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Modeling BRDF by a Probability Distribution

GraphiCon’2010. 20th International Conference on Computer Graphics and Vision September 20-24, 2010, St.Petersburg , Russia. Modeling BRDF by a Probability Distribution. Aydın ÖZTÜRK Murat KURT Ahmet BİLGİLİ. Bidirectional Reflectance Distribution Function (BRDF).

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Modeling BRDF by a Probability Distribution

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  1. GraphiCon’2010 20th International Conference on Computer Graphics and Vision September 20-24, 2010, St.Petersburg, Russia Modeling BRDF by a ProbabilityDistribution Aydın ÖZTÜRK Murat KURT Ahmet BİLGİLİ

  2. BidirectionalReflectanceDistributionFunction (BRDF) • BRDF functionsdefinesthesurfacereflectionbehaviourthrough a mathematical model. • BRDF is firstformulatedbyNicadamususingthefollowingrelationship: Incoming, outgoingandsurface normal vectors, Differentialoutgoingradiance, Differentialirradiance, Incomingradiance, Differentialsolidangle.

  3. PhysicalProperties of BRDF • A good BRDF shouldobeythefollowingprinciples: Reciprocity EnergyConservation

  4. Modeling BRDF byProbabilityDistributions • Ward (1992) BRDF model, • GaussianDistribution. • Edwards et al. (2006) BRDF model, • HalfwayVector Disk distribution. • Öztürk et al. (2010) Copula-Based BRDF model, • ArchimedeanCopuladistributions.

  5. CopulaDistributions • Copula is a multivariate cumulative distribution function of theuniform random variables on the interval [0,1]. • They provide asimple and general structure for modeling multivariate distributions through univariate marginal distributions.

  6. CopulaDistributions (2) RandomVariables CumulativeDistribution MarginalCumulativeDistributions CopulaDistribution

  7. CopulaDistributions (3) ProbabilityDensityFunction (pdf) MarginalDensityfunction Copulapdf

  8. ArchimedeanCopulaDistributions GeneratorFunction Inverse of GeneratorFunction Properties of GeneratorFunctions ArchimedeanCopulaDistribution

  9. ArchimedeanCopulaDistributions (2) Frank GeneratorFunction CopulaProbabilityDistributionFunction

  10. Copula-Based BRDF Model Scalingcoefficient: MarginalCumulativeDistributions:

  11. Copula-Based BRDF Model (2) Scaled BRDF values Measured BRDF values Sum of measured BRDF values MarginalDensityfunction MarginalCumulativeDistribution

  12. Copula-Based BRDF Model (3) Empiricalcumulativemarginaldistributions of measured BRDF

  13. Copula-Based BRDF Model (3) • We have observed that the marginal distributions of for specularmaterials are extremely skewed. Our empirical results showedthat copula distributions do not provide satisfactory approximationsfor these cases. • We overcome this difficulty by dividing dataalonginto subsamples and fitting the BRDF model to each of thesesubsamples. • We defined6 subsamples by dividing into 6 non-overlapping intervals each with a length of 90º/6 = 15º.

  14. PhysicalProperties of Copula-Based BRDF model • Reciprocity • Copula-Based BRDF Model satisfiesreciprocityifweusetheidentity

  15. PhysicalProperties of Copula-Based BRDF model (2) • EnergyConservation • Our BRDF model depends on a multivariate probability distribution function but it is scaled by adifferentcoefficientKforeachsubsumple. In this sense,our model may not be considered as an energy conserving model. • However, forallsamplesconsidered in thisstudy, our BRDF model empiricallysatisfiestheenergyconservationproperty. Albedofor 3 isotropicmaterials. Nickel, Yellow-Matte-Plastic, Blue-Metallic-Paint.

  16. ImportanceSampling • Importancesampling is a variancereductiontechnique in Monte Carlorendering RenderingEquation Monte CarloEstimator of RenderingEquation

  17. ImportanceSampling • If BRDF model is a probabilitydistribution, it can be simplifiedto: • Importantforreal time renderingif is provided.

  18. ImportanceSampling • If BRDF can be modeledby a 4 dimensionalArchimedeanCopuladistribution, incominglightvectors can be sampledfrom

  19. Results Various spheres were rendered with our Frank copula model using different materials. Columns left to right: alum-bronze, black-oxidized-steel, dark-specular-fabric, green-metallic-paint, pvc and silver-metallic-paint. Rows top to bottom: Reference images were renderedusing measured data; images were rendered using our Frank copula model and color-coded difference images (Color-coded differences arescaled by a factor of five to improve the visibility of differences between the real and approximated images).

  20. Results The PSNR values of the Ashikhmin-Shirley, the Cook-Torrance, the Ward and our Frank copula models. The BRDFs are sorted in the PSNRs of the Ashikhmin-Shirley model (Blue) for visualization purpose. Image is takenfromfollowingpaper: Öztürk, A., Kurt, M., Bilgili, A., A Copula-Based BRDF Model, ComputerGraphics Forum, 29(6), 1795-1806, 2010.

  21. GraphiCon’2010 20th International Conference on Computer Graphics and Vision September 20-24, 2010, St.Petersburg, Russia Thankyou ! Questions ?

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