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Adaptive Progressive Photon Mapping

Adaptive Progressive Photon Mapping. Adaptive PPM. Original PPM. Anton S. Kaplanyan Karlsruhe Institute of Technology, Germany. Progressive Photon Mapping in Essence. Pixel estimate using eye and light subpaths Generate full path by joining subpaths. Photon radiance. Eye subpath

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Adaptive Progressive Photon Mapping

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  1. Adaptive Progressive Photon Mapping • Adaptive PPM • Original PPM Anton S. Kaplanyan Karlsruhe Institute of Technology, Germany

  2. Progressive Photon Mapping in Essence Pixel estimate using eye and light subpaths Generate full path by joining subpaths Photon radiance Eye subpath importance Kernel-regularized connection of subpaths

  3. Reformulation of Photon Mapping PPM = recursive (online) estimator [Yamato71] Rearrange the sum to see that Kernel estimation Path contribution

  4. Radius Shrinkage Shrink radius (bandwidth) for th photon map User-defined parameters and Problem: Optimal value of and areunknown Usually globally constant / k-NN defined

  5. User Parameters Example Box scene (reference)

  6. User Parameters Example Larger Difference image Larger 𝛼

  7. Radius Shrinkage Parameters …

  8. Optimal Convergence of Progressive Photon Mapping

  9. Optimal Asymptotic Convergence Rate …

  10. Optimal Convergence Rate • Variance and bias depend on [KZ11] Optimal rate is with • Asymptotic convergence Unbiased Monte Carlo is faster:

  11. Convergence Rate of Kernel Estimation Convergence rate for dimensions Suffers from curse of dimensionality Adding a dimension reduces the rate! Shutter time kernel estimation – not recommended Wavelength kernel estimation – not recommended Volumetric photon mapping

  12. Adaptive Bandwidth Selection

  13. Optimal Asymptotic Convergence Rate …

  14. Adaptive Bandwidth Selection might not yield minimal Minimize with respect to • Achieve variance ↔ bias tradeoff • Select optimal using past samples

  15. Estimation Error • Mean Squared Error [Hachisuka et al. 2010]

  16. Estimation Error • Variance is two-fold: • Path measurement contribution • Kernel estimation

  17. Estimation Error • Measurement variance is higher

  18. Estimation Error So, MSE has noise(path variance) and bias Variance Bias

  19. Adaptive Bandwidth Selection • Both variance and bias depend on • Where is a pixel Laplacian Laplacian is unknown

  20. Estimating Pixel Laplacian consists of Laplacians at all shading points • Weighted per-vertex Laplacians

  21. Estimating Per-Vertex Laplacian Estimate per-vertex Laplacian at a point Recursive finite differences [Ngen11] • Yet another recursive estimator • Another shrinking bandwidth • Robust estimation on discontinuities

  22. Adaptive Bandwidth Selection Estimate all unknowns Path variance Pixel Laplacian Minimize MSE as MSE(r) Lower initial error • Keeps noise-bias balance • Data-driven bandwidth selector

  23. Results Progressive Photon Mapping Adaptive PPM 20 seconds!

  24. Results Progressive Photon Mapping Adaptive PPM 3 seconds!

  25. Conclusion Optimal asymptotic convergence rate • Asymptotically slower than unbiased methods • Not always optimal in finite time Adaptive bandwidth selection • Based on previous samples • Balances variance-bias • Speeds up convergence • Attractive for interactive preview

  26. Thank you for your attention.

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