Optical Filtering for Near Field Photometry with High Order Basis

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Order Basis

Xavier Granier, Michael Goesele, Wolfgang Heidrich, Hans-Peter Seidel

To cite this version:

Xavier Granier, Michael Goesele, Wolfgang Heidrich, Hans-Peter Seidel.

Optical Filtering for

Near Field Photometry with High Order Basis. [Research Report] RR-6000, INRIA. 2006.

�inria-00107463v2�

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inria-00107463, version 2 - 23 Oct 2006

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Thème COG

Optical Filtering for Near Field Photometry with

High Order Basis

Xavier Granier — Michael Goesele — Wolfgang Heidrich — Hans-Peter Seidel

N° 6000

October 2006

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Unité de recherche INRIA Futurs

XavierGranier

∗

, Mi hael Goesele

†

,Wolfgang Heidri h

‡

,Hans-Peter Seidel

§

ThèmeCOG Systèmes ognitifs ProjetIPARLA

Rapportdere her he n°6000O tober200623pages

Abstra t: A urately apturing the near eld emission of omplex luminaires is still verydi ult. Inthispaper,wedes ribeanewa quisition pipelineofsu hluminaires that performs anorthogonalproje tiononagivenbasisin atwo-steppro edure. First, weuse an opti al low-pass lter that orresponds to the re onstru tion basis to guarantee high pre isionmeasurements. These ondstepisanumeri alpro essonthea quireddata that nalizesthe proje tion. Based onthis on ept, we introdu e newexperimental setupsfor automati a quisition andperformadetailederroranalysisofthea quisitionpro ess.

Key-words: LightSour esA quisition;Opti alFiltering; ErrorAnalysis

∗

IPARLAproje t(LaBRI-INRIAfuturs)

†

UniversityofWashington

‡

TheUniversityofBritishColumbia

§

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Résumé: Lamesurepré isedu hamppro hedesluminairesesttoujourstrèsdi ile. Dans erapport,nousprésentonsunenouvellepro édured'a quisitionde essour esdelumières qui ee tue une proje tionorthogonaledans une base prédénie, et e par une pro édure en deux étapes. Premièrement, nousutilisons un ltreoptique passe-bas orrespondant à labase dere onstru tion. Ce i garantitune mesurede hautepré ision. Lase onde étape estune pro essus numériquesurlesdonnéesa quisesqui naliselaproje tionorthogonale danslebase re onstru tion. Basésur e on ept, nousintroduisonsde nouveaux systèmes expérimentauxd'a quisitionetnousprésentonsuneétudedétailléedeserreurs orrespondantes.

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1 Introdu tion

1.1 Motivation

For lighting simulation and design, the light sour e model has a major inuen e on the a ura yofthesimulation[CJM02℄. Unfortunately,thedire tsimulationapproa hisoften notfeasible,sin etherequireddataisrarelyavailableforreal lightsour es,sin ea urate modelingofarealluminaireisdi ultandina urateparameterssu hastheglassthi kness ofahalogenlightbulbs anhaveasigni antimpa tontheluminaire'semission pattern; andnally,a ompleterenderingofas enein ludingafullsimulationofthelightemission andtransportinsidetheluminaireisgenerally omplexandthereforeverytimeandresour e onsuming.

In orporatingmeasuredlightemissiondatafrom omplexluminairesintoarendering sys-temprovidesasolutiontothisproblem. Theluminaireisgenerallyapproximatedbyapoint light sour e [SB90℄ and therefore many ompanies provide goniometri diagrams [VG84℄, whi hdes ribetheirfareld(i.e.,lightemissionfromapointlightsour eparameterizedby emissiondire tion).

Unfortunately, the far eld is only a faithful approximation of the emitted light when the light sour e is su iently far away from the obje t to be illuminated as a rule of thumb atleast vetimesthemaximumluminairedimension[Ash95℄. All ee tsrequiring knowledgeofthespatialanddire tionallightemission anonlybemodeledbyanear eld representationsu hasalighteld[GGSC96,LH96℄. Itrepresentstheluminaire'semission withoutknowledgeofitsgeometry orinternalstru ture.

1.2 Previous Work

Several approa hesto measure thenear eld emission ofaluminaire havebeendeveloped in re ent years. Ashdown [Ash93, Ash95℄ pointed a number of ameras at a luminaire (or, more pra ti ally, a single amerais movedaround on arobot arm) and re ordedthe irradian ein identonanimagingsensor. BothRykowskiandWooley[RW97℄,andJenkins and Mön h [JM00℄ employ asimilar setup to a quire the light eld of a luminaire, while SiegelandSto k[SS96℄repla edthe ameralenswithapinhole.

Inall theseapproa hes,the amerapositions orrespondto asamplingof somevirtual surfa e

S

. Inpra ti e,theluminairemayprodu earbitrarilyhighspatialfrequen ieson

S

. Evenwithagoodsamplingstrategy[CCST00℄,aliasingisintrodu edunlessalow-passlter is applied before the sampling step. This issue has beenraised by Halle [Hal94℄ and was notedby Levoyand Hanrahan[LH96℄. Theyall proposeto usethe nite aperture of the ameralensasalow-passlterby hoosing theaperturesize equaltothesize ofasample onthe ameraplane. Thepropertiesofthelow-passlteraredependentonthelenssystem, and annotbeuser-dened. Nofurther ontrol onthepro essispossible. Asanexample, we annotguaranteethe ontinuityofthenalre onstru tionwhenusingtheseapproa hes.

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Toaddressthisproblem,Goeseleetal.[GGHS03℄introdu edanewa quisitionapproa h basedonanopti allow-passlter. This lterperformsaproje tionoftheluminaire'snear eldonapredened basis.

