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

© Lastra/Machiraju/Möller/Weiskopf

Fundamentals of Rendering -Radiometry / Photometry

CMPT 461/761Image SynthesisTorsten Möller


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

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 FAQhttp://www.optics.arizona.edu/Palmer/rpfaq/rpfaq.htm


The physics of light

4


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

11


Radiometric quantities

12


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

€

θ

€

r

€

l

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

© Wikipedia


€

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

S

A

θ


Other Lights


Other kinds of lights


€

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


€

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


x

xs

θ

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

r

ω

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

d

θ

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

area.


• 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( )


€

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( )


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

€

dω2


so

Radiance doesn’t change with distance!

€

dω1

Conservation Law !

€

dω2


€

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


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


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

46


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).


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

© Lastra/Machiraju/Möller/Weiskopf49

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

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

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