[PPT] - Advanced 3D computer graphics for movies and games (NPGR010) PowerPoint Presentation

SLIDE 1

Advanced 3D computer graphics for movies and games (NPGR010) – Radiometry

Jiří Vorba, MFF UK/Weta Digital jirka@cgg.mff.cuni.cz

Slides of prof. Jaroslav Křivánek, minor edits by Jiří Vorba

SLIDE 2

Summary of basic radiometric quantities

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

Image: Wojciech Jarosz

SLIDE 3

Direction, solid angle, spherical integrals

SLIDE 4

Direction in 3D

◼ Direction = unit vector in 3D

❑ Cartesian coordinates ❑ Spherical coordinates ❑ q … polar angle – angle from the Z axis ❑ f ... azimuth – angle measured counter-clockwise from the X

axis

], , , [ z y x = 

1

2 2 2

= + + z y x

] 2 , [ ] , [ ] , [    q  q    = x y z arctan arccos = =  q q  q  q cos sin sin cos sin = = = z y x

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 5

Function on a unit sphere

◼ Function as any other, except that its argument is a

direction in 3D

◼ Notation

❑ F() ❑ F(x,y,z) ❑ F(q,f) ❑ … ❑ Depends in the chosen representation of directions in 3D

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 6

Solid angle

◼ Planar angle

❑ Arc length on a unit circle ❑ A full circle has 2 radians (unit circle has the length of 2)

◼ Solid angle (steradian, sr)

❑ Surface area on an unit sphere ❑ Full sphere has 4 steradians

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 7

Differential solid angle

◼ “Infinitesimally small” solid angle around a given

direction

◼ By convention, represented as a 3D vector

❑ Magnitude … d ◼

Size of a differential area on the unit sphere

❑ Direction …  ◼

Center of the projection of the differential area

n the unit sphere

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 8

Differential solid angle

◼ (Differential) solid angle subtended by a differential area 2

cos d d r A q  =

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 9

Differential solid angle

r

f q

f q q f q q  d d sin ) d (sin ) d ( d = =

df dq

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 10

Radiometry and photometry

SLIDE 11

Radiometry and photometry

◼ “Radiometry is a set of techniques for measuring

electromagnetic radiation, including visible light.

◼ Radiometric techniques in optics characterize the

distribution of the radiation's power in space, as opposed to photometric techniques, which characterize the light's interaction with the human eye.” (Wikipedia)

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 12

Radiometry and photometry

◼

Radiometric quantities

◼

Radiant energy (zářivá energie) – Joule

◼

Radiant flux (zářivý tok) – Watt

◼

Radiant intensity (zářivost) – Watt/sr

◼

Denoted by subscript e „energy“

◼

Photometric quantities

◼

Luminous energy (světelná energie) – Lumen- second, a.k.a. Talbot

◼

Luminous flux (světelný tok) – Lumen

◼

Luminous intensity (svítivost) – candela

◼

Denoted by subscript v „visual“

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 13

Relation between photo- and radiometric quantities

◼ Spectral luminous efficiency K(l) Source: M. Procházka: Optika pro počítačovou grafiku

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

l l

l

e

) (   = d d K

SLIDE 14

Human eye

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

Source: Encyclopedia Britanica, 1994

SLIDE 15

Relation between photo- and radiometric quantities

◼ Spectrum to luminous flux (visual response):

l l l d ) ( ) (

nm 770 nm 380 e



 =  K

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 16

Relation between photo- and radiometric quantities

◼ Relative spectral luminous efficiency V(l)

❑ Sensitivity of the eye to light of wavelength l relative to the

peak sensitivity at lmax = 555 nm (for photopic vision).

❑ CIE standard 1924

Source: M. Procházka: Optika pro počítačovou grafiku

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 17

Relation between photo- and radiometric quantities

◼ Spectrum to luminous flux (visual response):

Φ𝑤 = 638.002 lm W × න

380 nm 770 nm

𝑊 𝜇 Φ𝑓 𝜇 𝑒𝜇

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 18

Relation between photo- and radiometric quantities

◼ Radiometry

❑ More fundamental – photometric quantities can all be

derived from the radiometric ones

◼ Photometry

❑ Longer history – studied through psychophysical

(empirical) studies long before Maxwell equations came into being.