1.3 Overview and Contributions

This paper introdu es two a quisition setups for a new a quisition pipeline and the new designofthe orrespondingopti allters. Thea quisitionpipelineisatwo-steppro ess: the lightsour eisrstopti allyltered. Thenthea quiredimagesarenumeri allytransformed, tonish theproje tionofthelightsour einthepre-dened basis. Thedesignofthelter isbasedonadetailederrorstudy,alsopresentedinthispaper. Itshowstheresultingerrors forthedierentapproximationsthatareneededtoprodu eafeasiblesetup.

The remainder of this paper is organized as follows: rst, we summarize the general approa h forlight sour ea quisition [GGHS03℄. Then, weintrodu eour newltering ap-proa hinSe tion3thatsimpliesthea quisitionbutrequiresabasistransformationofthe measureddata. InSe tion 4,wedes ribethesetups ofourexperimentsin ludingaplaner and a ylindri al onguration. Then we analyze the error introdu ed by the ylindri al onguration(Se tion5). Wenally showsomea quisitionsof realluminaires(Se tion 6) beforewe on lude. 2 Theory 2.1 Basi Approa h

light source

measurement surface

sampling surface

M S

filter

Φ

∗

(u, v)

Φ

∗

(s, t)

θ

S

θ

M

Figure1: Crossse tionthroughthe on eptuala quisitionsetup.

The on eptualsetup forourapproa hto lteredlightsour emeasurementisdepi ted in Figure 1. Light rays are emitted from the light sour e and hit a lter in asurfa e

S

. Thissurfa eisopaqueex eptforthenitesupportareaofthelterthat anbemovedtoa numberofdis retepositionsonaregulargrid. Thelterisasemi-transparentlm,similar to aslide, ontainingthe 2Dimage ofa fun tion

Φ

∗

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Table1: Notation(overview)seealsoFigure1. Symbol Meaning

Ψ

ijkl

(u, v, s, t)

basisfun tionforapproximatingthelighteld

Φ

i

1Dbasisusedforre onstru tion

Φ

ijkl

(u, v, s, t)

4Dtensorprodu tbasisforre onstru tion

Φ

∗

i

biorthogonal1Dbasisusedformeasurement

Φ

∗

ijkl

(u, v, s, t)

4Dtensorprodu tbasisformeasurement

M

surfa eonwhi htheirradian eismeasured(

(s, t)

-surfa e)

S

surfa eonwhi htheopti alltersarepla ed(

(u, v)

-surfa e)

L(u, v, s, t)

radian epassingthrough

(u, v)

on

S

and

(s, t)

on

M

˜

L(u, v, s, t)

proje tionof

L(u, v, s, t)

intobasis

{Ψ

ijkl

(u, v, s, t)}

L

mn

(u, v, s, t)

radian e

L(u, v, s, t)Φ

∗

mn

(u, v)

lteredwith

Φ

∗

mn

(u, v)

E

mn

(s, t)

irradian e ausedby

L

mn

(u, v, s, t)

onthesurfa e

M

to allow a physi al setup. The light falling through this lter is attenuateda ording to

Φ

∗

ij

(uv)

andhitsase ondsurfa e

M

, onwhi hweareabletomeasure theirradian ewith ahighspatial resolution.

The fundamental dieren e to previous work [Ash93, SS96℄ is that we an sele t an arbitrarylter

Φ

∗

ij

(u, v)

dependingon thearbitrarily hosenfun tion spa einto whi h we wantto proje tthelighteldemittedbytheluminaire. Thisresultsin anopti allow-pass lteringofthelighteldbeforesampling,andthushelpsavoidingaliasing artifa ts.

Inprin iple,onealsohasto performlow-passlteringonthemeasurementsurfa e

M

. This, however, is opti ally verydi ult oreven impossible to implement when the lter kernels overlap. Nevertheless, it is te hni ally easy to a hieve a very high measurement resolutionon

M

,sothatwe an onsiderthemeasureddataaspointsamples,andimplement an arbitrarylter kernel while downsampling to the nal resolution. Note that thelter kernelsfor

S

and

M

anbe hosenindependentlyasrequiredbythere onstru tionpro ess.

2.2 Light Sour e Representation

Beforewe dis ussthetheory behindtheproposed method in detail,werstintrodu ethe mathemati al notation used throughout this do ument (summarized in Figure 1 and Ta-ble1).

For the measurement, we assume that the light eld emitted by the light sour e is well represented by a proje tioninto a basis

{Ψ

ijkl

(u, v, s, t)}

ijkl∈ZZ

(similar to the

lumi-graph[GGSC96℄):

L(u, v, s, t) ≈ ˜

L(u, v, s, t) :=

X

i,j,k,l

L

ijkl

Ψ

ijkl

(u, v, s, t) .

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Weassumethat

Ψ

ijkl

haslo alsupport,and

i

,

j

,

k

,and

l

roughly orrespondto transla-tionsin

u, v, s,

and

t

,respe tively. Wealsodenetwoadditionalsetsofbasisfun tions,one

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for measuring and one for re onstru tion. Forre onstru tion, we use a 1Dbasis

{Φ

i

}

i∈ZZ

withtheproperty

Φ

i

(x) = Φ (x + i)

. The4Dre onstru tionbasisisthengivenasthetensor produ tbasis

Φ

ijkl

(u, v, s, t) := Φ

i

(u) Φ

j

(v) Φ

k

(s) Φ

l

(t) .

