Fundamentals of Rendering - Radiometry / Photometry

Image Synthesis Torsten Möller

• The physics of light • Radiometric quantities • Photometry vs/ Radiometry

• Chapter 5 of “Physically Based Rendering” by Pharr&Humphreys • Chapter 2 (by Hanrahan) “Radiosity and Realistic Image Synthesis,” in Cohen and Wallace. • pp. 648 – 659 and Chapter 13 in “Principles of Digital Image Synthesis”, by Glassner. • Radiometry FAQ https://employeepages.scad.edu/~kwitte/ documents/Photometry_FAQ.PDF

• Determination of Intensity • Mechanisms – Emittance (+) – Absorption (-) – Scattering (+) (single vs. multiple) • Cameras or retinas record quantity of light

• Nature of light and how it is: – Measured – Characterized / recorded • (local) reﬂection of light • (global) spatial distribution of light • Steady state light transport?

• Light - particle model (Newton) – Light travels in straight lines – Light can travel through a vacuum (waves need a medium to travel) • Light – wave model (Huygens): electromagnetic radiation: sinusoidal wave formed coupled electric (E) and magnetic (H) ﬁelds – Dispersion / Polarization – Transmitted/reﬂected components

– Pink Floyd © Lastra/Machiraju/Möller/Weiskopf 7 Electromagnetic spectrum

• Geometrical or ray – Traditional graphics – Reﬂection, refraction – Optical system design • Physical or wave – Diffraction, interference – Interaction of objects of size comparable to wavelength • Quantum or photon optics

– Interaction of© Lastra/Machiraju/Möller/Weiskopf light with atoms and molecules 9 Nature of Light

• Wave-particle duality – Wave packets which emulate particle transport. Explain quantum phenomena. • Incoherent as opposed to laser. Waves are multiple frequencies, and random in phase. • Polarization – Ignore. Un-polarized light has many waves summed all with random orientation

• Science of measuring light • Analogous science called photometry is based on human perception.

• Function of wavelength, time, position, direction, polarization. g(⇥, t, r, ⇤, )

• Assume wavelength independence g(t, r, ⇥, ) – No phosphorescence, ﬂuorescence

– Incident wavelength λ1 exit λ1

• Steady State g(r, ⇥, ) – Light travels fast – No luminescence phenomena - Ignore it © Lastra/Machiraju/Möller/Weiskopf 13 Result – ﬁve dimensions

• Adieu Polarization g(r, ) – Would likely need wave optics to simulate • Two quantities – Position (3 components) – Direction (2 components)

• Energy Q • Power Φ - J/s or W – Energy per time (radiant power or ﬂux) • Irradiance E and Radiosity B - W/m2 – Power per area • Intensity I - W/sr – Power per solid angle • Radiance L - W/sr/m2 – Power per projected area and solid angle © Lastra/Machiraju/Möller/Weiskopf 15 Radiant Energy - Q

• Think of a photon as carrying quantum of energy hc/λ = hf (photoelectric effect) – c is speed of light – h is Planck’s constant – f is frequency of radiation • Wave packets • Total energy, Q, is then energy of the total number of photons • Units - joules or eV © Lastra/Machiraju/Möller/Weiskopf 16 Radiation Blackbody Tungsten

• Flow of energy (important for transport) • Also known as radiant power or ﬂux • Energy per unit time (joules/s = eV/s) • Unit: W — watts • Φ = dQ/dt

• Flux density per unit solid angle dΦ I = dω • Units – watts per steradian • “intensity” is heavily overloaded. Why? – Power€ of light source – Perceived brightness

• Size of a patch, dA, is dA = (rsinθ dφ)(r dθ) • Solid angle is dA dω = = sinθdθdφ r2 € • Compare with 2D angle l θ = € r l θ r φ - azimuth angle

€ © Lastra/Machiraju/Möller/Weiskopf θ - polar angle 22 € € € Solid Angle (contd.)

• Solid angle generalizes angle! • Steradian • Sphere has 4π steradians! Why? © Wikipedia • Dodecahedron – 12 faces, each pentagon. • One steradian approx equal to solid angle subtended by a single face of dodecahedron

dΦ Φ I = = dω 4π • Even distribution over sphere • How do you get this? • Quiz:€ Warn’s spot-light? Determine ﬂux

• Quiz:€ Determine ﬂux

• Area density of ﬂux (W/m2) • u = Energy arriving/leaving a surface per unit time interval per unit area • dA can be any 2D surface in space dΦ u = dA dA

€ © Lastra/Machiraju/Möller/Weiskopf 28 Radiant Flux Area Density

• E.g. sphere / point light source: dΦ Φ u = = dA 4πr2 • Flux falls off with square of the distance • Bouguer’s classic experiment: – Compare a light source and a candle € – Intensity is proportional to ratio of distances squared

• Power per unit area incident on a surface dΦ E = dA

€

• Power per unit area leaving surface • Also known as Radiosity • Same unit as irradiance, just direction changes dΦ B = dA

€ © Lastra/Machiraju/Möller/Weiskopf 31 Irradiance on Differential Patch from a Point Light Source

θ • Inverse square law xs • Key observation:

2 x − x s dω = cosθdA

• Use a hemisphere Η over surface to measure incoming/outgoing ﬂux • Replace objects and points with their hemispherical projection

• Power per unit projected area per unit solid angle. d 2Φ L = dAp dω

• Units – watts per (steradian m2) • We have€ now introduced projected area, a cosine term. d 2Φ L = dAcosθdω

€ Why the Cosine Term?

