Lighting affects appearance

What is the Question ?(based on work of Basri and Jacobs, ICCV 2001)

Given an object described by its normal at each surface point and its albedo (we will focus on Lambertian surfaces)

1.What is the dimension of the space of images that this object can generate given any set of lighting conditions ?

2. How to generate a basis for this space ?

90.797.296.399.5#9

88.596.395.399.1#7

84.794.193.597.9#5

76.388.290.294.4#3

42.867.953.748.2#1

ParrotPhoneFaceBall

(Epstein, Hallinan and Yuille; see also Hallinan; Belhumeur and Kriegman)

5 2DDimension:

Empirical Study

DomainDomain

Lambertian

No cast shadows

(“convex” objects)

Lights are distant

nl

Lambert Law

k(max (cos, 0)

Lighting to Reflectance: Intuition

1

0 1 2 30

0.5

1

0 1 2 30

0.5

1

1.5

2

r Three point-light sources, l), Illuminating a sphere and its reflection r). Profiles of l and r

Lighting to Reflectance: Intuition

Images

...

Lighting Reflectance

lllllS llrrrr ddlkr sin),(),;,(),(2

where )0),max(cos(1

)(),;,( lrlrllrr kk

...

Spherical Harmonics (S.H.)

Orthonormal basis, , for functions on the sphere.

n’th order harmonics have 2n+1 components.

Rotation = phase shift (same n, different m).

In space coordinates: polynomials of degree n.

imnmnm eP

mn

mnnY )(cos

)!(

)!(

4

)12(),(

nmn

mn

n

m

nm zdz

d

n

zzP )1(

!2

)1()( 2

22

nmY

S.H. analog to convolution theorem

• Funk-Hecke theorem: “Convolution” in function domain is multiplication in spherical harmonic domain.filter.

k

nn

nmnnm

kn

YYk

12

4

Harmonic Transform of Kernel

00

( ) max(cos ,0) n nn

k k h

12

( 2)! (2 1)

13

( 1) 2, even2 ( 1)!( 1)!

0 2, od2

2

d2

0

nn

n

n nn

n n

kn

n

n

0nY 2cos16

5cos

2

1

4

1

0.886

0.591

0.222

0.037 0.014 0.0070

0.5

1

1.5

0 1 2 3 4 5 6 7 8

Amplitudes of Kernel

n

nA

n

nmnmn k

nA 2

12

1

Energy of Lambertian Kernel in low order harmonics

Accumulated Energy

37.5

87.599.2 99.81 99.93 99.97

0

20

40

60

80

100

120

0 1 2 4 6 8

kis a low pass filter 2cos16

5cos

2

1

4

1)( k

Reflectance Functions Near Low-dimensional Linear Subspace

0

( )n

nm nm nmn m n

r k l K L h

Yields 9D linear subspace.

2

0

)(n

n

nmnmnmnm hLK

nmY

nmY

Forming Harmonic Imagesnm nmb (p)=λr (X,Y,Z)

Z YX

2 2 22λ(Z -X -Y ) XZ YZ2 2λ(X -Y ) XY

How accurate is approximation?Point light source

Amplitude of k

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 4 6 8

Amplitude of l = point source

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 4 6 8

9D space captures 99.2% of energy

2

00

)()(n

n

nmnmnmnm

n

n

nmnmnmnm hLKhLKlkr nmY nmY

How accurate is approximation? Worst case.

9D space captures 98% of energy

DC component as big as any other. 1st and 2nd harmonics of light could have zero energy

Amplitude of k

-0.50

0.51

1.5

22.5

33.5

0 1 2 4 6 8

Amplitude of l = point source

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 4 6 8

How Accurate Is Approximation?

Accuracy depends on lighting.

For point source: 9D space captures 99.2% of energy

For any lighting: 9D space captures >98% of energy.

Accuracy of Approximation of Images

Normals present to varying amounts.

Albedo makes some pixels more important.

Worst case approximation arbitrarily bad.

“Average” case approximation should be good.

Summary

Convex, Lambertian objects: 9D linear space captures >98% of reflectance.

Explains previous empirical results.

For lighting, justifies low-dim methods.

Models

Query

Find Pose

Compare

Vector: IMatrix: B

Harmonic Images

Recognition

Experiments (Basri&Jacobs) 3-D Models of 42 faces

acquired with scanner.

30 query images for each of 10 faces (300 images).

Pose automatically computed using manually selected features (Blicher and Roy).

Best lighting found for each model; best fitting model wins.

Results

9D Linear Method: 90% correct.

9D Non-negative light: 88% correct.

Ongoing work: Most errors seem due to pose problems. With better poses, results seem near 100%.