Formeasurement, we use the biorthogonal (or dual)

{Φ

∗

i

(x)}

_i∈ZZ

of the re onstru tion basisdenedby

hΦ

∗

i

· Φ

i

′

i =

Z

∞

−∞

Φ

∗

i

(x) Φ

i

′

(x) dx = δ

i,i

′

(2)

- where the Krone ker'ssymbol

δ

i,i

′

= 1

if

i = i

′

, and

δ

i,i

′

= 0

otherwise- and again we performtensor-produ t onstru tionforthe4Dmeasurementbasis.

2.3 Measured Irradian e

Ourapproa his basedonmeasuring theirradian e

E

mn

(s, t)

on themeasurementsurfa e

M

(parameterizedby

(s, t)

)thatis aused bythein identradian e

L(u, v, s, t)Φ

∗

mn

(u, v)

:

E

mn

(s, t) =

Z

∞

−∞

Z

∞

−∞

g (u, v, s, t) L(u, v, s, t)Φ

∗

mn

(u, v) du dv

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≈

X

i,j,k,l

L

ijkl

Z

∞

−∞

Z

∞

−∞

g (u, v, s, t) Ψ

ijkl

(u, v, s, t) Φ

∗

mn

(u, v) du dv.

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Thegeometri term

g (u, v, s, t)

isdenedas

g (u, v, s, t) =

cos θ

S

cos θ

M

d

2 .

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where

d

isthe distan e between thepoint

(u, v)

on thesampling surfa e

S

and the point

(s, t)

onthesampling surfa e

M

.

θ

S

and

θ

M

aretheanglesbetweenthenormalsof

S

and

M

andtheve tor onne ting

(u, v)

and

(s, t)

,respe tively. Notethat

g

alsoa ountsforany dieren esin theparameterizationonthetwosurfa es(i.e., dierentgrid spa ing). When

S

and

M

areparallel,

g = cos

2 _θ/d

2

,sin e

θ

S

=

θ

M

=

θ

.

2.4 Re onstru tion

Wenowdes ribe anexa tre onstru tion algorithm from themeasurements

E

mn

. Tothis end, werstdene what therelationshipbetweenthebasisfun tions

Ψ

ijkl

andthe re on-stru tionandmeasurementbasesshould be. Wedene

Ψ

ijkl

(u, v, s, t) :=

1 g (u, v, s, t)

Φ

ij

(u, v) Φ

kl

(s, t) .

(6) FromEquation4andthebiorthogonalityrelationship(Equation 2)follows

E

mn

(s, t) =

X

k

X

l

L

mnkl

Φ

kl

(s, t) .

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In order to determine whi h re onstru tion lter to use, we rewrite Equation 1 using Equations6and7:

˜

L(u, v, s, t) =

X

m,n,k,l

L

mnkl

Ψ

mnkl

(u, v, s, t) =

X

m,n

1 g (u, v, s, t)

Φ

mn

(u, v) E

mn

(s, t).

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This indi ates that we an exa tly re onstru t

L

˜

, the proje tion of

L

into the basis

{Ψ

ijkl

}

, byusingthere onstru tionlter

Φ

mn

(u, v) /g (u, v, s, t)

.

Based on Equations8 and 3, aswell asthe biorthogonality relationship(Equation 2), the oe ients

L

mnkl

are denedas

L

mnkl

=

Z

∞

−∞

Z

∞

−∞

E

mn

(s, t) Φ

∗

kl

(s, t) ds dt

=

Z

∞

−∞

Z

∞

−∞

Z

∞

−∞

Z

∞

−∞

g (u, v, s, t) L(u, v, s, t)Φ

∗

mn

(u, v) Φ

∗

kl

(s, t) du dv ds dt

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orrespondingtotheenergymeasuredataposition onthemeasurementsurfa e.

3 Filter design

3.1 Criteria for Sele ting Bases and Filters

Weproposetwomain riteriaforthelterdesign. First,forpra ti alreasons,itispreferable touseasinglelterforallpositionson

S

(i.e.,

Φ

∗

i

= Φ

∗

i

′

).

Furthermore,the hoi eofabasis(andthenof alter)should dependontheintended appli ation. Forexample, using the a quired light sour es for image synthesis, the basis should be at least

C

1

ontinuous, sin e the human'svisual system is sensible to both

C

0

-and

C

1

-dis ontinuities[LaDP99℄. Thisisourse ond riteria.

The posssibility of sele ting ontext-based proterty for both our lter (and thus our re onstru tionbasis)isthemain dieren ewiththepreviouswork.

3.2 Dual Basis for an Orthogonal Proje tion

The theoreti alsetup aspresentedin Se tion 2.2 usesa dual basis aslter. A dual basis

{Φ

∗

i

}

asdes ribed in Equation 2 denes a proje tioninto the basis

{Φ

i

}

. Thus, the proje tion

f

˜

of

f

is:

f =

˜

P

i

hΦ

∗

i

· f i Φ

i

Sin e abasisandits dual spanthesamefun tion spa e,theindividualbasisfun tions

Φ

∗

i

ofthedual anbeexpressedaslinear ombinations

oftheprimarybasis:

Φ

∗

i

=

P

j

a

ij

Φ

j

UsingEquation2,weobtainthefollowing onditions:

hΦ

∗

i

· Φ

i

i =

P

j

a

ij

hΦ

j

· Φ

k

i = δ

ik

Thus,the

(a

ij

)

oe ients anbe omputedbyinverting thesymmetri matrix

(hΦ

i

· Φ

j

i)

ij

.

Unfortunately,using dire tlythedual fun tion asalter hasseveraldownsidesforthe realizationofaworkingsetup. First,forapositiveprimarybasis,thedualfun tionusually hasbothpositiveandnegativefun tionvalues(seeFigure2). Onesolutionwouldbeto reate

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Figure 2: Four quadrati B-splines (dotted line) and one dual fun tion (plain line). The dual orrespondstotherightmostB-splinefun tion(bothin bold).

twolters,one representingthepositivepart ofthe dual only,and the other representing thenegativepart[GGHS03℄. A measurementsession will then needtwopasses, with some possiblealignmentproblemsbetweenthetwoa quisitions.

