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Light, Reflectance,and Global Illumination

TOPICS:

• Survey of Representations for Light and Visibility

• Color, Pereption, and Light

• Refletane

• Cost of Ray Traing

• !lobal Illu"ination Overvie#• Radiosity

• $u%s Inverse Global Illumination paper 

&'()*+, I"age(ased -odeling and Rendering

Paul .e/bert, &) Ot0 &+++

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Representations for Light and Visibility

fi1ed vie#point or  planar sene

2(3 i"age 4r,g,b5

diffuse surfae 20'(36

7(3

te1tured range data 48,r,g,b5

or pre-lit polygons

speular surfae in

lear air, unoluded

9(3 light field

speular surfae in fog,

or oluded0 plenopti funtion I41,y,8,θ,φ5

'(3 '(3 light field

or surface/volume model with general reflectance properties

italics: not image-based technique

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Physics of Light and Color

Light o"es fro" eletro"agneti radiation0The a"plitude of radiation is a funtion of #avelength λ0

ny ;uantity related to #avelength is alled re;ueny  ν ? 2 π 6 λ

@- spetru":

radio, infrared, visible, ultraviolet, 1(ray, ga""a(ray

  long wavelength short wavelength

  low frequency high frequency

• 3on%t onfuse @- AAspetru"%% #ith AAspetru"%% of signal in signal6i"age proessingBthey%re usually oneptually distint0

• Li/e#ise, don%t onfuse spetral #avelength fre;ueny of @- spetru" #ith spatial

#avelength fre;ueny in i"age proessing0

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Perception of Light and Color

Light is electromagnetic radiation visible to the human eye..u"ans have evolved to sense the do"inant #avelengths e"itted by

the sunB spae aliens have presu"ably evolved differently0

In lo# light, #e pereive pri"arily #ith rods in the retina, #hih

have broad spetral sensitivity 4

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What is Color

Color is human perception of light.Our pereption is i"perfetB #e don%t see spetral po#er P4λ5,

instead #e see three salars:

at long #avelength 4≈ red5: L ? ∫ P4λ5sL4λ5dλ

at "iddle #avelength 4≈ yello#5: - ? ∫ P4λ5s-4λ5dλat short #avelength 4≈ blue5: S ? ∫ P4λ5sS4λ5dλ

  #here sL4λ5, s-4λ5, and sS4λ5 are the sensitivity urves of our threesets of ones0

Color is three di"ensional beause any light #e see isindistinguishable to our eyes fro" so"e "i1ture of three spetral4"onohro"ati, single(#avelength5 primaries0

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Color !paces

Color an be desribed in various olor spaes:

spectrum ( allo#s non(visible radiation to be desribed, but usuallyi"pratial6unneessary0

RG" ( CRT(oriented olor spae

good for o"puter storage0

#!V ( a "ore intuitive olor spaeB good for user interfaes.?hue (( the olor #heel, the spetral olors

S?saturation 4purity5 (( ho# grayF

V?value 4related to brightness, lu"inane5 (( ho# brightF

a non(linear transfor" of R!, sine . is yli0

CI$ %&' ( used by olor sientists, a linear transfor" of R!0

other olor spaes, less o""only used000

>or e1tre"ely realisti i"age synthesis, use of four or "ore sa"ples of the spetru" "ay beneessary, but for "ost purposes, the three sa"ples used by R! olor spae is Gust fine0

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Light is a (unction of )any Variables

Light is a funtion of •  position 1,y,8

• diretion   θ,φ• #avelength λ

• ti"e t•  polari8ation

•  phase

In o"puter graphis, #e typially ignore the last three by assu"ingstati senes, unpolari8ed, inoherent light, and assu"e that the speedof light is infinite0

ut light is still a o"pliated funtion of "any variables0

.o# do #e "easure light, #hat are the unitsF

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*nits of Light

quantity dimension units

solid angle   solid angle Hsteradian

po+er   energy6ti"e H#att?HGoule6se

radiance L   po#er64areaJsolid angle5 H#att64"2Jsteradian5a0/0a0 intensity I

In vauu" or as an appro1i"ation for air, radiance is constant along

a ray0

piture is an array of ino"ing radiane values at i"aginary proGetion planeB beause of radiane(onstany, these are e;ual tooutgoing radianes at intersetion of ray #ith first surfae hit

