adaptive progressive photon mapping
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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