Ase ondproblemisthatthedualsusuallyhavealargesupport( orrespondingtoalarge numberof non-zero oe ients

(a

ij

)

). Inpra ti e,this anbeaddressedby restri tingthe supportof thedualbasisfun tions,ee tivelysetting mostof the

(a

ij

)

to zero. Obviously thisintrodu esanapproximationerror.

Thenalproblemisthat,foranitebasis,thedualsareingeneralnotexa tandshifted opiesofea hother,eveniftheprimarybasisfun tionsare. Thiswouldrequirethe reation ofdierentopti alltersfordierentpositions,whi hisnotfeasiblein pra ti e.

3.3 Using the Primary Basis as a Filter

Toover omealltheseissues,wedire tlyusetheprimarybasisasalterforthemeasurement pro ess. Thedual representationisthen omputedsoftwareas

hΦ

∗

i

· f i =

X

j

a

i,j

hΦ

j

· f i .

This approa h has several advantages. First, we have omplete ontrol overthe lter support. Se ond, for any primary basis dened by a positive fun tion, only one lter is required. Then,allltersarenowexa t opiesofea hotherandthustourrst riterion (seeSe tion3.1). Inparti ular,theyarealsoindependentofthemeasurementdomain,whi h meansthatwe annowextendthedomainwithouthavingtorepeatpreviousmeasurements withnewlters. Finally,thisapproa hredu estheltersupportforagivenre onstru tion resolution.

With this new experimental setup, we are now free to use a wide variety of dierent lters. In our experimentswe use quadrati B-Splines, sin e there are simple lters that fulllthe ontinuity riterionoutlinedinSe tion3.1.

Unfortunately,thelinear ombinationintrodu edwiththisnewlterdesign anbetime onsuming. Whenthesizeofthere onstru tionbasisonthelterplanesis

N

,amaximum of

O (N )

linear ombinations is required to ompute the measure orresponding to ea h

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basisfun tion. Hen e, the omplexityofthis stepis then

O

(N × M )

2

where

N × M

is thenumberofa quiredimages.

Fortunately,theabsolutevaluesofthedualbasis oe ients(i.e.,the

|a

ij

|

des ribedin Se tion3.2) de reasequi kly. With only14 oe ientstheresultingerroronthea quired dataisbelow0.5%forthe1Ddualofaquadrati B-Spline(theequivalentofa1%errorfor the2D ase). Bykeepingthenumberof oe ientsunder14,the omplexityofthisstepis linear.

Note that this error orresponds to ameasure of a onstant light sour e through the equivalent dual lter (after the linear ombination). This is a very onservative upper bound. Itislowerforthedualbasisattheborderofthedomain.

Hen e,thelinear ombinationintrodu esalsosomeexibilityin thenal qualityofthe a quisition: arstapproximation anbedonequi klywithalownumberof oe ientsand thehighqualitysolution anbe omputedlater.

4 Automati A quisition Setup

4.1 Generi Planar Conguration

Filter

Diffuse White Screen

Black Curtains

Digital Camera

Filter

Linear Stage

Light Source

Figure3: Physi alsetup

Asamanuala quisitionofalargenumberofmeasures anbeverytime onsumingand pronetoerrors,wedesignedasetupbasedontheACME[PLL

+

99℄fa ilityforanautomati a quisition(seeFigure3). Wehavedevelopedtwodierent ongurations,therstonewith twoparallel planesas measurementandlterplanes, andthese ond onewithtwo oaxial ylindri alplanes.

Due tothedesign,thexed omponentofthesetupisthemeasurementplane(awhite s reen, mate painted in order to reate a diuse s reen). A amera fa ing this s reen is our measurement devi e. Due to the build-in limitation in intensity range of a digital amera, wehave to take several pi tureswith dierent exposure time for one position of the lter and of the light sour e, in order to re onstru t "High Dynami Range" (HDR) images[DM97, RBS99℄. With thisapproa h,we anre overthe realintensityofthe light fallingonthemeasurementplane.

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For ea h new measure, the lter has to be shifted horizontally and verti ally. The verti altranslationisa hievedbyaverti allinearstage,onwhi hthesupportofthelteris xed. Themovementofthemeasuredlightsour e,mountedonahorizontallylinearstage, simulatesthehorizontaltranslation.

Thea quisitionsetupisalsoisolatedfromanyotherin ominglight. Somedark urtains arepla edto reateabla kboxaroundthewhites reen. Inthisbla kbox,a ameraisfa ing thes reen. This guarantees that the measure datais perturbed by someinter-ree tions. Also,inorder toremovealllightsotherthanthemeasuredlightsour e,themeasurements takepla ein adarkroom.

After the physi al a quisition, a linear ombination of the images is performed (see Se tion3.3)inorderto nalizetheproje tiononthesele tedbasis.

4.2 Cylindri alConguration

Still,aplanar ongurationhasabuilt-in limitation: whenthelteris movingawayfrom theopti alaxis,themeasuredsolidangleisde reasing. Alightsour ewithanangularrange largerthan180

o

annotbe aptured ompletely.

filter

Φ

∗

(u, v)

filter

Φ

∗

(s, t)

measurement surface

light source

sampling surface

Figure4: Thetheoreti al ylindri alsetup

Thenatural ongurationto over omethisproblem (aspheri alsetuparoundthelight sour e)isverydi ultto design: thelterhastobe urvedand annotbeeasilydesigned to beuniqueand tobeshiftedoverthespherewith aguaranteeof

C

1

ontinuity. Instead, weintrodu ea ylindri alsetup(seeFigure4). Thankstothedistortion-freemappingfrom aplanetoa ylinder, thelterdesignissimilarto theplanar ase. With thisapproa h,we an apturealltheemittingdire tions ofalightsour earoundthe ylinderaxis.

