Appearance Models for Graphics

COMS 6998-3

Brief Overview of Reflection Models

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Assignments

• E-mail me name, status, Grade/PF. If you don’t do this, you won’t be on class list. [and give me e-mails now]

• Let me know if you don’t receive e-mail by tomorrow

• E-mail me list of papers to present (rank 4 in descending order). Must receive by Fri or you might be randomly assigned.

• Next week, e-mail brief descriptions of proposed projects. Think about this when picking papers

Today

Appearance models

– Physical/Structural (Microfacet: Torrance-Sparrow, Oren-Nayar)

– Phenomenological (Koenderink van Doorn)

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Symmetric Microfacets

Shadowing Masking Interreflection

Brdf of grooves simple: specular/Lambertian

Torrance-Sparrow: Specular Grooves. Specular direction bisects (half-angle) incident, outgoing directions

'( ) ( , ) ( )

4 cos( )cos( )

i i r h

i r

F G Df

Oren-Nayar: Lambertian Grooves.Analysis more complicated. Lambertian plus a correction

Phenomenological BRDF model Koenderink and van Doorn

• General compact representation

• Domain is product of hemispheres

• Same topology as unit disk, adapt basis

– Zernike Polynomials

Paper presentations

• Torrance-Sparrow (Kshitiz)

• Oren-Nayar (Aner)

• Koenderink van Doorn (me, briefly)

Phenomenological BRDF model Koenderink and van Doorn

• General compact representation

• Preserve reciprocity/isotropy if desired

• Domain is product of hemispheres

• Same topology as unit disk, adapt basis

• Outline– Zernike Polynomials– Brdf Representation– Applications

Zernike Polynomials

• Optics, complete orthogonal basis on unit disk using polynomials of radius

• R has terms of degree at least m. Even or odd depending on m even or odd

• Orthonormal, using measure dd

immn

mn eR

nZ )(

1),(

n-|m| even|m|n

Cool Demo: http://wyant.opt-sci.arizona.edu/zernikes/zernikes.htm

|m| 0 1 2

n

0

1

2

n-|m| mustbe even

|m| n

|m| n

|m| n

n-|m| mustbe even

n-|m| mustbe even

n-|m| mustbe even

Hemispherical Zernike Basis

• Measure Disk: Hemisphere: sin()dd

• Set

dd

)2sin(2

dddd )sin(2

)())2sin(2(1

),(

mmn

mn azR

nK

azm=

m>0:cos(m)m=0:sqrt(2)m<0:sin(m)

dd )2cos(2

1

dd )2cos()2sin(

BRDF representation

• Reciprocity: aklmn=amnkl

),(),(),,,( rrmnii

kl

klmnklmnrrii KKaf

BRDF representation

• Reciprocity: aklmn=amnkl

• Isotropy: Dep. only on = |i-r| Expand as a function series of form cos(m[i-r])

• Can define new isotropic functions

• Symmetry (Reciprocity): alnm= anl

m

),(),(),,,( rrmnii

kl

klmnklmnrrii KKaf

)cos())2sin(2())2sin(2(),,( mRARI rmli

mnri

mnl

),,(),,( rimnl

lmn

mnlri Iaf

BRDF Representation: Properties

• First two terms in series

• 5 terms to order 2,14 to order 4, 55 order 8

• Lambertian: First term only

• Retroreflection: ln

• Mirror Reflection: (-1)m ln

• Very similar to Fourier Series

)cos()2sin()2sin(22

2

1

111

000

riI

I

alnm = l0 n0 m0

alnm = ln

alnm = (-1)m ln

Applications

• Interpolating, Smoothing BRDFs• Fitting coarse BRDFs (e.g. CURET).

Authors: Order 2 often sufficient• Extrapolation• Some BRDF models can be exactly

represented (Lambertian, Opik)• Others to low order after filtering/truncation• High-order terms are typically noisy

Discussion/Analysis

• Strong unified foundation

• Spectral analysis interesting in own right

• Ringing!! Must filter

• Don’t handle BRDF features well

• Specularity requires many terms

• Theoretically superior to spherical harmonics but in practice?