Last TimeTodayIrradianceRadianceDirect LightingBRDFBRDF PropertiesHemispherical-Directional ReflectanceBTDFWhere Do You Get BRDFs?Common BSDF ModelsPBRT ReflectanceSome TerminologyLambertian BRDFLambertian ExampleSample Lambertian BRDFOption 1: Uniform Sampling (PBR 14.5.1)Option 2: Cosine Weighted (PBR 14.5.3)Specular Reflection/TransmissionTotal Reflection and TransmissionFresnel ReflectanceFresnel IssuesFresnel for DielectricsFresnel for ConductorsFresnel ExampleSpecular Reflection BRDFSampling for Specular ReflectionSpecular Direct LightingSpecular Transmission BRDFNext Time01/31/05 © 2005 University of WisconsinLast Time•Cameras•Assignment 101/31/05 © 2005 University of WisconsinToday•Reflectance Functions Part 101/31/05 © 2005 University of WisconsinIrradiance•Irradiance is the power arriving at a surface, per unit area on the surface–Denoted E in PBR, or sometimes Ir or sometimes I–Units: Wm-2rE(p)= /4r2AE(p)= cos /App01/31/05 © 2005 University of WisconsinRadiance•Flux density per unit area perpendicular to the direction of travel, per unit solid angle:–Units Wm-2sr-1, power per unit area per unit solid angledAddL),(),(pp01/31/05 © 2005 University of WisconsinDirect Lighting•The simplest context for reflectance is direct lighting•Compute the outgoing radiance due to reflection of all incoming radiance•Integrate the proportion of light reflected from each incoming direction toward the outgoing direction•fr: Bidirectional Reflectance Distribution Function (BRDF) nxxx2cos),(),,(),(HiorodLfL01/31/05 © 2005 University of WisconsinBRDF•The ratio of the radiance in the outgoing direction to the incident irradiance•The BRDF is like a probability distribution function–The use of irradiance in the form on the bottom means the BRDF has no use unless integrated over a set of directions•The BRDF has units: sr-1dLLELfiiiooooiorcos),(),()(),(),,(xxxxx 01/31/05 © 2005 University of WisconsinBRDF Properties•Must be symmetric:–Many aspects of illumination depend on this•Must integrate to < 1 over the incoming hemisphere–Cannot be large in too many places–Light cannot be created1cos),,()(2nxHordf),,(),,(oiriorffxx 01/31/05 © 2005 University of WisconsinHemispherical-Directional Reflectance•Directional Hemispheric Reflectance is the fraction of the incident irradiance in a given direction that is reflected by the surface–Or, the total reflection in a given outgoing direction for constant illumination in the incoming direction•Useful for computation of some reflectance functions and for some algorithmsooioridhdfcos),,(),( xx01/31/05 © 2005 University of WisconsinBTDF•Bidirectional Transmission Distribution Function, ft–For light coming from inside an object, like glass•Defined similar to BRDF•PBR combines them into the bidirectional scattering distribution function, f2cos),(),,(),(SioodLfLxxx01/31/05 © 2005 University of WisconsinWhere Do You Get BRDFs?•Measured – repository at Cornell Graphics Group•Phenomenological – qualitatively similar to reality•Simulated – reflectance arises from physics of material and surface properties; these can be simulated•Physical (wave) Optics – solve Maxwell’s equations•Geometric Optics – like simulation, but analytic01/31/05 © 2005 University of WisconsinCommon BSDF Models•Lambertian (perfectly diffuse)•Specular (perfect mirror)–Fresnel reflectance•Microfacet models–Oren-Nayar–Torrance-Sparrow–Ward–Schlick•Measured Models–Lafortune01/31/05 © 2005 University of WisconsinPBRT Reflectance•Given an incoming and outgoing ray direction, evaluate f•Given an incoming ray direction, sample an outgoing ray and return f, the sampled direction , and the probability with which the sample was chosen, p–The p value is required for importance sampling–Also need a routine that, given an incoming and outgoing, returns p•To evaluate the direct lighting integral, call Sample_f many times to get sample outgoing rays and weight according to f and p01/31/05 © 2005 University of WisconsinSome Terminology•Classifying Reflectance:–Diffuse – matte paint–Perfect Specular – mirror–Glossy Specular – metals–Retro-reflective – moon, velvet•Symmetry?–Isotropic reflectance does not depend on –Anisotropic reflectance depends on viewing angle around the normal – grooved surfaces•Require a consistent, surface parameterization to orient grooves01/31/05 © 2005 University of WisconsinLambertian BRDF•For ideal diffuse surfaces, the BRDF has no directional dependence:•Easiest to specify hh, total reflected for uniform total incoming:   xxriorff ,,      xxxn nrH Hioiorhhfddf 2 2coscos101/31/05 © 2005 University of WisconsinLambertian Example01/31/05 © 2005 University of WisconsinSample Lambertian BRDF•Say we want to evaluate the direct lighting equation with a Lambertian BRDF:•In general, the incoming radiance, Li, is non-uniform, so we will use Monte Carlo methods–Sample N directions, (i)–Evaluate•The performance depends on the distribution used to sample directions, p nxxx2cos),()(),(HirodLfL   NiiiiirpLfN1)()()(cos,1xx01/31/05 © 2005 University of WisconsinOption 1: Uniform Sampling (PBR 14.5.1)•Sample 1 and 1 uniform on [0,1]•Convert to a direction in the hemisphere with•Then–2 is because we want to sample according to solid angle  1)(212)(212)(12sin12cosiziyix   NiiiiroLfNL1)()(21cos,1,xxx01/31/05 © 2005 University of WisconsinOption 2: Cosine Weighted (PBR 14.5.3)•Sample 1 and 1 uniform on [0,1]•Convert to a uniform sample on the disk, xd, yd•Project up onto the hemisphere•Then, with lower variance     NiiirNiiiiiroLfNLfNL1)(1)()()(,coscos,1,xxxxx22)(1,,ddddiyxyx 01/31/05 © 2005 University of WisconsinSpecular Reflection/Transmission•Reflection: arriving energy from one direction goes out only in reflection direction:•Refraction: Outgoing direction based on Snell’s law: is index of