Unfortunately,a ylindri almeasurements reenisalsomore omplexto reatethanthe orrespondingplanarones. Forourexperiments,weapproximate the ylindersbyaset of tangentplanes(seeFigure5). Toa hievethis onguration,thelightsour eismountedon arotationstage,insteadof thehorizontal linearstagein theplanar ase(seeFigure 3).

Similarly,wehavetoapproximatethe ylindri allterwithaplanarone. Thedesignof thisnewlterandtheresultingerrorsaredetailed inthefollowingse tion.

After the physi al a quisition, we reproje t the HDR images on the ylinder in order to orre t the planar approximation of the measurement s reen. The resulting error are

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light source

filter

measurement screen

theoretical cylinders

Figure 5: Thepra ti al ylindri alsetup

detailedin Se tion5.2. Then,likefortheplanar onguration,alinear ombinationofthe imagesisperformed.

5 Error Study for the Cylindri al Conguration

5.1 Filter Approximation

Therearetwomainaspe tsinthedesignofthisnewlter: theproje tionfromthe ylinder to the approximating plane, and the position of this plane relative to the ylinder. We presentnowthesedierentaspe ts.

5.1.1 Expressionof the Parallax Error

The errorresulting from the planarapproximation

ψ (u, v)

of the lteris due to the par-allax. For a given ray, emitted from the light sour e and traversing the lter, the error is

δ

P

(u, v, s, t) = ψ (P (u, v, s, t)) − ψ (u, v) ,

where

P

isthe parallaxdue to theproje tion s heme. Hen e,theerroronthemeasuredirradian e(seetheEquation3)is

E

′

(s, t) − E (s, t) =

Z

∞

−∞

Z

∞

−∞

cos

2 _{θ (u, v, s, t)}

R (u, v, s, t)

2 δ

P

(u, v, s, t) L (u, v, s, t) D dv,

This error an bebounded by

|E

′

(s, t) − E (s, t) | ≤ δ

max

P

(s, t) ¯

E (s, t)

where

δ

max

P

(s, t) =

max

L(u,v,s,t)>0

{|δ

P

(u, v, s, t) |}

representthemaximumparallaxdeviationfromallin oming

dire tionfromthelightsour eand

E (s, t)

¯

isthemeasuredirradian ewitha onstantlter. Notethat

δ

max

P

dependsontherelativesizeofthelightsour e omparedtothemeasurement

setup. Intuitively,forapointlightsour e, theparallaxerrorhaslessinuen e thanforan extendedone.

Then,theresultingerroronthemeasured oe ientsofthelightsour eis

|L − L

′

| =

Z

∞

−∞

Z

∞

−∞

Φ

∗

(s, t) (E (s, t) − E

′

(s, t)) ds dt

≤ kδk max

Φ

∗

(s,t)>0

E {s, t) .

¯

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InthisEquation,

kδk = max

Φ

∗

(s,t)>0

{δ

max

P

(s, t)}

.

Hen e,studyingtheerrordue totheapproximateplanarlterleadstostudythe maxi-mumparallaxerror. Thisisthemainfo usofthefollowingse tions.

Notethat,foragivenre onstru tionbasis

Φ

,usingthebasisasalterwillgreatlyredu e theparallaxerror,sin ethesupportsizewillbesmaller(about14timessmallerforagood approximationofthedual-see Se tion3.3).

5.1.2 Dierent proje tion s hemes

approximated filter

Radial

Perpendicular

curved filter

theoretical cylinders

Figure6: Perpendi ularandradialproje tions

Sin e we are usinga planarapproximationof the lter,wehave toproje tthe urved lterontheplanarone. Westudytwoproje tions: aperpendi ularproje tionandaradial proje tion(seeFigure 6). As areferen e, we also add thelter forthe planarsetup used herein a ylindri al ontext.

Theproje tedltersandthenon-proje tedone(seeTable2)lters onvergetoasimilar behaviorwhen the lter size is de reasing: the hoi e of the proje tion does not hange signi antlytheerrorforsmalllters. Theradialproje tionisthemosta urateforallthe ongurations. Itisourgeneri hoi eforthelterdesign.

5.1.3 Dierent lterpositions

We study here two extreme positions for the approximate planar lter, as des ribed in Figure 7. The rst position is tangentto the theoreti al ylinder; the se ond - the inner position- orrespondtothe asewheretheextremitiesoftheapproximateplanarlterare onthe ylinder.

The inner position requires a smaller lter (for a radial proje tion) so it seems to be interesting. However, the distan e between the urved lter and the approximate planar lterismaximalin the enter,wherethevaluesofthelterarethemostimportant. Thus,

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Table2: Comparisonofthemaximumparallaxerror lteraperture

3π/5

3π/10

3π/20

3π/40

3π/80

tangentposition radialproj. 21% 5% 3.8% 2.3% 1.22% un-proje tedlter 17.6% 7% 4.2% 2.4% 1.23% perpendi ularproj. 8% 4.3% 2.5% 1.24% inner position radialproj. 84% 28% 16.8% 9.4% 5% perpendi ularproj. 56% 32% 17.8% 9.7% 5.1%

The omparisonisdoneforperpendi ularandradialproje tion,andfornon-proje tedlter. Theinner ylinderhasaradiusof70 mandthedistan ebetweenthetwo ylindri alsurfa es is30 m. Onlytheapertureofthetheoreti allterisvarying.Wesupposethatinnerradius bounds thesize ofthelightsour e.