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 19

Radiometric quantities

SLIDE 20

Transport theory

◼ Empirical theory describing flow of “energy” in space ◼ Assumption:

❑ Energy is continuous, infinitesimally divisible ❑ Needs to be taken so we can use derivatives to define

quantities

◼ Intuition of the “energy flow”

❑ Particles flying through space ❑ No mutual interactions (implies linear superposition) ❑ Energy density proportional to the density of particles ❑ This intuition is abstract, empirical, and has nothing to do

with photons and quantum theory

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 21

Radiant energy – Q [J]

◼ Unit: Joule, J

Q (S, <t1, t2>, <l1, l2>)

Time interval Surface in 3D (imaginary or real)

S

Wavelength interval

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 22

Spectral radiant energy – Q [J]

◼ Energy of light at a specific wavelength

❑ „Density of energy w.r.t wavelength“

◼ We will leave out the subscript and argument l for brevity

❑ We always consider spectral quantities in image synthesis

◼ Photometric quantity:

❑ Luminous energy, unit Lumen-second aka Talbot

( ) ( )

l l l  l l l

l l l l l l

d d formally , , , , , lim , , ,

2 1 2 1 2 1 , ) , ( 2 1

2 1 2 1

Q t t S Q t t S Q

d

= = =

 →

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 23

Radiant flux (power) – Φ [W]

◼ How quickly does energy „flow“ from/to surface S?

❑ „Energy density w.r.t. time“

◼ Unit: Watt – W ◼ Photometric quantity:

❑ Luminous flux, unit Lumen

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 24

Irradiance– E [W.m-2]

◼ What is the spatial flux density at a given point x on a

surface S?

◼ Always defined w.r.t some point x on S with a specified

surface normal N(x).

❑ Irradiance DOES depend on N(x) (Lambert law)

◼ We’re only interested in light arriving from the “outside”

f the surface (given by the orientation of the normal).

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 25

Irradiance – E [W.m-2]

◼ Unit: Watt per meter squared – W.m-2 ◼ Photometric quantity:

❑ Illuminance, unit Lux = lumen.m-2

light meter (cz: expozimetr)

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 26

Lambert cosine law

◼ Johan Heindrich Lambert, Photometria, 1760

A E  =



A

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 27

Lambert cosine law

◼ Johan Heindrich Lambert, Photometria, 1760

A

q cos ' ' A A E  =  =

 q

A’=A / cosq

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 28

Lambert cosine law

◼ Another way of looking at the same situation

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

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Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 30

Radiant exitance – B [W.m-2]

◼ Same as irradiance, except that it describes exitant

radiation.

❑ The exitant radiation can either be directly emitted (if

the surface is a light source) or reflected.

◼ Common name: radiosity ◼ Denoted: B, M ◼ Unit: Watt per meter squared – W.m-2 ◼ Photometric quantity:

❑ Luminosity, unit Lux = lumen.m-2

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 31

Radiant intensity – I [W.sr-1]

◼ Angular flux density in direction  ◼ Definition: Radiant intensity is the power per unit solid

angle emitted by a point source.

◼ Unit: Watt per steradian – W.sr-1 ◼ Photometric quantity

❑ Luminous intensity,

unit Candela (cd = lumen.sr-1), SI base unit

   d d I ) ( ) (  =

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 32

Point light sources

◼ Light emitted from a single point

❑ Mathematical idealization, does not exist in nature

◼ Emission completely described by the radiant intensity as

a function of the direction of emission: I()

❑ Isotropic point source ◼

Radiant intensity independent of direction

❑ Spot light ◼

Constant radiant intensity inside a cone, zero elsewhere

❑ General point source ◼

Can be described by a goniometric diagram

❑ Tabulated expression for I() as a function of the direction  ❑ Extensively used in illumination engineering

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 33

Spot Light

◼ Point source with a directionally-

dependent radiant intensity

◼ Intensity is a function of the

deviation from a reference direction d :

◼ E.g. ◼ What is the total flux emitted by

the source in the cases (1) a (2)? (See exercises.)

d 

) , ( ) ( d    = f I      =  =  =

therwise

) , ( ) ( ) ( ) , ( cos ) (       d d d

I

I I I I

(2) (1)

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 34

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 35

◼ Spatial and directional flux density

at a given location x and direction .

