Adaptive Progressive Photon Mapping
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Transcript of Adaptive Progressive Photon Mapping
Anton S. Kaplanyan
Karlsruhe Institute of Technology, Germany
Adaptive Progressive Photon Mapping
Adaptive PPM Original PPM
2
Progressive Photon Mapping in Essence
Pixel estimate using eye and light subpaths
Generate full path by joining subpathsEye subpathimportance
Photonradiance
𝛾𝑖+1
Kernel-regularized connection of subpaths
𝑊 𝑁 𝛾𝑖
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Reformulation of Photon Mapping
PPM = recursive (online) estimator [Yamato71]
Rearrange the sum to see that
Kernelestimation
Pathcontribution
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Radius Shrinkage
Shrink radius (bandwidth) for th photon map
User-defined parameters and
Problem:
Optimal value of and are unknown
Usually globally constant / k-NN defined
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𝛼 𝐨𝐩𝐭
Optimal Convergence Rate
Variance and bias depend on [KZ11]
Optimal rate is with Asymptotic convergence
Unbiased Monte Carlo is faster:
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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
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Adaptive Bandwidth Selection
might not yield minimal
Minimize with respect to Achieve variance ↔ bias tradeoff
Select optimal using past samples
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Adaptive Bandwidth Selection
Both variance and bias depend on
Where is a pixel Laplacian
Laplacian is unknown
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Estimating Pixel Laplacian
consists of Laplacians at all shading pointsWeighted per-vertex Laplacians
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𝑥 𝑥+h𝑢𝑥− h𝑢
∆ 𝐿𝑢=𝐿𝑥+ h𝑢 +𝐿𝑥− h𝑢 −2𝐿𝑥
h2
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
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Adaptive Bandwidth Selection
Estimate all unknownsPath variance
Pixel Laplacian
Minimize MSE as MSE(r)
Lower initial error Keeps noise-bias balance
Data-driven bandwidth selector
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Conclusion
Optimal asymptotic convergence rateAsymptotically slower than unbiased methods
Not always optimal in finite time
Adaptive bandwidth selectionBased on previous samples
Balances variance-bias
Speeds up convergence
Attractive for interactive preview