Radial

tangent position

inner position

Perpendicular

R

sin (θ

max

)

R

tan (θ

max

)

θ

max

Figure7: Filterposition

theresultingparallaxerroristhenlarger(seeTable2). Ontheotherhand,withthetangent position,thedistan e betweenthe twoltersissmall in the enter and highertowardthe endswherethevaluesofthelterarealmostzero.

Consequentlythetangentpositionshouldbe hosen omparedtitheotherone,andthis fa tis onrmedbythenumeri alresults(seeTable2).

5.2 Reproje tion on the Measurement Surfa e

The ylindri almeasurementsurfa eisalsoapproximatedbyaplanars reen. Thea quired measuresneedthento bereproje tedinorder reate areal ylindri al onguration,while

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introdu inganothererrorthatisdetailed inthisse tion. Werstpresenthowtherepro je -tionis performed.

5.2.1 Corre tive Fa tor and Error Expression

Real Conguration

θ

₁

θ

S

θ

_M

d

₁

d

CentralApproximation

θ

′

₁

θ

′

_M

d

′

₁

d

′

Figure8: Reproje tion ongurationonthemeasurements reen

Insteadofthetheoreti almeasured oe ients(seeEquation9),theplanar approxima-tionofthemeasurements reenprovides(usingthenotionsof Figure8,and assumingthat thebasisfun tions onthemeasurementsurfa earethesameinboth ases):

L

′

mnpq

=

Z

∞

−∞

Z

∞

−∞

Z

∞

−∞

Z

∞

−∞

cos θ

1 cos θ

S

d

1

2 L(u, v, s, t)Φ

∗

mn

(u, v) Φ

∗

pq

(s, t) du dv ds dt.

Forthereproje tionstep,ea ha quired oe ientiss aledbyageometri al orre tion term

λ

mnpq

. theresultingerrorbetweenthe orre ted oe ientandthetheoreti aloneis:

L

mnpq

− λL

′

mnpq

≤ max

uv∈f ilter,st∈pixel

|1 − λ

mnpq

d

1

2 cos θ

M

d

2 _{cos θ}

1 | |L

mnpq

| = ǫ |L

mnpq

|

Whentheltersizeandthepixelsizeofameasurearerelativelysmall omparedtothe distan ebetweenthesampling planeandthemeasurementplane,thevalueof

d

1

(resp.

d

,

θ

1

,

θ

S

,

θ

M

)is notvaryingalot omparedto the entral value

d

′

1

(reps.

d

′

,

θ

′

1

,

θ

′

1

,

θ

′

M

).

λ

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Table3: Inuen eofdierentfa torsfortheerroronthereproje tion. Original Light

=

R

2

Light

= 0

D = R

D = 4R

aperture

=

3π

20

aperture

=

3π

80 5%

6.2%

3.8%

10.4%

2.4%

20%

3.2%

Theoriginal ongurationisthe following: theradius of thelightsour eis

R/8

(where

R

istheradiusoftheltering ylinder),thelterapertureis

Π

20

,thedistan e

D

betweenthe lterandthemeasurements reenis

2R

. Theother olumn representthe hange ompared totheoriginalsetup.

issele tedthatthereisnoerrorontheraydenedbythe enterofthelterandthe enter ofthepixel:

λ

mnpq

=

d

′ 2

cos θ

′

1 d

′

1

2 cos θ

′

M

.

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Wedesignthereproje tionstepasfollows. Ea hpixelofthea quiredimageis orre ted by the s aling fa tor dened in Equation 10. The resulting image orresponds to an a -quisitionona ylindersurfa e,but withnon-regular pixels(ea h one orrespondingtothe proje tionoftheoriginalplanarpixelonthe ylinder). Thus,theimageisnallyresampled onaregulargridonthe ylindri alsurfa e.

5.2.2 SomeNumeri al Results

The results shown in Table 3 onrm some of our intuitions. First, the error in reases signi antly withthe width ofthe lter(see thetwolast olumns). On e again usingthe duallter wedramati allyin reasesthenal error,asshownin Se tion 3.3, thedualis at least14timeslargerthanthere onstru tionbasis. Thesolutionistoredu ethesizeofthe duallter,leadingtosmallerbasisfun tionandtoanin reaseofthemeasurementnumber to overthe same lightsour e. Using the basis is still the best hoi e for the ylindri al a quisition.

Se ond,in reasingthedistan ebetweenthelterandthes reen anhelpalsoinredu ing the error, asit relativelyredu es the size of the pixel and of the lter. But for the same dire tional overage,thesize of thes reen hasto in reasealso, leadingto non-pra ti able setups.

Then,byadjustingthesetwofa tors,we anrea hanerror

ǫ

around

6%

withareasonable setup: thelterapertureis

π

10

,thedistan ebetweenthelterandthes reenisaroundtwi e

thedistan ebetweenthelightandthelter,andtheradiusofthelightsour eisbelowhalf oftheradiusofthelter.

5.3 Con lusion

Inthe ylindri al ase, the planarapproximation an reate largeerrors, but they anbe redu edbya arefuldesign. Thereareprin ipallythree ontrols: (i)theltersize/aperture

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Table4: Common ongurationforourexperiments distan elightsour elter: 30 m

distan elters reen: 75 m s reensize(width

×

height): 120 m

×

100 m resolutionofa quiredimages (width

×

height): 960

×

840

anberedu ed-thisredu es boththeparallaxandthereproje tionerror,(ii)therelative sizeofthelightsour e anberedu edbys alingupthe ompletesetup-thatwouldredu e theerroronthelterbutnotsigni antlytheerroronthes reen-,(iii)thedistan ebetween thelterandthes reen anbein reased-thisredu esmostlythereproje tionerror-.

