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Fundamentals of Rendering -Radiometry / Photometry
CMPT 461/761Image SynthesisTorsten Möller
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Today• The physics of light• Radiometric quantities• Photometry vs/ Radiometry
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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 FAQhttp://www.optics.arizona.edu/Palmer/rpfaq/rpfaq.htm
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The physics of light
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Realistic Rendering• Determination of Intensity• Mechanisms
– Emittance (+)– Absorption (-)– Scattering (+) (single vs. multiple)
• Cameras or retinas record quantity of light
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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?
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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
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Electromagnetic spectrum
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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 light with atoms and molecules
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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
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Today• The physics of light• Radiometric quantities• Photometry vs/ Radiometry
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Radiometric quantities
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Radiometry• Science of measuring light• Analogous science called photometry is
based on human perception.
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Radiometric Quantities• Function of wavelength, time, position, direction,
polarization.
• Assume wavelength independence– No phosphorescence, fluorescence– Incident wavelength λ1 exit λ1
• Steady State– Light travels fast– No luminescence phenomena - Ignore it
g(⇥, t, r,⇤, �)
g(t, r,⇥, �)
g(r,⇥, �)
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Result – five dimensions• Adieu Polarization
– Would likely need wave optics to simulate• Two quantities
– Position (3 components)– Direction (2 components)
g(r,�)
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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
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Radiant Energy - Q• Think of 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
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RadiationTungstenBlackbody
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Consequence
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Fluorescent Lamps
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Power / Flux - Φ
• Flow of energy (important for transport)• Also - radiant power or flux.• Energy per unit time (joules/s = eV/s)• Unit: W - watts• Φ = dQ/dt
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€
I =dΦdω
Intensity I• Flux density per unit solid angle
• Units – watts per steradian• “intensity” is heavily overloaded. Why?
– Power of light source– Perceived brightness
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Solid Angle• Size of a patch, dA, is
• Solid angle is
• Compare with 2D - angle
€
dω =dAr2
= sinθdθdφ€
dA = (rsinθ dφ)(r dθ)
€
θ =lr
€
θ
€
r
€
l
φ - azimuth angleθ - polar angle
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Solid Angle (contd.)• Solid angle generalizes angle!• Steradian• Sphere has 4π steradians! Why?• Dodecahedron – 12 faces, each pentagon.• One steradian approx equal to solid angle
subtended by a single face of dodecahedron
© Wikipedia
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€
I =dΦdω
=Φ4π
Isotropic Point Source
• Even distribution over sphere• How do you get this?• Quiz: Warn’s spot-light? Determine flux
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I(θ) = (S ⋅ A)s = cossθ
Warns Spot Light
• Quiz: Determine flux
S
A
θ
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Other Lights
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Other kinds of lights
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€
u =dΦdA
dA
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
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€
u =dΦdA
=Φ4πr2
Radiant Flux Area Density• E.g. sphere / point light source:
• 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
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Irradiance E• Power per unit area incident on a surface
€
E =dΦdA
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Radiosity or Radiant Exitance B• Power per unit area leaving surface• Also known as Radiosity• Same unit as irradiance, just direction
changes
€
B =dΦdA
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x
xs
θ
Irradiance on Differential Patch from a Point Light Source
• Inverse square law• Key observation:
€
x − x s2dω = cosθdA
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• Use a hemisphere Η over surface to measure incoming/outgoing flux
• Replace objects and points with their hemispherical projection
r
ω
Hemispherical Projection
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€
L =d2Φ
dAp dω
€
L =d2Φ
dAcosθdω
Radiance L• Power per unit projected area per unit solid
angle.• Units – watts per (steradian m2)• We have now introduced projected area, a
cosine term.
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d cosθ
d
θ
Why the Cosine Term?• projected into the solid angle!• Foreshortening is by cosine of angle.• Radiance gives energy by effective surface
area.
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• Incident Radiance: Li(p, ω)
• Exitant Radiance: Lo(p, ω)• In general:• At a point p - no surface, no participating
media:
Incident and Exitant Radiance
p
€
Li p,ω( )p
€
Lo p,ω( )€
Li p,ω( ) ≠ Lo p,ω( )
€
Li p,ω( ) = Lo p,−ω( )
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€
E p,n( ) = Li p,ω( )cosθΩ
∫ dω
Irradiance from Radiance
• |cosθ|dω is projection of a differential area• We take |cosθ| in order to integrate over the
whole sphere
r
n
p
€
Li p,ω( )
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Radiance• Fundamental quantity, everything else
derived from it.• Law 1: Radiance is invariant along a ray • Law 2: Sensor response is proportional to
radiance
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L1dω1dA1 = L2dω2dA2Total flux leaving one side = flux arriving other side, so
Law1: Conservation Law !
€
dω1
€
dω2
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dω1 = dA2 /r2 and
€
dω2 = dA1 /r2
therefore
Conservation Law !
€
dω1
€
dω2
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so
Radiance doesn’t change with distance!
€
dω1
Conservation Law !
€
dω2
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€
dω = dA /r2
Radiance at a sensor• Sensor of a fixed area sees more of a surface
that is farther away.• However, the solid angle is inversely
proportional to distance.
• Response of a sensor is proportional to radiance.
Ω
A
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Radiance at a sensor• Response of a sensor is proportional
to radiance.
• T depends on geometry of sensor
€
Φ = LdωdAΩ
∫A∫ = LT
T = dωdAΩ
∫A∫
A
Ω
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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.
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Radiometry vs. Photometry
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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).
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0
0.200
0.400
0.600
0.800
1.000
1.200
350 450 550 650 750
Violet Green Red© CIE, 1924, many more curves available, see http://cvision.ucsd.edu/lumindex.htm
CIE curve• Response is integral over all wavelengths
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Radiometric Quantities Physicaldescription
Definition Symbol Units Radiometric quantity(also “spectral *” [*/m])
Energy Qe [J=Ws] Joule Radiant energy
Power / flux dQ/dt Φe [W=J/s] Radiant power / flux
Flux density dQ/dAdt Ee [W/m2] Irradiance
Flux density dQ/dAdt Me= Be [W/m2] Radiosity
Angular flux density
dQ/dAdωdt Lv [W/m2sr] Radiance
Intensity dQ/dωdt Iv [W/sr] Radiant intensity
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Photometric Quantities
Quantities weighted by luminous efficiency function
Physicaldescription
Definition Symbol Units Photometric quantity(also “spectral *” [*/m])
Energy Qv [talbot] Luminous energy
Power / flux dQ/dt Φv [lm (lumens)= talbot/s]
Luminous power / flux
Flux density dQ/dAdt Ev [lux= lm/m2] Illuminance
Flux density dQ/dAdt [Mv=] Bv [lux] Luminosity
Angular flux density
dQ/dAΦdωdt Lv [lm/m2sr] Luminance
Intensity dQ/dωdt Iv [cd (candela)
= lm/sr]
Radiant / luminous intensity
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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