Fundamentals of Rendering - Radiometry / Photometry
Image Synthesis Torsten Möller
© Lastra/Machiraju/Möller/Weiskopf Today
• The physics of light • Radiometric quantities • Photometry vs/ Radiometry
© Lastra/Machiraju/Möller/Weiskopf 2 Reading
• 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
© Lastra/Machiraju/Möller/Weiskopf 3 The physics of light
© Lastra/Machiraju/Möller/Weiskopf 4 Realistic Rendering
• Determination of Intensity • Mechanisms – Emittance (+) – Absorption (-) – Scattering (+) (single vs. multiple) • Cameras or retinas record quantity of light
© Lastra/Machiraju/Möller/Weiskopf 5 Pertinent Questions
• Nature of light and how it is: – Measured – Characterized / recorded • (local) reflection of light • (global) spatial distribution of light • Steady state light transport?
© Lastra/Machiraju/Möller/Weiskopf 6 What Is Light ?
• 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
© Lastra/Machiraju/Möller/Weiskopf 8 Optics
• 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
© Lastra/Machiraju/Möller/Weiskopf 10 Radiometric quantities
© Lastra/Machiraju/Möller/Weiskopf 11 Radiometry
• Science of measuring light • Analogous science called photometry is based on human perception.
© Lastra/Machiraju/Möller/Weiskopf 12 Radiometric Quantities
• 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)
© Lastra/Machiraju/Möller/Weiskopf 14 Radiometry - Quantities
• 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
© Lastra/Machiraju/Möller/Weiskopf 17 Consequence
© Lastra/Machiraju/Möller/Weiskopf 18 Fluorescent Lamps
© Lastra/Machiraju/Möller/Weiskopf 19 Power / Flux - Φ
• 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
© Lastra/Machiraju/Möller/Weiskopf 20 Intensity I
• Flux density per unit solid angle dΦ I = dω • Units – watts per steradian • “intensity” is heavily overloaded. Why? – Power€ of light source – Perceived brightness
© Lastra/Machiraju/Möller/Weiskopf 21 Solid Angle
• 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
© Lastra/Machiraju/Möller/Weiskopf 23 Isotropic Point Source
dΦ Φ I = = dω 4π • Even distribution over sphere • How do you get this? • Quiz:€ Warn’s spot-light? Determine flux
© Lastra/Machiraju/Möller/Weiskopf 24 Warns Spot Light S θ I(θ) = (S ⋅ A)s = coss θ
A
• Quiz:€ Determine flux
© Lastra/Machiraju/Möller/Weiskopf 25 Other Lights
© Lastra/Machiraju/Möller/Weiskopf 26 Other kinds of lights
© Lastra/Machiraju/Möller/Weiskopf 27 Radiant Flux Area Density
• 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
© Lastra/Machiraju/Möller/Weiskopf 29 Irradiance E
• Power per unit area incident on a surface dΦ E = dA
€
© Lastra/Machiraju/Möller/Weiskopf 30 Radiosity or Radiant Exitance B
• 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
x
θ • Inverse square law xs • Key observation:
2 x − x s dω = cosθdA
© Lastra/Machiraju/Möller/Weiskopf 32 € Hemispherical Projection
• Use a hemisphere Η over surface to measure incoming/outgoing flux • Replace objects and points with their hemispherical projection
ω
r
© Lastra/Machiraju/Möller/Weiskopf 33 Radiance L
• 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
© Lastra/Machiraju/Möller/Weiskopf 34
€ 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
© Lastra/Machiraju/Möller/Weiskopf 36 € € Irradiance from Radiance
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
© Lastra/Machiraju/Möller/Weiskopf 37 Radiance
• Fundamental quantity, everything else derived from it. • Law 1: Radiance is invariant along a ray • Law 2: Sensor response is proportional to radiance
© Lastra/Machiraju/Möller/Weiskopf 38 Law1: Conservation Law !
dω1
dω2 €
Total flux leaving one side = flux arriving other side, so
€ L1dω1dA1 = L2dω2dA2
© Lastra/Machiraju/Möller/Weiskopf 39
€ 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!
© Lastra/Machiraju/Möller/Weiskopf 41 Radiance at a sensor
• 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
© Lastra/Machiraju/Möller/Weiskopf 42 Radiance at a sensor
• 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
€ © Lastra/Machiraju/Möller/Weiskopf 43 Thus …
• 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.
© Lastra/Machiraju/Möller/Weiskopf 44 Radiometry vs. Photometry
© Lastra/Machiraju/Möller/Weiskopf 45 Photometry and Radiometry
• 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
800 nm). © Lastra/Machiraju/Möller/Weiskopf 46 CIE curve
• 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
© Lastra/Machiraju/Möller/Weiskopf 47 Radiometric Quantities
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
© Lastra/Machiraju/Möller/Weiskopf 48 Photometric Quantities
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
function © Lastra/Machiraju/Möller/Weiskopf 49 Typical Values of Illuminance • Solar constant: 1,37 kW/m2
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