Radiance Computer Graphics (Fall 2008) Computer Graphics (Fall - PDF document

Radiance Computer Graphics (Fall 2008) Computer Graphics (Fall 2008) • Power per unit projected area perpendicular to the ray per unit solid angle in the direction of the ray COMS 4160, Lecture 19: Illumination and Shading 2 http://www.cs.columbia.edu/~cs4160 • Symbol: L(x, ω ) (W/m 2 sr) • Flux given by d Φ = L(x, ω ) cos θ d ω dA Radiance properties • Radiance is constant as it propagates along ray – Derived from conservation of flux – Fundamental in Light Transport. Φ = ω = ω = Φ d L d dA L d dA d 1 1 1 2 2 2 1 2 ω = 2 ω = 2 d dA r d dA r 1 2 2 1 dA dA ω = = ω d dA 1 2 d dA 1 1 2 2 r 2 ∴ = L L 1 2 Radiance properties • Sensor response proportional to radiance (constant of proportionality is throughput) – Far away surface: See more, but subtends smaller angle – Wall equally bright across viewing distances Consequences – Radiance associated with rays in a ray tracer – Other radiometric quants derived from radiance

Irradiance, Radiosity Building up the BRDF • Irradiance E is radiant power per unit area • Bi-Directional Reflectance Distribution Function • Integrate incoming radiance over hemisphere [Nicodemus 77] – Projected solid angle (cos θ d ω ) – Uniform illumination: • Function based on incident, view direction Irradiance = π [CW 24,25] – Units: W/m 2 • Relates incoming light energy to outgoing light energy • Radiosity – Power per unit area leaving • We have already seen special cases: Lambertian, Phong surface (like irradiance) • In this lecture, we study all this abstractly BRDF • Reflected Radiance proportional to Irradiance • Constant proportionality: BRDF [CW pp 28,29] – Ratio of outgoing light (radiance) to incoming light (irradiance) – Bidirectional Reflection Distribution Function – (4 Vars) units 1/sr ω L ( ) r r ω ω = f ( , ) i r ω θ ω L ( )cos d i i i i ω = ω ω ω θ ω L ( ) L ( ) ( f , )cos d r r i i i r i i

Isotropic vs vs Anisotropic Anisotropic Radiometry Isotropic Radiometry � Isotropic: Most materials (you can rotate about � Physical measurement of electromagnetic energy normal without changing reflections) � We consider light field � Anisotropic: brushed metal etc. preferred tangential � Radiance, Irradiance direction � Reflection functions: Bi-Directional Reflectance Distribution Function or BRDF � Reflection Equation � Simple BRDF models Anisotropic Isotropic Reflection Equation Reflection Equation ω ω ω ω i i r r Sum over all light sources = ∑ ω ω ω ω ω ω = ω ω ω ω i L ( ) L ( ) ( f , )( i n ) L ( ) L ( ) ( f , )( n ) r r i i i r i r r i i i r i i Reflected Radiance Incident BRDF Cosine of Reflected Radiance Incident BRDF Cosine of radiance (from Incident angle radiance (from Incident angle (Output Image) (Output Image) light source) light source) Reflection Equation Radiometry Radiometry � Physical measurement of electromagnetic energy � We consider light field � Radiance, Irradiance ω d ω � Reflection functions: Bi-Directional Reflectance ω i r i Distribution Function or BRDF � Reflection Equation � Simple BRDF models = ∫ Replace sum with integral ω ω ω ω ω ω L ( ) L ( ) ( f , )( i n d ) r r i i i r i i Ω Incident BRDF Cosine of Reflected Radiance (Output Image) radiance (from Incident angle light source)

Brdf Viewer plots Viewer plots Brdf Demo Demo Diffuse Torrance-Sparrow Anisotropic bv written by Szymon Rusinkiewicz

Analytical BRDF: TS example Torrance- -Sparrow Sparrow Analytical BRDF: TS example Torrance � One famous analytically derived BRDF is the � Assume the surface is made up grooves at the microscopic level. Torrance-Sparrow model. � T-S is used to model specular surface, like the Phong model. � more accurate than Phong � Assume the faces of these grooves (called microfacets) are � has more parameters that can be set to match different perfect reflectors. materials � Take into account 3 phenomena � derived based on assumptions of underlying geometry. (instead of ‘because it works well’) Shadowing Masking Interreflection Torrance- Torrance -Sparrow Result Sparrow Result Other BRDF models Other BRDF models Geometric Attenuation: Fresnel term: reduces the output based on the allows for wavelength amount of shadowing or masking dependency � Empirical: Measure and build a 4D table that occurs. � Anisotropic models for hair, brushed steel Distribution: θ ω ω θ F ( ) ( G , ) D ( ) distribution function i i r h = f � Cartoon shaders, funky BRDFs determines what θ θ 4cos( )cos( ) percentage of i r � Capturing spatial variation microfacets are How much of the oriented to reflect in macroscopic surface How much of the the viewer direction. � Very active area of research is visible to the light macroscopic source surface is visible to the viewer Complex Lighting Complex Lighting Environment Maps Environment Maps � Instead of determining the lighting direction by knowing � So far we’ve looked at simple, discrete light sources. what lights exist, determine what light exists by knowing the lighting direction. � Real environments contribute many colors of light from many directions. � The complex lighting of a scene can be captured in an Environment map. � Just paint the environment on a sphere. Blinn and Newell 1976, Miller and Hoffman, 1984 Later, Greene 86, Cabral et al. 87

Conclusion Demo Demo Conclusion � All this (OpenGL, physically based) are local illumination and shading models � Good lighting, BRDFs produce convincing results � Matrix movies, modern realistic computer graphics � Do not consider global effects like shadows, interreflections (from one surface on another) � Subject of next unit (global illumination) What’ What ’s Next s Next � Have finished basic material for the class � Texture mapping lecture later today � Review of illumination and Shading � Remaining topics are global illumination (written assignment 2): Lectures on rendering eq, radiosity � Historical movie: Story of Computer Graphics � Likely to finish these by Dec 1: No class Dec 8, � Work instead on HW 4, written assignments � Dec 10? will be demo session for HW 4