Radiometry and Photometry

Measuring spatial properties of light ￭ Radiant power ￭ Radiant intensity ￭ Irradiance ￭ Inverse square law and cosine law ￭ Radiance From London and Upton ￭ Radiant exitance (radiosity)

Goal is to perform lighting calculations in the physically correct way

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Radiant Energy and Power

2 4 6 8 1 10 12 14 16 28 20 22 24 6 1 1 10 10 10 10 10 10 10 10 10 10 10 10 0

P o w e r Heat Radio Infra- Ultra- X-Rays Gamma Cosmic Power: Watts (radiometry) Red Violet R a y s R a y s 1 16 14 12 80 6 4 2 - -2 -4 -6 8 1 10 10 10 10 10 10 10 10 1 10 10 10 0 vs. Lumens (photometry) Wavelength (NM) Φ IR R G B UV 7 0 0 6 0 0 5 0 0 4 0 0 ￭ Spectral effcacy ￭ Energy effciency

Energy: Joules vs. Talbot ￭ Exposure Film response ￭ Photometric luminance ￭ Skin - sunburn Y = ∫V(λ)L(λ)dλ CS348b Lecture 4 Pat Hanrahan, Spring 2016 Radiant Intensity Radiant Intensity

Defnition: The radiant (luminous) intensity is the power per unit solid angle emanating from a point source.

dΦ I(ω) ≡ dω

!W " !lm " = cd = candela %# sr &$ %# sr &$

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Angles and Solid Angles

l Angle θ = r ⇒ circle has 2 π radians

A Solid angle Ω = R2 ⇒ sphere has 4 π steradians

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Differential Solid Angles

r sinθ

dφ r dA = (r dθ )(r sinθ dφ) dθ θ = r 2 sinθ dθ dφ

φ

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Differential Solid Angles

r sinθ

dφ r dA = (r dθ )(r sinθ dφ) dθ θ = r 2 sinθ dθ dφ

φ dA dω = = sinθ dθ dφ r 2

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Differential Solid Angles

dω = sinθ dθ dφ r sinθ

dφ r Ω = ∫ dω dθ S2 θ π 2π = ∫ ∫ sinθ dθ dφ φ 0 0 1 2π = ∫ ∫ d cosθ dφ −1 0 = 4π

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Isotropic Point Source

Sphere S2 Φ = ∫ I dω S2 = 4π I

Φ I = 4π

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Warn’s Spotlight ! ω ˆ θ A s ! s I(ω) = cos θ = (ω • Aˆ )

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Warn’s Spotlight ! ω ˆ θ A s ! s I(ω) = cos θ = (ω • Aˆ )

2π 1 Φ = ∫ ∫ I(ω)d cosθ dϕ 0 0

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Warn’s Spotlight ! ω ˆ θ A s ! s I(ω) = cos θ = (ω • Aˆ )

2π 1 1 2π Φ = ∫ ∫ I(ω)d cosθ dϕ = 2π ∫ coss θ d cosθ = 0 0 0 s +1

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Warn’s Spotlight ! ω ˆ θ A s ! s I(ω) = cos θ = (ω • Aˆ )

2π 1 1 2π Φ = ∫ ∫ I(ω)d cosθ dϕ = 2π ∫ coss θ d cosθ = 0 0 0 s +1 s +1 I(ω) = Φ coss θ 2π

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Light Source Goniometric Diagrams

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Irradiance Irradiance

Defnition: The irradiance (illuminance) is the power per unit area incident on a surface. dΦ E(x) ≡ i dA

! W " ! lm " 2 2 = lux %#m &$ %#m &$

Sometimes referred to as the radiant (luminous) incidence.

