 868+66 = 934 s (1.2 × ) ( fi t to 18M photons) 26 Saturday, August 4, 12

  27. BRE : 4M Photons 89 + 638 = 727 s 27 Saturday, August 4, 12

  28. Our method : 16K Gaussians 330 + 127 = 457 s (1.6 × ) 28 Saturday, August 4, 12

  29. Temporal Coherence Build Progressive EM yes Shoot E Hierarchy photons converged? Shoot more Re fi ne M photons cut Initial Render guess no • Feed the result of the current frame into the next one Faster fi tting, no temporal noise Saturday, August 4, 12

  30. [ Video ] Saturday, August 4, 12

  31. [ Video ] GPU-based rasterizer: • Anisotropic Gaussian splat shader: 30 lines of GLSL • Gaussian representation is very compact (4096-term GMM requires only ~240KB of storage) Saturday, August 4, 12

  32. Conclusion Contributions • Rendering technique based on parametric density estimation • Uses a progressive and optimized variant of accelerated EM • Compact & hierarchical representation of volumetric radiance • Extensions for temporal coherence and real-time visualization Saturday, August 4, 12

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