◼ Definition: Radiance is the power per unit area

perpendicular to the ray and per unit solid angle in the direction of the ray.

Radiance – L [W.m-2.sr-1]

 q  d d cos ) , (

2

A d L  = x

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 36

◼ Spatial and directional flux density

at a given location x and direction .

◼ Unit: W. m-2.sr-1 ◼ Photometric quantity

❑ Luminance, unit candela.m-2 (a.k.a. Nit – used only in

English)

Radiance – L [W.m-2.sr-1]

 q  d d cos ) , (

2

A d L  = x

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 37

The cosine factor cos q in the definition of radiance

◼ cos q compensates for the decrease of irradiance with

increasing q

❑ The idea is that we do not want radiance to depend on

the mutual orientation of the ray and the reference surface

◼ If you illuminate some surface while rotating it, then:

❑ Irradiance does change with the rotation (because

the actual spatial flux density changes).

❑ Radiance does not change (because the flux density

change is exactly compensated by the cos q factor in the definition of radiance). And that’s what we want.

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

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Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

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Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

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Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

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Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 42

Env maps – Terminator II

◼ https://www.youtube.com/watch?v=BVE-7x9Usvw

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 43

Calculation of the remaining quantities from radiance

 q  d cos ) , ( ) (

) (



=

x

x x

H

L E  q d cos

= projected solid angle

x x x

x x A L A E

A H A

d d cos ) , ( d ) (

) (

  

= =   q  ) (x H

= hemisphere above the point x

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 44

Area light sources

◼ Emission of an area light source is fully described by the

emitted radiance Le(x,) for all positions on the source x and all directions .

◼ The total emitted power (flux) is given by an integral of

Le(x,) over the surface of the light source and all directions.

A L

A H e

d d cos ) , (

) (

 

=   q 

x

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 45

Properties of radiance (1)

◼ Radiance is constant along a ray in vacuum

❑ Fundamental property for light transport simulation ❑ This is why radiance is the quantity associated with

rays in a ray tracer

❑ Derived from energy conservation (next two slides)

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 46

Energy conservation along a ray

L d dA L d dA

1 1 1 2 2 2

  =

d2 dA2 L2() d1 dA1 L1() r emitted flux received flux

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 47

Energy conservation along a ray

L d dA L d dA

1 1 1 2 2 2

  =

d2 dA2 L2() T d dA d dA dA dA r = = = =  

1 1 2 2 1 2 2

ray throughput d1 dA1 L1() r

L L

1 2

=

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 48

Properties of radiance (2)

◼ Sensor response (i.e. camera or human eye) is directly

proportional to the value of radiance reflected by the surface visible to the sensor.

( )

R L A d dA L T

in A in

=  = 

 

, cos  q 



2

Sensor area A2 Aperture area A1

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 49

Incoming / outgoing radiance

◼ Radiance is discontinuous at an interface between

materials

❑ Incoming radiance – Li(x,) ◼

radiance just before the interaction (reflection/transmission)

❑ Outgoing radiance – Lo(x,) ◼

radiance just after the interaction

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 50

Radiometric and photometric terminology

Fyzika Physics Radiometrie Radiometry Fotometrie Photometry Energie Energy Zářivá energie Radiant energy Světelná energie Luminous energy Výkon (tok) Power (flux) Zářivý tok Radiant flux (power) Světelný tok (výkon) Luminous power Hustota toku Flux density Ozáření Irradiance Osvětlení Illuminance dtto Intenzita vyzařování Radiosity ??? Luminosity Úhlová hustota toku Angular flux density Zář Radiance Jas Luminance ??? Intensity Zářivost Radiant Intensity Svítivost Luminous intensity

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 51

Next lecture

◼ Light reflection on surfaces, BRDF

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015

SLIDE 52

Study references

◼ Petr Olšák: Základní radiometrické veličiny

http://petr.olsak.net/ftp/olsak/grafika/svetlo.pdf

◼ Petr Olšák: Proč je 𝑒𝜕 = sin 𝜄 𝑒𝜄 𝑒𝜚

http://petr.olsak.net/ftp/olsak/grafika/domega.pdf

◼ Wiki

https://cs.wikipedia.org/wiki/Radiometrie

Advanced 3D Graphics (NPGR010) - J. Vorba 2020, created by J. Křivánek 2015