Illumination and Shading - IIFrom last time…Today’s topicsOpenGL’s illumination modelDesiderataBetter reflectance modelsCook-Torrance illuminationMicrofacet distribution functionBeckman’s distributionGeometric attenuation factorBlocked reflectionBlocked beamGeometric attenuation factorFresnel reflectionFresnel reflection (dialectric)Fresnel reflectionFresnel factor (conductor)Results of Cook-TorranceMore Cook-Torrance resultsEnergy conserving approachesDefinitionsIrradianceWhat does Irradiance look like?RadianceBRDFType of BRDFsProperties of BRDFsProperties of BRDFsMeasuring BRDFsHow to use BRDF Data?BRDF approachesBRDF approachesRemaining hard problemsHW2 – Rendering pipelineHW2 – Rendering pipelineNext time2/19/071Illumination and Shading - IIComputer GraphicsCOMP 770 (236)Spring 2007Instructor: Brandon Lloyd2/19/072From last time…■ Light Sources ■ Empirical Illumination■ Shading■ Local vs Global Illumination2/19/073Today’s topics■ Cook-Torrance illumination model° Microfacets° Geometry term° Fresnel reflection■ Radiance and irradiance■ BRDFs■ Homework #22/19/074OpenGL’s illumination model==+ ⋅+ ⋅∑slightsnraadd ssi1ˆˆˆI k I k I max((N L),0) k I max((R ),0)ˆV■ Problems with empirical models: ° What are the coefficients for copper? ° What are ka, ks, and ns? Are they measurable quantities? ° How does the incoming light at a pointrelate to the outgoing light? Is energy conserved? ° What exactly is light intensity? ° Is my picture accurate?2/19/075Desiderata■ A model that uses physical properties that canbe looked up in the CRC Handbook of Chemistry and Physics (indices of refraction, reflectivity,conductivity, etc.) ■ Parameters that that have clear physicalanalogies (how rough or polished a surface is) ■ Models that are predictive (the simulation attempts to model the real scene) ■ Models that conserve energy ■ Complex surface substructures (crystals, amorphous materials, boundary-layer behavior) ■ If it was easy... everyone would do it.2/19/076Better reflectance models■ Blinn-Torrance-Sparrow (1977) ° isotropic reflectors with smooth microstructure ■ Cook-Torrance (1982) ° wavelength dependent Fresnel term ■ He-Torrance-Sillion-Greenberg (1991)° adds polarization, statistical microstructure, self-reflectance ■ Very little of this work has made its way into graphics H/W. Why?2/19/077Cook-Torrance illuminationλλλ λ λ=⎛⎞θ=+ −−ρ⋅+⎜⎟⎜⎟π⋅⎝⎠∑lightsi,r ,a a ,i a s i si1DGF ( )ˆˆIIk I(1kk)(LN)kˆˆ(V N)Definitions: • Iλ,a- Ambient light intensity (same old hack) • ka- Ambient surface reflectance (hacks beget hacks) • Iλ,i- Luminous intensity of light source i • ks- percentage of light reflected specularly (notice terms sum to one) ρλ- diffuse reflectivity • Li- vector to light source • N - average surface normal at point • V - vector to viewer• D - microfacet distribution function • G - geometric attenuation Factor • Fλ(θi) - Fresnel reflection term2/19/078Microfacet distribution function ■ Statistical model of the variation in normal direction ■ Based on a Beckman distribution function° Consistent with the surface variations of rough surfaces■ β - the angle between and■m- the root-mean-square slope of the the microfacets.° large m indicates steep slopes and the reflections spread out over the surfaceˆNˆH2/19/079Beckman’s distribution2/19/0710Geometric attenuation factor■ There are many different ways that an incoming beam of light can interact with the surface locally.2/19/0711Blocked reflection■ A portion of the out-going beam can be blocked.2/19/0712Blocked beam■ A portion of the incoming beam can be blocked.2/19/0713Geometric attenuation factor■ In each case, the geometric configurations can be analyzed to compute the percentage of light that actually escapes from the surface:° The geometric factor, chooses the smallest amount of light that is lost as the local self-shadowing model2/19/0714Fresnel reflection■ The Fresnel term results from a complete analysis of the reflection process while considering light as an electromagnetic wave. ■ When the electric field is oriented consistent it is said to be polarized■ The behavior of reflection depends on the orientation of the electric field relative to the surface normal at the point of incidence2/19/0715Fresnel reflection (dialectric)parallel perpendicularairglassglassair2/19/0716Fresnel reflection■ The Fresnel reflection is wavelength dependent° depends on index-of-refraction of the material■ Dialectrics transmit light that is not reflected■ Conductors absorb light that is not reflected° index-of-refraction is complex to account for absorption° leads to variation in colors seen in highlights■ This version of the equation ignores the polarization of the incoming and reflected rays.2/19/0717Fresnel factor (conductor)2/19/0718Results of Cook-TorranceA Copper Vase with a more metallicappearancePlastic-looking copper rendered using Phong model2/19/0719More Cook-Torrance results2/19/0720Energy conserving approaches■ Physically based models must obey the conservation of energy:absorbedinemittedoutLightLightLightLight −+=2/19/0721Definitions■Radiant flux (W) ° the rate at which light energy is emitted ■Irradiance (W/m2) ° the rate of incident or incoming energy at a surface point per unit surface area. ■Radiant intensity (W/sr) ° the rate that light energy is radiated through a given solid angle ° a steradian(sr ) a unit of solid angle (4π steradians on a sphere)■Radiance (W/(sr m2)) ° the rate of energy radiated through a given solid angle as seenreflected from a surface (i.e. the hemisphere is projected onto the surface) ° remains constant along a ray° most important for computer graphics2/19/0722IrradianceThe irradiance function is a two dimensional function describing the incoming light energy impinging on a given point.2/19/0723What does Irradiance look like?2/19/0724Radiance■ Radiance is a two dimensional function representing the light reflected from a surface. ° We’ve already used plots like these to visualize illumination equations for a single light direction ° Radiance response integrates over all incoming directions2/19/0725BRDF■ The Bi-directional Reflectance Distribution Function (BRDF) describes the transport of irradiance to radiance2/19/0726Type of BRDFs■ Isotropic° Reflectance independent of rotation about