Anisotropic Density Estimation in Global Illumination Lars Schjth - - PowerPoint PPT Presentation

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Anisotropic Density Estimation in Global Illumination Lars Schjth - - PowerPoint PPT Presentation

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Introduction Three different density estimators Comparison and evaluation Anisotropic Density Estimation in Global Illumination Lars Schjth University of Copenhagen Department of Computer Science 29. Maj 2009 Introduction Three different

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Introduction Three different density estimators Comparison and evaluation

Anisotropic Density Estimation in Global Illumination

Lars Schjøth

University of Copenhagen Department of Computer Science

  • 29. Maj 2009
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Introduction Three different density estimators Comparison and evaluation

Caustic by reflection

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Introduction Three different density estimators Comparison and evaluation

Photon distribution

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Introduction Three different density estimators Comparison and evaluation

Density Estimation

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Introduction Three different density estimators Comparison and evaluation

Density Estimation

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Introduction Three different density estimators Comparison and evaluation

Density Estimation

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Introduction Three different density estimators Comparison and evaluation

Kernel Density Estimation

−0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7 8 Kernel Denisty Estimator, h = 0.008

  • f(x) = 1

nh

n

  • i=1

K x − xi h

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Introduction Three different density estimators Comparison and evaluation

Kernel Density Estimation

−0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7 8 Kernel Denisty Estimator, h = 0.008 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 0.5 1 1.5 2 2.5 3 3.5 4 Kernel Denisty Estimator, h = 0.04

  • f(x) = 1

nh

n

  • i=1

K x − xi h

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Introduction Three different density estimators Comparison and evaluation

KNN Kernel Estimator

−0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 2 4 6 8 10 12 KNN Kernel Denisty Estimator, k = 25

  • f(x) = 1

n

n

  • i=1

1 h(x)K x − xi h(x)

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Introduction Three different density estimators Comparison and evaluation

Variable Kernel Estimator

−0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 2 4 6 8 10 12 Variable Kernel Denisty Estimator, k = 25

  • f(x) = 1

n

n

  • i=1

1 h(xi)K x − xi h(xi)

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Introduction Three different density estimators Comparison and evaluation

Mean Integrated Square Error

MISE( f) = E f(x) − f(x) 2dx +

  • var

f(x)dx

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Introduction Three different density estimators Comparison and evaluation

Mean Integrated Square Error

MISE( f) = E f(x) − f(x) 2dx +

  • var

f(x)dx AMISE( f) = 1 4h4

  • f ′′(x)2dx
  • y2K(y)dy

2 + (nh)−1

  • K(y)2dy
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Introduction Three different density estimators Comparison and evaluation

Trade-off problem in CG

Effect of bandwidth Low High

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Introduction Three different density estimators Comparison and evaluation

Three radiance estimates

Jensen’s radiance estimate Lr(x, ω) ≈ Lr(x, ω) = 1 πhk(x)2

k

  • i=1

fr(x, ωi, ω)Φi

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Introduction Three different density estimators Comparison and evaluation

Three radiance estimates

Jensen’s radiance estimate Lr(x, ω) ≈ Lr(x, ω) = 1 πhk(x)2

k

  • i=1

fr(x, ωi, ω)Φi Diffusion based radiance estimate

  • Lr(x,

ω) = 1 πh2√ det D

n

  • i=1

K (x − xi)TD−1(x − xi) h2

  • fr(x,

ωi, ω)Φi

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Introduction Three different density estimators Comparison and evaluation

Three radiance estimates

Jensen’s radiance estimate Lr(x, ω) ≈ Lr(x, ω) = 1 πhk(x)2

k

  • i=1

fr(x, ωi, ω)Φi Diffusion based radiance estimate

  • Lr(x,

ω) = 1 πh2√ det D

n

  • i=1

K (x − xi)TD−1(x − xi) h2

  • fr(x,

ωi, ω)Φi Photon differentials

  • Lr(x, ω) =

n

  • pd=1

fr(x, ωpd, ω)K

  • (x − xpd)TMT

pdMpd(x − xpd)

  • ∆Epd(x, ωpd)
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Introduction Three different density estimators Comparison and evaluation

Difference in methods

Three different adaptive kernel estimators

(a) Regular photon mapping (b) Diffusion based PM (c) Photon differentials

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Introduction Three different density estimators Comparison and evaluation

Case studies

(d) Refraction (e) Reflection

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Introduction Three different density estimators Comparison and evaluation

Case studies

Reflection case study - reference image

(f) Rendering (g) Distribution

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Introduction Three different density estimators Comparison and evaluation

Optimal bandwidth

Finding the optimal bandwidth using two different image quality measures; the Mean Square Error and Structural SIMilarity index.

50 100 150 200 250 300 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Bandwidth [k] MSE 100 200 300 400 500 600 700 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Bandwidth [k] SSIM index

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Introduction Three different density estimators Comparison and evaluation

Optimal bandwidth

Renderings for Regular photon mapping, Diffusion based photon mapping and Photon differentials at optimal bandwidth.

(h) k = 40, MSE = 0.0405 (i) k = 335, SSIM = 0.8650 (j) h = 0.006, q = 0.18, MSE = 0.0224 (k) h = 0.012, q = 0.29, SSIM = 0.8912 (l) s = 0.23, MSE = 0.0232, SSIM = 0.9013

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Introduction Three different density estimators Comparison and evaluation

Render example

Figure: Underwater view of sea floor. Right image rendered using regular photon mapping and (b) using photon differentials. Both images were rendered using a photon map containing 20000 photons.