Adaptive PPM Original PPM Adaptive Progressive Photon Mapping Anton S. Kaplanyan Karlsruhe Institute of Technology, Germany Progressive Photon Mapping in Essence Pixel estimate using eye and light subpaths πΌβ ππ π(π₯π β π¦π )πΎπ π,π Eye subpath importance Photon radiance Generate full path by joining subpaths Kernel-regularized connection of subpaths ππ πΎπ πΎπ+1 2 Reformulation of Photon Mapping PPM = recursive (online) estimator [Yamato71] πβ1 πΌπ = πΌπβ1 + ππ π(π₯ β π₯π ) πΎπ π π Rearrange the sum to see that πβ1 πΌπ = πΌπβ1 + π π₯ β π₯π [ππ πΎπ ] π Kernel Path π estimation contribution 3 Radius Shrinkage Shrink radius (bandwidth) for πth photon map ππ2 = π02 π πΌβ1 , πΌ β 0; 1 User-defined parameters π0 and πΌ Problem: Optimal value π0 of and πΌ are unknown Usually globally constant / k-NN defined 4 User Parameters Example Box scene (reference) 5 Larger ππ User Parameters Example Difference image π π π π Larger πΌ 6 Radius Shrinkage Parameters πΌ π0 β¦ π0 π0 7 Optimal Convergence of Progressive Photon Mapping Optimal Asymptotic Convergence Rate πΌ π0 β¦ π0 π0 10 Optimal Convergence Rate Variance and bias depend on πΌ [KZ11] VarMeas πΌ ~π βπΌ 2 BiasKernel (πΌ)~π πΌβ1 πΌ π¨π©π Optimal rate is MSE β π β 2/3 with πΌopt = 2/3 Asymptotic convergence Unbiased Monte Carlo is faster: MSE β π β1 11 Convergence Rate of Kernel Estimation Convergence rate for π dimensions MSE β π β 4/(π+4) Suffers from curse of dimensionality Adding a dimension reduces the rate! Shutter time kernel estimation β not recommended Wavelength kernel estimation β not recommended Volumetric photon mapping MSE β π β 4/π 12 Adaptive Bandwidth Selection Optimal Asymptotic Convergence Rate πΌ π0 β¦ π0 π0 14 Adaptive Bandwidth Selection πΌopt might not yield minimal MSE Minimize MSE with respect to π0 Achieve variance β bias tradeoff Select optimal π using past samples 15 Estimation Error Mean Squared Error [Hachisuka et al. 2010] 2 MSE = VarEst + BiasKernel 16 Estimation Error MSE = 2 VarMeas + VarKernel + BiasKernel Variance is two-fold: Path measurement contribution Kernel estimation 17 Estimation Error MSE β VarMeas + 2 BiasKernel Measurement variance is higher VarMeas β« VarKernel 18 Estimation Error So, MSE has noise (path variance) and bias 2 MSE β VarMeas + BiasKernel Variance Bias 19 Adaptive Bandwidth Selection Both variance and bias depend on π VarMeas π ~π β2 BiasKernel r ~βπΌ π 2 Where βπΌ = β(ππ πΎπ ) is a pixel Laplacian Laplacian βπΌ is unknown 20 Estimating Pixel Laplacian βπΌ consists of Laplacians at all shading points Weighted per-vertex Laplacians βπΎπ = βπΏ 21 Estimating Per-Vertex Laplacian Estimate per-vertex Laplacian at a point Recursive finite differences [Ngen11] Yet another recursive estimator Another shrinking bandwidth β Robust estimation on discontinuities πΏπ₯+π’β + πΏπ₯βπ’β β 2πΏπ₯ βπΏπ’ = β2 π₯ β π’β π₯ π₯ + βπ’ 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.
© Copyright 2026 Paperzz