Fundamentals of Rendering - Radiometry / Photometry and Radiometry • Photometry (begun 1700s by...

Fundamentals of Rendering -Radiometry / Photometry

CMPT 461/761Image SynthesisTorsten Möller

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

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

The physics of light

Realistic Rendering• Determination of Intensity• Mechanisms

– Emittance (+)– Absorption (-)– Scattering (+) (single vs. multiple)

• Cameras or retinas record quantity of light

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?

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

Electromagnetic spectrum

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

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

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

Radiometric quantities

Radiometry• Science of measuring light• Analogous science called photometry is

based on human perception.

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,⇥, �)

Result – five dimensions• Adieu Polarization

– Would likely need wave optics to simulate• Two quantities

– Position (3 components)– Direction (2 components)

g(r,�)

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

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

RadiationTungstenBlackbody

Consequence

Fluorescent Lamps

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

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

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

φ - azimuth angleθ - polar angle

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

I =dΦdω

=Φ4π

Isotropic Point Source

• Even distribution over sphere• How do you get this?• Quiz: Warn’s spot-light? Determine flux

I(θ) = (S ⋅ A)s = cossθ

Warns Spot Light

• Quiz: Determine flux

Other Lights

Other kinds of lights

u =dΦdA

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

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

Irradiance E• Power per unit area incident on a surface

E =dΦdA

Radiosity or Radiant Exitance B• Power per unit area leaving surface• Also known as Radiosity• Same unit as irradiance, just direction

changes

B =dΦdA

Irradiance on Differential Patch from a Point Light Source

• Inverse square law• Key observation:

x − x s2dω = cosθdA

• Use a hemisphere Η over surface to measure incoming/outgoing flux

• Replace objects and points with their hemispherical projection

Hemispherical Projection

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.

d cosθ

Why the Cosine Term?• projected into the solid angle!• Foreshortening is by cosine of angle.• Radiance gives energy by effective surface

• 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

Li p,ω( )p

Lo p,ω( )€

Li p,ω( ) ≠ Lo p,ω( )

Li p,ω( ) = Lo p,−ω( )

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

Li p,ω( )

Radiance• Fundamental quantity, everything else

derived from it.• Law 1: Radiance is invariant along a ray • Law 2: Sensor response is proportional to

radiance

L1dω1dA1 = L2dω2dA2Total flux leaving one side = flux arriving other side, so

Law1: Conservation Law !

dω1 = dA2 /r2 and

dω2 = dA1 /r2

therefore

Conservation Law !

Radiance doesn’t change with distance!

Conservation Law !

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.

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∫

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.

Radiometry vs. Photometry

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).

350 450 550 650 750

CIE curve• Response is integral over all wavelengths

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

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

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

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