Whenthepositionsofthelightsour e,thepositionofthes reenandthesizeofthelter are given, theonly parameteris thelter position. On onehand,movingthe lter loser to the s reenhas the three following ee ts. (i) Sin e this redu es theaperture, and this in reasestherequirednumberofmeasures. Both errorson thes reen andonthelterare redu ed. (ii)This alsoredu es thesize ofthelight ompared to thesizeof thetheoreti al ylinderofthelter,whileredu ingbotherrorsaswell. (iii)But,thisredu esthedistan e between the lter and the s reen, and that in reases the reproje tionerror (whi h is the largerofthetwoerrors).

On the other hand, when the lter is loser to the light sour e, the parallax error is in reasedandrea hesitsmaximumvalue. Butthis erroris thesmallerofthetwo. Conse-quently, the best solutionis to put thelter loserto the lightsour e thanto the s reen. Thishasbeentakenintoa ountfordesigningtheexperimentalsetupdes ribedinthenext Se tion.

Hen e, the basis lter seemseven more to bethe best hoi e in the ylindri al setup, sin e it redu es the lter support for a given re onstru tion resolution (see Se tion 3.3). Withsmalllters,alltheapproximationsarevalid.

6 A quired Data

In order to validate our results, we a quired a light sour e (a ar headlight) with both geometri ongurations ( ylindri al and planar) and varying lter sizes. Table 4 shows the resolution of the a quired images and the geometri onguration of the setup. The setupwasdeterminedbytheresultsoftheerrorstudypresentedinthepreviousse tionand onstrainedbythedimension oftheACMEmeasurementfa ility. Thelterswereprinted with1016dpiresolutiononhigh-densitylmusinga alibratedLightJet2080DigitalFilm Re order. Thisprodu es slideswith adynami rangeof morethan1:50,000. Theimages were aptured with aJenoptik C14digital amera. Thelter (seeSe tion 3) used forthe experimentsisbasedonaquadrati B-Splinefun tion,withtwodierentsizes: alargeone (136.5mm)andasmallerone(49.14mm).

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6.1 Cylindri alA quisition

Forthe ylindri ala quisition,werstexperimentthelargelter(136.5mm). Basedonthe errorstudy,wesele taradialproje tionandatangentpositionofthelter(seeSe tion5.1). With thislargelter,onlyanarrayof9

×

7(angle

×

height)imageswasneededto overthe ompletelightsour e. Thisresultsin194MBytesofdata.

Sin etheboundingsphereofthelightsour ehasaradiuslessthan20 m,thetheoreti al maximumparallaxerroris3.7%. Fortheperpendi ularproje tion,thetheoreti alerroris 4.4%and4.2%foranon-proje tedlter. On eagain,theradialproje tionisbetter.

Figure 9: Some a quired data (left) and the orresponding images after the ylindri al proje tion(middle)andnally thelinear ombination(right).

Forthe reproje tionstep, the theoreti al error is about 15%. This maximum will be rea hedattheborderofthea quiredimages. Aftera loserlookonthedata(seeFigure 9-left), this orresponds to pixelswith lowvaluein thea quired image, redu ing thea tual error. It an beseen on theproje ted images (see Figure 9-middle), where no noti eable dieren e is visible. For this light sour e, the a quisition system does not introdu e a signi anterror.

Forthelinear ombinationstep,wedonotuseanapproximatedualrepresentation,sin e thenumberofdata is su ientlylow. We annoti e inFigure 9-rightthat thedata after the linear ombination an be signi antly dierent from the original one, as somelarge negative oe ientsare introdu ed. Su h approa hguarantiesa orre trepresentation of theoriginalradian evariationfromtheoriginallightsour e, omparedtotraditional near-eldphotometrybasedonpositiveonlydata[Ash93,Ash95,RW97,JM00,SS96℄.

In order to show the a ura y, we ompare the isolines of the proje ted pattern for both the real light sour e and thea quired onein Figure 10. The resultsare similar and moreregular, due to the hoi e of there onstru tion basis. Theapproximationsfrom the ylindri al asedonotintrodu eanerrorthat anbevisuallynoti eableerrors.

Wealso omputetheproje tedpatternof thelightattwodierentdistan es,asshown inFigure11,in ordertoshowthatwemanageto apturesomeneareld ee t. Withonly agoniometri diagram,thepattern wouldnot hange. Here,thepattern is hangingwith thedistan e.

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Reallight From ylindri ala quisition Fromplanara quisiton

Figure10: Comparisonontheproje tedpatternofa arheadlight(distan esour e-plane= 120 m). Alogarithmi s aledhasbeenappliedto theoriginalintensitiesinorderto reate legibleset of isolines. Forea h proje tedpattern, 15 isolineshave been sele ted, in order toremovepossibledieren einmaximumintensitybetweenimages, andfo usonlyonthe shapevariation.

Figure11: Proje tiononawallparalleltothe ylinderaxis. Theintensitieshavebeens aled res aleto

[0, 1]

in orderto showthe patternvariationwith thedistan e (near-eldee t). Left,distan esour eplane=120 m;right,distan esour eplane=300 m.

6.2 Planar A quisition

Intheplanara quisition,wetestthea quisitionwithasmallerlter(2.73mm). Notethat with su h a dimension, an error of alignment between two measures would have a larger

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Figure12: Globalilluminationsolution

inuen e onthe nal results. Usinga dual asalter(and using onelter forits positive omponentandoneforitsnegativeone)is thusnotbewellsuitedforthis onguration.