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Typical Values of Illuminance [lm/m2]

Sunlight plus skylight 100,000 lux Sunlight plus skylight (overcast) 10,000 Interior near window (daylight) 1,000 Artifcial light (minimum) 100 Moonlight (full) 0.02 Starlight 0.0003

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Beam Power in Terms of Irradiance

Φ E = A

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Beam Power Falling on the Surface

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Projected Area

A

θ

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Lambert’s Cosine Law

A

θ

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Irradiance: Isotropic Point Source

Φ I = 4π θ h r

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Irradiance: Isotropic Point Source

Φ I = dω 4π θ h r dA

dΦ = I dω

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Irradiance: Isotropic Point Source

Φ I = dω 4π θ h r dA

cosθ dω = dA r 2

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Irradiance: Isotropic Point Source

Φ I = dω 4π θ h r dA Φ cosθ I dω = dA 4π r 2

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Irradiance: Isotropic Point Source

Φ I = dω 4π θ h r dA Φ cosθ I dω = dA = E dA 4π r 2 Φ cosθ E = 2 CS348b Lecture 4 4π r Pat Hanrahan, Spring 2016 The Invention of Photometry

Bouguer’s classic experiment ■ Compare a light source and a candle ■ Move until they both appear equally bright ■ Intensity is proportional to ratio of distances squared

Defnition of a candela ■ Originally a “standard” candle ■ Currently 550 nm laser with 1/683 W/sr ■ 1 of 6 fundamental SI units

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Radiance Area Lights – Surface Radiance

Defnition: The surface radiance (luminance) is the intensity per unit area leaving a surface L(x,ω) dI(x,ω) L(x,ω) ≡ dA d 2 ω d Φ(x,ω) = dωdA ! W " ! cd lm " # 2 $ # 2 = 2 = nit$ dA %sr m & %m sr m &

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Typical Values of Luminance [cd/m2]

Surface of the sun 2,000,000,000 nit Sunlight clouds 30,000 Clear sky 3,000 Overcast sky 300 Moon 0.03

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Directional Power Leaving a Surface

2 d Φo (x,ω) = Lo (x,ω)cosθdAdω

Lo (x,ω) dω

θ

dA

Same dA for all directions

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Radiant Exitance

(Radiosity) Radiant Exitance

Defnition: The radiant (luminous) exitance is the energy per unit area leaving a surface. dΦ M(x) ≡ o dA W lm ! " ! " 2 2 = lux %#m &$ %#m &$

In computer graphics, this quantity is usually referred to as the radiosity (B)

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Area Light Source

2 d Φo (x,ω) = Lo (x,ω)cosθdAdω

Lo (x,ω) dω

θ

dA

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Area Light Source

d 2Φ (x,ω) dM (x,ω) = o = L (x,ω)cosθ dω dA o

Lo (x,ω) dω

θ

dA

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Area Light Source

dM (x,ω) = Lo (x,ω)cosθ dω

Lo (x,ω) dω

θ

dA

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Area Light Source

M = dM (x,ω) = L (x,ω)cosθ dω ∫ ∫ o H 2 H 2

Lo (x,ω) dω

θ

dA H 2 Hemisphere

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Uniform Diffuse Emitter

M = L cosθ dω ∫ o 2 H Lo (x,ω) = Lo = L cosθ dω o ∫ dω H2

Uniform means Lo is not θ a function of direction dA

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Projected Solid Angle Ω! ≡ ∫ cosθ dω Ω dω θ

cosθ dω

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Projected Solid Angle Ω! ≡ ∫ cosθ dω Ω dω θ

cosθ dω Ω! = ∫ cosθ dω = π 2 CS348b Lecture 4 H Pat Hanrahan, Spring 2016 Uniform Diffuse Emitter

M = L cosθ dω ∫ o H2 Lo = L cosθ dω M o ∫ H2 dω

= πLo θ

M L = o π dA

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Radiometry and Photometry

Summary Radiometric and Photometric Terms

Physics Radiometry Photometry

Energy Radiant Energy Luminous Energy

Flux (Power) Radiant Power Luminous Power

Flux Density Irradiance Illuminance

Radiosity Luminosity

Angular Flux Density Radiance Luminance

Intensity Radiant Intensity Luminous Intensity

CS348b Lecture 4 Pat Hanrahan, Spring 2016 Photometric Units

Photometry Units MKS CGS British

Luminous Energy Talbot

Luminous Power Lumen

Illuminance Lux Phot Footcandle Luminosity

Luminance Nit Stilb Apostilb, Blondel Lambert Footlambert

Luminous Intensity Candela (Candle, Candlepower, Carcel, Hefner) “Thus one nit is one lux per steradian is one candela per square meter is one lumen per square meter per steradian. Got it?”, James Kajiya

CS348b Lecture 4 Pat Hanrahan, Spring 2016