# Overview. Radiometry and Photometry. Foundations of Computer Graphics (Spring 2012)

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

2 2

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 dφ = L(x,ω) cos θ dω da Radiance properties Radiance constant as propagates along ray Derived from conservation of flux Fundamental in Light Transport. dφ = L dω da = L dω da = dφ dω = da r dω = da r da da dω da = = dω da r L = L

5 Radiance properties Sensor response proportional to radiance (constant of proportionality is throughput) Far surface: See more, but subtend 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 Irradiance E is radiant power per unit area Integrate incoming radiance over hemisphere Projected solid angle (cos θ dω) Uniform illumination: Irradiance = π [CW 24,25] Units: W/m 2 Radiant Exitance (radiosity) Power per unit area leaving surface (like irradiance) 5

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 Lr(ω r) f (ω i,ω r) = Li(ω i)cosθ idω i Lr(ω r) = Li(ω i)f (ω i,ω r)cosθ idω i 7

8 Isotropic vs Anisotropic Isotropic: Most materials (you can rotate about normal without changing reflections) Anisotropic: brushed metal etc. preferred tangential direction Isotropic Anisotropic 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 (and simple BRDF models) Reflected Light (Output Image) Reflection Equation ω i x ω r 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) Emission Incident BRDF Light (from light source) Cosine of Incident angle Reflection Equation Reflection Equation Reflected Light (Output Image) ω i x ω r Sum over all light sources 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) Emission Incident BRDF Light (from light source) Cosine of Incident angle dω i ω i x ω r L ( x, ω ) L ( x, ω ) L( x, ω ) f( x, ω, ω )cosθdω Reflected Light (Output Image) = + Replace sum with integral r r e r i i i r i i Ω Emission Incident BRDF Cosine of Light (from Incident angle light source) 8

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

10 Analytical BRDF: TS example One famous analytically derived BRDF is the Torrance-Sparrow model T-S is used to model specular surface, like Phong more accurate than Phong has more parameters that can be set to match different materials derived based on assumptions of underlying geometry. (instead of because it works well ) Torrance-Sparrow Assume the surface is made up grooves at microscopic level. Assume the faces of these grooves (called microfacets) are perfect reflectors. Take into account 3 phenomena Shadowing Masking Interreflection Fresnel term: allows for wavelength dependency How much of the macroscopic surface is visible to the light source Torrance-Sparrow Result f = F(θ i)g(ω i,ω r)d(θ h) 4cos(θ i)cos(θ r) How much of the macroscopic surface is visible to the viewer Geometric Attenuation: reduces the output based on the amount of shadowing or masking that occurs. Distribution: distribution function determines what percentage of microfacets are oriented to reflect in the viewer direction. Other BRDF models Empirical: Measure and build a 4D table Anisotropic models for hair, brushed steel Cartoon shaders, funky BRDFs Capturing spatial variation Very active area of research 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) Blinn and Newell 1976, Miller and Hoffman, 1984 Later, Greene 86, Cabral et al. 87 dω i Reflection Equation ω i x ω r L ( x, ω ) L ( x, ω ) L( x, ω ) f( x, ω, ω )cosθdω Reflected Light (Output Image) = + Replace sum with integral r r e r i i i r i i Ω Emission Environment BRDF Cosine of Map (continuous) Incident angle 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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