Motivation: BRDFs, Sampling and Reconstruction of Visual Appearance § Basics of Illumination, § Formal radiometric analysis (not ad-hoc) CSE 274 [Fall 2018], Lecture 2 § Reflection Equation Ravi Ramamoorthi § Monte Carlo Rendering next week http://www.cs.ucsd.edu/~ravir

§ Appreciate formal analysis in a graduate course, even if not absolutely essential in practice

§ Physical measurement of electromagnetic energy

§ Measure spatial (and angular) properties of § , § Reflection functions: Bi-Directional Distribution Function or BRDF § Reflection Equation § Simple BRDF models

1 Radiance Radiance properties ? Radiance constant as propagates along ray ? per unit projected area perpendicular – Derived from conservation of to the ray per unit in the direction of the ray – Fundamental in Light Transport. dΦ == L dωω dA L d dA = dΦ 1 1112 2 2 2

2 ωω==22 ? Symbol: L(x,ω) (W/m sr) d12 dA r d 21 dA r dA dA dωω dA==12 d dA ? Flux given by 11 r 2 2 2 dΦ = L(x,ω) cos θ dω dA ∴LL12=

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

§ Physical measurement of electromagnetic energy

§ Measure spatial (and angular) properties of light § Radiance, Irradiance § Reflection functions: Bi-Directional Reflectance Distribution Function or BRDF § Reflection Equation § Simple BRDF models

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

4 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 Lrr()ω f (,ωωir )= Ldii(ω )cos θω i i

Lrr(ω )= Lf ii ( ω ) ( ωω ir , )cos θ i d ω i

Isotropic vs Anisotropic

§ Isotropic: Most materials (you can rotate about normal without changing reflections) § Anisotropic: brushed metal etc. preferred tangential direction

Isotropic Anisotropic

Reflection Equation

ω i ωr x

L (x,ω ) = L (x,ω ) + L (x,ω )f (x,ω ,ω )(ω i n) r r e r i i i r i Reflected Light Emission Incident BRDF Cosine of (Output Image) Light (from Incident angle light source)

5 Reflection Equation Reflection Equation

ω ω ω ω i r dωi i r x x Replace sum with integral Sum over all light sources Lx(,ω )=+ L (, xω ) Lx (,ωω )(,fx ωθ , )cos dω L (x,ω ) = L (x,ω ) + L (x,ω )f (x,ω ,ω )(ω i n) r r e r∫ i i ir i i r r e r ∑ i i i r i Ω Reflected Light Emission Incident BRDF Cosine of Reflected Light Emission Incident BRDF Cosine of (Output Image) Light (from Incident angle (Output Image) Light (from Incident angle light source) light source)

§ Physical measurement of electromagnetic energy

§ Measure spatial (and angular) properties of light § Radiance, Irradiance § Reflection functions: Bi-Directional Reflectance Diffuse Torrance-Sparrow Distribution Function or BRDF Anisotropic § Reflection Equation § Simple BRDF models

bv written by Szymon Rusinkiewicz

6 Specular Term (Phong)

Torrance-Sparrow

Fresnel term: Geometric Attenuation: allows for reduces the output based on the dependency amount of shadowing or masking that occurs.

Distribution: F(θ i)G(ω i,ω r)D(θ h) distribution function f = determines what 4cos(θ i)cos(θ r) percentage of How much of the microfacets are macroscopic surface oriented to reflect in How much of the the viewer direction. is visible to the light macroscopic source surface is visible to the viewer

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

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