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

5

Box scene(reference)

User Parameters Example

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User Parameters Example

Larger 𝛼

Larg

er

𝒓 𝒓

𝒓 𝒓Differenceimage

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Radius Shrinkage Parameters

𝑟0

𝑟0

𝑟0

…𝛼

Optimal Convergence of Progressive Photon Mapping

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Optimal Asymptotic Convergence Rate

𝑟0

𝑟0

𝑟0

…𝛼

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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

Adaptive Bandwidth Selection

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Optimal Asymptotic Convergence Rate

𝑟0

𝑟0

𝑟0

…𝛼

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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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Estimation ErrorMean Squared Error [Hachisuka et al. 2010]

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Estimation Error

Variance is two-fold: Path measurement contribution

Kernel estimation

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Estimation Error

Measurement variance is higher

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Estimation Error

So, MSE has noise (path variance) and bias

Variance Bias

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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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Progressive Photon Mapping Adaptive PPM

20 seconds!

Results

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Progressive Photon Mapping Adaptive PPM

3 seconds!

Results

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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

Thank you for your attention.