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Overview Foundations of Computer Graphics Lighting and shading key in computer graphics (Spring 2012) HW 2 etc. ad-hoc shading models, no units CS 184, Lecture 21: Radiometry Really, want to match physical light reflection

  1. Overview Foundations of Computer Graphics  Lighting and shading key in computer graphics (Spring 2012)  HW 2 etc. ad-hoc shading models, no units CS 184, Lecture 21: Radiometry  Really, want to match physical light reflection http://inst.eecs.berkeley.edu/~cs184  Next 3 lectures look at this formally  Today: physical measurement of light: radiometry  Formal reflection equation, reflectance models  Global Illumination (later) Many slides courtesy Pat Hanrahan Radiometry and Photometry  Physical measurement of electromagnetic energy  Measure spatial (and angular) properties of light  Radiant Power  Radiant Intensity  Irradiance  Inverse square and cosine law  Radiance  Radiant Exitance (Radiosity)  Reflection functions: Bi-Directional Reflectance Distribution Function or BRDF  Reflection Equation 1

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  3. Radiometry and Photometry  Physical measurement of electromagnetic energy  Measure spatial (and angular) properties of light  Radiant Power  Radiant Intensity  Irradiance  Inverse square and cosine law  Radiance  Radiant Exitance (Radiosity)  Reflection functions: Bi-Directional Reflectance Distribution Function or BRDF  Reflection Equation 3

  4. Radiometry and Photometry  Physical measurement of electromagnetic energy  Measure spatial (and angular) properties of light  Radiant Power  Radiant Intensity  Irradiance  Inverse square and cosine law  Radiance  Radiant Exitance (Radiosity)  Reflection functions: Bi-Directional Reflectance Distribution Function or BRDF  Reflection Equation Radiance � Power per unit projected area perpendicular to the ray per unit solid angle in the direction of the ray � Symbol: L(x, ω ) (W/m 2 sr) � Flux given by d Φ = L(x, ω ) cos θ d ω dA Radiance properties � Radiance constant as 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 = ω = d dA r 2 d dA r 2 1 2 2 1 dAdA ω = = ω d dA 1 2 d dA 1 1 2 2 2 r ∴ = L L 1 2 4

  5. Radiance properties Irradiance, Radiosity � Irradiance E is radiant power per unit area � Sensor response proportional to radiance (constant of proportionality is throughput) � Integrate incoming radiance over hemisphere – Far surface: See more, but subtend smaller angle – Projected solid angle (cos θ d ω ) – Wall equally bright across viewing distances – Uniform illumination: Consequences Irradiance = π [CW 24,25] – Radiance associated with rays in a ray tracer – Units: W/m 2 – Other radiometric quants derived from radiance � Radiant Exitance (radiosity) – Power per unit area leaving surface (like irradiance) 5

  6. Irradiance Environment Maps R N Incident Radiance Irradiance Environment Map (Illumination Environment Map) Radiometry and Photometry  Physical measurement of electromagnetic energy  Measure spatial (and angular) properties of light  Radiant Power  Radiant Intensity  Irradiance  Inverse square and cosine law  Radiance  Radiant Exitance (Radiosity)  Reflection functions: Bi-Directional Reflectance Distribution Function or BRDF  Reflection Equation 6

  7. Building up the BRDF � Bi-Directional Reflectance Distribution Function [Nicodemus 77] � Function based on incident, view direction � Relates incoming light energy to outgoing � Unifying framework for many materials BRDF � Reflected Radiance proportional Irradiance � Constant proportionality: BRDF � Ratio of outgoing light (radiance) to incoming light (irradiance) – Bidirectional Reflection Distribution Function – (4 Vars) units 1/sr L r ( ω r ) f ( ω i , ω r ) = L i ( ω i )cos θ i d ω i L r ( ω r ) = L i ( ω i ) f ( ω i , ω r )cos θ i d ω i 7

  8. Isotropic vs Anisotropic  Isotropic: Most materials (you can rotate about normal without changing reflections)  Anisotropic: brushed metal etc. preferred tangential direction Anisotropic Isotropic Reflection Equation Radiometry and Photometry  Physical measurement of electromagnetic energy  Measure spatial (and angular) properties of light  Radiant Power  Radiant Intensity ω i ω  Irradiance r  Inverse square and cosine law x  Radiance  Radiant Exitance (Radiosity)  Reflection functions: Bi-Directional Reflectance L r ( x , ω r ) = L e ( x , ω r ) + L i ( x , ω i ) f ( x , ω i , ω r )( ω i i n ) Distribution Function or BRDF Cosine of Reflected Light Emission Incident BRDF (Output Image) Incident angle Light (from  Reflection Equation (and simple BRDF models) light source) Reflection Equation Reflection Equation ω i ω i d ω ω ω r r i x x Replace sum with integral Sum over all light sources + ∫ ω = ω ω ω ω θ ω L x ( , ) L x ( , ) L x ( , ) ( , f x , ) cos d L r ( x , ω r ) = L e ( x , ω r ) + ∑ L i ( x , ω i ) f ( x , ω i , ω r )( ω i i n ) r r e r i i i r i i Ω Reflected Light Emission Incident BRDF Cosine of Reflected Light Emission Incident BRDF Cosine of Incident angle Incident angle (Output Image) Light (from (Output Image) Light (from light source) light source) 8

  9. BRDF Viewer plots Diffuse Torrance-Sparrow Anisotropic bv written by Szymon Rusinkiewicz 9

  10. Analytical BRDF: TS example Torrance-Sparrow  One famous analytically derived BRDF is the  Assume the surface is made up grooves at microscopic level. Torrance-Sparrow model  T-S is used to model specular surface, like Phong  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-Sparrow Result Other BRDF models Geometric Attenuation: Fresnel term:  Empirical: Measure and build a 4D table reduces the output based on the allows for amount of shadowing or masking wavelength  Anisotropic models for hair, brushed steel that occurs. dependency  Cartoon shaders, funky BRDFs Distribution: f = F ( θ i ) G ( ω i , ω r ) D ( θ h ) distribution  Capturing spatial variation function 4cos( θ i )cos( θ r ) determines what  Very active area of research percentage of How much of the microfacets are macroscopic How much of oriented to reflect surface is visible the macroscopic in the viewer to the light source surface is visible direction. to the viewer Reflection Equation Environment Maps  Light as a function of direction, from entire environment  Captured by photographing a chrome steel or mirror sphere  Accurate only for one point, but distant lighting same at other scene locations (typically use only one env. map) ω i d ω ω r i x Replace sum with integral + ∫ ω = ω ω ω ω θ ω L x ( , ) L x ( , ) L x ( , ) ( , f x , ) cos d r r e r i i i r i i Ω Reflected Light Emission Environment BRDF Cosine of Incident angle (Output Image) Map (continuous) Blinn and Newell 1976, Miller and Hoffman, 1984 Later, Greene 86, Cabral et al. 87 10

  11. Environment Maps Demo  Environment maps widely used as lighting representation  Many modern methods deal with offline and real-time rendering with environment maps  Image-based complex lighting + complex BRDFs 11

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