In general, light is absorbed and sattered along a ray0

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General Reflection -ransmission

t a surface, reflectance is the fration of inident 4ino"ing5 light thatis refleted, transmittance is the fration that is trans"itted into the"aterial0 Opa;ue "aterials have 8ero trans"ittane0

In so"e boo/s, refletane is denoted ρ, and trans"ittane τ0 Other boo/s use / dr  and / sr  for diffuse and speular refletane, and / dt and

/ st for trans"ittane0

general refletane funtion has the for" of a bidirectionalreflectance distribution function /"R0(1: ρ4θi,φi,θo,φo5, #here thediretion of ino"ing light is 4θi,φi5 and the diretion of outgoing light

is 4θo,φo5, θ is the polar angle "easured fro" perpendiular, and φ isthe a8i"uth0

There is a si"ilar funtion for bidiretional trans"ittane, τ4θi,φi,θo,φo50

Light is absorbed and sattered by so"e "edia 4e0g0 fog50

Phong%s Illu"ination "odel is an appro1i"ation to general refletane0

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Phong Illumination )odel

point light soure #ith radiane I l, illu"inating an opa;ue surfae,reflets light of the follo#ing radiane:

If surfae is perfetly diffuse 4La"bertian50 It is independent of vie+ing directionK

I ? Il / dr  "a1M0L,EN6r 2

#here / dr  ? oeffiient of diffuse refletion H&6steradian

 M ? unit nor"al vetor 

L ? unit diretion vetor to light, r is distane to light

If surfae is perfetly specular0 It is not independent of vie+ing direction0

I ? Il / sr  "a1M0.,ENe6r 2

#here / sr  ? oeffiient of speular refletion H&6steradian. ? unit

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Cost of "asic Rendering lgorithms

 s ? Qsurfaes 4e0g0 polygons5 t  s ? ti"e per surfae 4transfor"ing, 0005

 p ? Qpi1els t  p ? ti"e per pi1el 4#riting, inre"enting, 0005 ? Qlights t  ? ti"e to light surfae point #0r0t0 one light

a ? Σ sreen areas of surfs t i ? ti"e for one ray6surfae intersetion test

Painter%s or (buffer algorith", #ith flat shading

#assuming no sorting in painter’s algorithm$

#orst ase ost ? s4t  s  t 5 at  p≈ at  p if polygons big

Painter%s or (buffer algorith", #ith per pi1el shading 4e0g0 Phong5

#orst ase ost ? st  s  a4t  t  p5 ≈ at   if polygons big

Ray asting #ith no shado#s, no spatial data strutures

#orst ase ost ? p4 st i  t   t  p5 ≈ pst i  if "any surfaesRay traing to "a1 depth d  #ith shado#s, refltran, no spat0 3S, no supersa"pling

%d − & intersections/pi'el( for each of which there are    shadow rays

#orst ase ost ? p42d −&5H4 &5 st i  t  ≈ 2d  p st i  if "any surfaes

 )ote: time constants vary( eg t  p is larger for *-buffer than for painter’s

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Interreflection

De typially si"ulate Gust direct illumination: light traveling on astraight, unoluded line fro" light soure to surfae, refletedthere, then traveling in a straight, unoluded line into eye0

Light travels by a variety of paths:

light soure → eye #+ bounces: loo,ing at light source$light soure → surfae& → eye #& bounce: direct illumination$light soure → surf& → surf2 → eye #% bounces$light soure → surf& → surf2 → surf7 → eye # bounces$

000

Illu"ination via a path of 2 or "ore bounes is alled indirectillumination or  interreflection0 It also happens #ith trans"ission0

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Global Illumination

.bservation: light o"es fro" other surfaes, not Gust designated lightsoures0

Goal: simulate interreflection of light in -! scenes

 Difficulty2 you can no longer shade surfaces one at a time, since they3reno+ interrelated4

T#o general lasses of algorith"s:

5. ray tracing methods: si"ulate "otion of photons one by one, traing photon paths either ba/#ards 4

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-he *nit of Radiosity

Radiance 4a0/0a0 intensity5 is po#er fro"6to an area in a given

diretion0units: po#er 6 4area × solid angle5

Radiosity is outgoing po#er per unit area due to e"ission or refletion

over a he"isphere of diretions0units: po#er 6 area

radiosity ? radiane × integral of Hos4polar angle5 × d4solid angle5 over ahe"isphere ? π × radiane

So radiosity and radiane are linearly interrelated0

Thus,

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Radiosity as an Integral $8uation

This is alled an integral e8uation beause the un/no#n funtion

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Classical Radiosity )ethod

3efinitions:

surfaes are divided into elements

radiosity ? integral of e"itted radiane plus refleted radiane over ahe"isphere0 units: Hpo#er6area

ssu"ptions:

   no partiipating "edia 4no fog5 →  shade surfaces only( not vols   opa;ue surfaes 4no trans"ission5

   reflection and emission are diffuse radiance is direction-indep. (radiance is a function of %-! surface parameters and λ

   refletion and e"ission are independent of λ #ithin eah of several

#avelength bandsB typially use 7 bands: R,!, →  solve linear systemsof equations   radiosity is onstant aross eah ele"ent → one G0 radiosity per element 

Typially 4but not e1lusively5:

   surfaes are polygons, ele"ents are ;uadrilaterals or triangles

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0eriving Radiosity $8uations, 5 

Let   Ai   = area of element i  (computable)  ei   = radiant emitted flux density of elementi  (given)   ρ i   = reflectance of element i  (given)  bi   = radiosity of elementi  (unknown)

  F ij   = form factor fromi to  j  = fraction of power leaving i that arrives at j  (computable)

So the equation above can be rewritten:

   Aibi =  Aiei + ρ i   F  ji j=1

n

∑  A jb j

outgoing

powerof elem i

 

 

 

   =

power

emittedby elem i

 

 

 

   +

power

reflectedby elem i

 

 

 

   

outgoingpower

of elem i

 

 

 

   =

poweremitted

by elem i

 

 

 

   +

reflectanceof elem i

    

  ×

fraction of powerleaving elem  j that

arrives at elem i

 

 

 

   

elem  j

∑ ×outgoingpower

of elem  j

 

 

 

   

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(orm (actors

3efine the form factor  1 i2 to be the fration of light leaving ele"ent i that arrives at ele"ent 2

Dherevi2 is a boolean visibility funtion: E if point on i is oluded #ith respet to

 point on G, & if unoluded0

This is a double area integral0 3iffiultK De end up appro1i"ating it0

d3i and d3 2 are infinitesi"al areas on ele"ents i and 2, respetively

θi and θ 2 are polar angles: the angles bet#een ray and nor"als on ele"ents i and 2, respetively

ProGeted area of d3i fro" 2 is os θi d3i, hene the osines

r  is distane fro" point on i to point on 2

Reiproity la#:  3i 1 i2 ? 3 2 1  2i0

F ij =1

 Ai

cosθ i cosθ  j

π r 2  vijdA jdAi

 A j

∫  Ai

∫ 

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0eriving Radiosity $8uations, 6

  Earlier, we had :  Aibi  =  Aiei + ρ i   F  ji j=1

n

∑   A jb j

Dividing by Ai : bi  = ei + ρ i   F  ji j=1

n

∑   A j Ai b j

By the reciprocity law, F  ji ( A j  /  Ai ) =  F ij , so bi  =  ei + ρ i   F ij j=1

n

∑   b j   for all elemsiOr, in matrix/vector notation:  b = e + Kbwhere b is the vector of unknown radiosities, e is the vector of known emissions,and K is a square matrix of reflectance times form factor: K 

ij

 =  ρ i

F ijSubtracting Kb from both sides, we get b - Kb = e, or

  (I - K)b = e where I is an identity matrix.This is a linear system ofn equations in n unknowns (the bi ).

There are three such systems of equations, one for the red channel, one for

green, and one for blue. The variables ρ i , ei , and bi  are RGB vectors.

