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Linear Color Representation for Full

Spectral Rendering

Mark Peercy

Presented by Weidong Shi, 2001

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Overview

Use full color distribution, not tri-stimulus RBG.

Decompose color distribution into a linear space of basisfunctions.

Represent color distribution as a coefficient vector.

Model surface reflectance as matrices.

Convert results into tri-stimulus RGB.

Find optimal set of basis functions.

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Why Full Spectral Information?

Non-uniform spectral distribution. Photo-realisticrendering needs more samples of wavelengths than tri-

stimulus.

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Existing Methods of Handling Full Spectrum

Point sampling.

- Sample surfaces and lights at certain number of

wavelengths.

- Numerically integration into RGB required for display.

Polynomial representation of light.

Linear models of surfaces and light.

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Point Sampling

wavelength

Spectral powerS3 SnS1 S2

i

N

i

iSRRd

1

i

N

i

iSGGrn

1

i

N

i

iSBBl

1

)(I

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Basis Function and Linear Representation

Orthonomal basis functions, a set of functionsf1(x),f2(x),

Given a function g(x), compute coefficient by,

g(x) is approximated by a vector of m-length.

dxxfxfx

ji )()(0

dxxfxfx

ii )()(1

dxxfxgx ii )()(

m

i

ii xfxg1

)()(

For any fi(x)

For any two different functions fi(x)andfj(x)

orthogonal

normal

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Linear Representation of Power Distribution

Describe spectral power distribution of light as a

linear combination of m orthonormal basis

functions

dEIii )()(

m

i

iiEI1

)()(

E1

E2

E3

m

i

iiE

1

)(

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Surface Reflectance

Lighting Model

)()()()()()()()(),(

ssssddaao IRGIRGIRI

)(s

I

)()()( sddIRG

)()()( sss IRG

reflected light as

a function of wavelength

and surface geometry

ambient light

input lightBRDF diffuse terms

BRDF specular terms

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Surface Reflectance- cont

Represent ambient light, directional light, and reflected

light as coefficient vectors.

a

m

a

1

s

m

s

1

om

o

1

),( oI)(sI)(aI

ambient light light source reflected light

= = =

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Surface Reflectance - cont

Through math deduction, surface reflectance can

be represented by m*m matrices

)(aR )()( dd RG )()( ss RG

aR11

dR11

sR11

)(d

G )(sG

dEERR jidd

ij )()()( dEERR jis

s

ij )()()( dEERRjia

a

ij )()()(

surface ambient property surface diffuse property surface specular property

)()()()()()()()(),( ssssddaao IRGIRGIRI

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Surface Reflectance - cont

New lighting equation

a

m

a

1

aR11

dR11

s

m

s

1

)(d

G

sR11

s

m

s

1

)(sG

o

m

o

1

= +

+

ambient termdiffuse term

specular term

)()()()()()()()(),( ssssddaao IRGIRGIRI

output light ambient light

light source

light source

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Light Distribution at Each Pixel

The next step is to convert it into RGB.

m

i

i

p

ip EI1

)()(

p

m

p

1

Lights arrive at each pixel is also represented as a m-lengthvector

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Convert Light at Each Pixel to RGB

dExdIxXi

m

i

p

ip

)()()()(1

dEydIyY i

m

i

p

ip )()()()(1

dEzdIzZ i

m

i

p

ip )()()()(1

Integrate lights at each pixel (m-length vector) over three

color matching functions X,Y, and Z.

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Selection of Basis Function

Consider computation complexity

Nature of the spectral power distribution in the

scene. Point sampling is sufficient if powerdistribution is smooth.

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Method2 - Character Vector Analysis (CVA)

Given a representative set of spectral distributionsin a scene (I1(), I2(), ), find m basis function

automatically

The set of basis functions minimizes

The representative set of lights may contain light sources,

once-reflected light, and multi-reflected light

dEIErrl

m

i

ilil

2

1

,)()(

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Examples

Scene under fluoresecnt light, CVA on light source, first

order, and second order surface reflectance from four

sample of the surface

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Examples - Result basis functions

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Rendering Result

2 3 4 5

4 9 16 25

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Summary of Rendering Procedure

Determine Basis Functions

Represent Lights as Coefficient Vector

Represent Surface Reflectance as Matrices

Render

Convert Results at Each Pixel into XYZ

Convert XYZ into RGB

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Conclusion

Provide a general efficient way of full spectral color

rendering

Represent color distribution by linear combination of basis

functions

Capture surface reflectance by matrices

Linear lighting equation

Better photo-realistic rendering result

Easy to implement