Asthelterissmaller(i.e.,wehavealargerresolution),thenumberofmeasureshasto bein reased. Thenaldatasetisthenanarrayof23x15(widthxheight)images,resulting in 1GBytes of data. Su h large data set would be very di ult to a quire without any automati setupsandwithoutasinglelter.

Wedonotuseanapproximationforthelinear ombinationstep,sin eittakesabout3.5 minutesto omputethelinear ombinationonea hresultingdata(notethatthe omputation wasperformedonan IntelXeon2.8GHz),and thefull omputation takesabout20hours. Byredu ingthenumberof oe ientsforthedualrepresentationto14,itwouldtakeonly about6.5hours.

We an see that theproje ted pattern (see Figure 10) is also similar to thereal light sour e,butmoreblurred. Thisleadstothefollowingremark: forrea hingthesamequality in the near eld representation, we need more measures at high resolution (i.e., with a smallerlter) forthe planar asethan for the ylindri al onguration, thanksto a more regularsamplingindire tion.

Thisdatahasalsobeenusedto omputeaglobalilluminationsolutionbasedonPhoton Mapalgorithm[Jen01℄-seeFigure12. Forthissimulation,thesamelighthasbeendupli ated inorderto reatethetwoheadlamps.

7 Con lusion

Inthis paper, wehave presented anewa quisition system for omplexlightsour es from thereal world. This systemisbased on arefullydesigned opti allter. This lter allows performinganorthogonalproje tionofthelightsour einasele tedbasis.

Thesystemhastwo omponents. Therststepisthephysi alsetupforthea quisition ofopti allylteredimages,usingthere onstru tionbasisfun tionasalter. Inthese ond step,thedataispro essedona omputertonalizetheproje tioninthesele tedbasis.

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Twodierentsetupshavebeenexperimented. Intherstone,weusetwoplanarsupports forboththeltersandtheimagingdevi e. These ondoneisbasedona ylindri alsupport. Thisin reasestheregularityofthea quisitionoveroneangulardire tion.

Inordertobindthedierenterrors(i.e.,inthelinear ombinationwithanapproximation ofthedualbasisfun tion,inthe ylindri alsetupwiththeplanarapproximationofthelter andthemeasurementplane),wehavepresentedatheoreti alerrorstudy. This helpsus in designingthenalexperimental setups.

Inorderto validatethisapproa h, someresultsonthetwosetupshavebeenpresented. They onsist in the a quisition of a ar head-light. The results show that we manage to a quire both the near eld and the far eld of a light sour e, that the omputation timeofthelinear ombinationisstillreasonable,and thattheresultingerrorof theplaner approximationof the ylindri alsetupislimited.

Allthesestudiesshowthatwe anrea hagooda ura yinlightsour ea quisitionusing thissimpleopti allterdesign,andahighorderbasis.

7.1 Future Work

There arethree main dire tions forfuture work. Therst one onsistsin a loserlook on the possible errors and a hara terization of su h a quisition devi e[RPN91℄. Re all that theresultspresentedherearevery onservativebounds. With a loseranalysis,webelieve thatthetheoreti alerrors anberedu ed.

These onddire tionistheextensionofthe ylindri alsetup. Thissetupallowsabetter regularityofthea quisitioninoneangulardimension. Itwouldbeusefultonotrestri tthis regularitytoonlyonedimension. Thebesttheoreti alsupportforthea quisitionwouldbe two on entri spheres. Nevertheless,thedesignofafeasiblesetupforthis ongurationis more omplex,andwouldrequiresome arefulapproximations.

The third dire tion is to nd a ompa t and memory-e ient representation. A om-pressions heme[AR97℄forsu hlightsour emodelswouldmaketheirusagemorepra ti al.

A knowledgments

This work was funded bythe PIMS (Pa i Institut for theMathemati al S ien es) Post-do toralFellowship program,and theINRIA (Institut National de Re her heen Informa-tiqueetenAutomatique)asso iatedlaboratoryprogram. TheauthorswishtothankDavid Roger,forhis workonsomeaspe tsoftheerrorstudy.

Theauthors anbe onta tedwiththefollowingemails: granierlabri.fr, goeselempi-sb.mpg.de,heidri h s.ub . aandhpseidelmpi-sb.mpg.de.

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Contents

1 Introdu tion 3

1.1 Motivation . . . 3

1.2 PreviousWork . . . 3

1.3 OverviewandContributions . . . 4

2 Theory 4 2.1 Basi Approa h . . . 4

2.2 LightSour eRepresentation . . . 5

2.3 MeasuredIrradian e . . . 6

2.4 Re onstru tion . . . 6

3 Filter design 7 3.1 CriteriaforSele tingBasesandFilters . . . 7

3.2 DualBasisforanOrthogonalProje tion . . . 7

3.3 UsingthePrimaryBasisasaFilter. . . 8

4 Automati A quisition Setup 9 4.1 Generi PlanarConguration . . . 9

4.2 Cylindri alConguration . . . 10

5 Error Study for the Cylindri alConguration 11 5.1 FilterApproximation. . . 11

5.1.1 ExpressionoftheParallaxError . . . 11

5.1.2 Dierentproje tions hemes. . . 12

5.1.3 Dierentlterpositions . . . 12

5.2 Reproje tionontheMeasurementSurfa e . . . 13

5.2.1 Corre tiveFa torandError Expression . . . 14

5.2.2 SomeNumeri alResults . . . 15

5.3 Con lusion . . . 15 6 A quired Data 16 6.1 Cylindri alA quisition . . . 17 6.2 PlanarA quisition . . . 18 7 Con lusion 19 7.1 FutureWork . . . 20

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