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Computing Visibility for (orm (actors

Co"puting visibility in the for" fator integral is li/e solving ahidden surfae proble" fro" the point of vie# of eah surfae inthe sene0

-+o methods2

ray tracing: easy to i"ple"ent, but an be slo# #ithout spatial subdivision"ethods 4grids, otrees, hierarhial bounding volu"es5 to speed up ray(surfae intersetion testing

hemicube: e1ploit speed of 8(buffer algorith", o"pute visibility bet#een oneele"ent and all other ele"ents0 !ood #hen you have 8(buffer hard#are, but

so"e tri/y issues regarding he"iube resolution

$ou end up appro1i"ating the double area integral #ith a double su""ation, Gustli/e nu"erial "ethods for appro1i"ating integrals0

Dhen t#o ele"ents are /no#n to be inter(visible 4no oluders5, you an use

analyti for" fator for"ulas and s/ip all this0

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Radiosity !ystems Issues

ll algorith"s re;uire the follo#ing operations:

&0 Input sene 4geo"etry, e"issions, refletanes50

20 Choose "esh 4i"portantK5, subdividing polygons into ele"ents0

70 Co"pute for" fators using ray traing or he"iube for visibility 4e1pensive50

90 Solve syste" of e;uations 4indiretly, if progressive radiosity50

'0 3isplay piture0

If "esh is hosen too oarse, appro1i"ation is poor, you get blo/y shado#s0

If "esh is hosen too fine, algorith" is slo#0 good "esh is ritialK

In radiosity si"ulations, beause sene is assu"ed diffuse, surfaes% radiane #ill bevie#(independent0

Changes in vie#point re;uire only visibility o"putations, not shading0 3o #ith 8( buffer hard#are for speed0 This is o""only used for

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!ummary of Radiosity lgorithms

Radiosity algorith"s allo# indiret lighting to be si"ulated0

Classical radiosity algorithms2

• !enerality: li"ited to diffuse, polygonal senes0

• Realis": aeptable for si"ple senesB blo/y shado#s on o"ple1 senes0 Trial

and error is used to find the right "esh0• Speed: good for si"ple senes0 If all for" fators are o"puted, .#n% $, but if

 progressive radiosity or ne#er

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Inverse Global Illumination2

Recovering Reflectance )odels ofReal !cenes from Photographs

$i8hou $u, Paul 3ebeve, itendra -ali/, and Ti" .a#/ins

SI!!RP. U++

&'()*+, I"age(ased -odeling and Rendering

Paul .e/bert, 2E Ot0 &+++

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&u2 )otivation

-ost I-R "ethods per"it novel vie#points but not novel lightingBthey assu"e surfaes are diffuse0

Dant to per"it hanges in lighting0

Dould li/e to to reover 4non(diffuse5 refletane "aps:speularity and roughness at eah surfae point

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&u2 !implifying ssumptions

• stati sene of opa;ue surfaes• /no#n shape

• /no#n light soure positions

• alibrated a"eras, /no#n a"era positions

• high dyna"i range photographs

• speular refletion para"eters 4speular refletion oeffiient androughness5 onstant over large surfae regions

• eah surfae point aptured in at least one i"age

• eah light soure aptured in at least one i"age

• i"age of a highlight in eah speular surfae region in at least one photograph

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&u lgorithm ;vervie+

5. !olve for shape 4use >3@ or si"ilar syste"5

6. !olve for coarse diffuse reflectances

   oarsely "esh the surfaes into triangles

   inverse radiosity proble": /no#n for" fators and radiosities 4fro" i"ageradianes5, solve for diffuse refletanes 4linear syste"5

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&u2 c8uisition 0etails

• spherial, frosted light bulbs• &)E volt 3C po#er to avoid *E.8 light fli/er 

•  bla/ strips of tape on #alls for digiti8ation

• shot &'E i"ages #ith digital a"era, "erged to reate 9E high

dyna"i range i"ages 4represent #ith floating point pi1el values5•  blo/ light soure to a"era in so"e of the i"ages

• orret for radial distortion and vignetting