• projected into the solid angle! • Foreshortening is by cosine of angle. • Radiance gives energy by effective surface area.

d cosθ θ

d © Lastra/Machiraju/Möller/Weiskopf 35 Incident and Exitant Radiance

• Incident Radiance: Li(p, ω)

• Exitant Radiance: Lo(p, ω)

• In general: Li (p,ω) ≠ Lo(p,ω) • At a point p - no surface, no participating media: Li (p,ω) = Lo(p,−ω) €

Lo p,ω Li (p,ω) ( ) € p p

E(p,n) = ∫ Li (p,ω)cosθ dω Ω • |cosθ|dω is projection of a differential area L p,ω € • We take |cosθ| in order i ( ) n to integrate over the

whole sphere € r p

• Fundamental quantity, everything else derived from it. • Law 1: Radiance is invariant along a ray • Law 2: Sensor response is proportional to radiance

dω1

dω2 €

Total ﬂux leaving one side = ﬂux arriving other side, so

€ L1dω1dA1 = L2dω2dA2

€ Conservation Law !

dω1

dω2 €

d dA /r2 and d dA /r2 € ω1 = 2 ω2 = 1 therefore

€ € © Lastra/Machiraju/Möller/Weiskopf 40 Conservation Law !

dω1

dω2 €

so €

Radiance doesn’t change with distance!

• Sensor of a ﬁxed area sees more of a surface that is farther away. Ω • However, the solid angle is inversely proportional to distance. dω = dA /r2 • Response of a sensor is proportional to radiance. € A

• Response of a sensor is proportional to radiance. Ω Φ = ∫ ∫ LdωdA = LT A Ω T = ∫ ∫ dωdA A Ω

• T depends on geometry of sensor A

• Radiance doesn’t change with distance – Therefore it’s the quantity we want to measure in a ray tracer. • Radiance proportional to what a sensor (camera, eye) measures. – Therefore it’s what we want to output.

• Photometry (begun 1700s by Bouguer) deals with how humans perceive light. • Bouguer compared every light with respect to candles and formulated the inverse square law ! • All measurements relative to perception of illumination • Units different from radiometric units but conversion is scale factor -- weighted by spectral response of eye (over about 360 to

• Response is integral over all wavelengths

1.200

1.000

0.800

0.600

0.400

0.200

0.000 350 450 550 650 750 Violet Green Red © CIE, 1924, many more curves available, see http://cvision.ucsd.edu/lumindex.htm

Physical Deﬁnition Symbol Units Radiometric quantity description (also “spectral *” [*/m])

Energy Qe [J=Ws] Joule Radiant energy

Power / ﬂux dQ/dt Φe [W=J/s] Radiant power / ﬂux

2 Flux density dQ/dAdt Ee [W/m ] Irradiance

2 Flux density dQ/dAdt Me= Be [W/m ] Radiosity

2 Angular ﬂux dQ/dAdωdt Le [W/m sr] Radiance density

Intensity dQ/dωdt Ie [W/sr] Radiant intensity

Physical Deﬁnition Symbol Units Photometric quantity description (also “spectral *” [*/m])

Energy Qv [talbot] Luminous energy

Power / ﬂux dQ/dt Φv [lm (lumens) Luminous power / ﬂux = talbot/s]

Flux density dQ/dAdt Ev [lux= lm/m2] Illuminance

Flux density dQ/dAdt [Mv=] Bv [lux] Luminosity

Angular ﬂux dQ/dAdωdt Lv [lm/m2sr] Luminance density

Intensity dQ/dωdt Iv [cd (candela) Radiant / luminous intensity = lm/sr] • Quantities weighted by luminous efﬁciency

Light source Illuminance[lux] Direct sun light 25,000 – 110,000 Day light 2,000 – 27,000 Sun set 1 – 108 Moon light 0.01 – 0.1 Stars 0.0001 – 0.001 TV studio 5,000 – 10,000 Lighting in shops/stores 1,000 – 5,500 Ofﬁce illumination 200 – 550 Home illumination 50 – 220 Street lighting 0.1 – 20 © Lastra/Machiraju/Möller/Weiskopf 50