Function Representation & Spherical Harmonics

Function Representation & Spherical HarmonicsFunction Representation & Spherical Harmonics

Function approximationFunction approximation

• G(x) ... function to represent

• B1(x), B2(x), … Bn(x) … basis functions

• G(x) is a linear combination of basis functions

• Storing a finite number of coefficients ci gives an

approximation of G(x)

)()(1

xBcxG i

n

ii

Examples of basis functionsExamples of basis functions

Tent function (linear interpolation) Associated Legendre polynomials

Function approximationFunction approximation

• Linear combination

– sum of scaled basis functions

1c

2c

3c

Function approximationFunction approximation

• Linear combination

– sum of scaled basis functions

xBcn

iii

1

Finding the coefficientsFinding the coefficients

• How to find coefficients ci?

– Minimize an error measure

• What error measure?

– L2 error2

])()([2

I iiiL xBcxGE

Approximated function

Original function

Finding the coefficientsFinding the coefficients

• Minimizing EL2 leads to

Where

nnnnnn

n

BG

BG

BG

c

c

c

BBBBBB

BBBB

BBBBBB

2

1

2

1

21

2212

12111

I

dxxHxFHF )()(

Finding the coefficientsFinding the coefficients

• Matrix

does not depend on G(x)

– Computed just once for a given basis

nnnn

n

BBBBBB

BBBB

BBBBBB

21

2212

12111

B

Finding the coefficientsFinding the coefficients

• Given a basis {Bi(x)}

1. Compute matrix B

2. Compute its inverse B-1

• Given a function G(x) to approximate

1. Compute dot products

2. … (next slide)

TnBGBGBG 21

Finding the coefficientsFinding the coefficients

2. Compute coefficients as

nn BG

BG

BG

c

c

c

2

1

12

1

B

Orthonormal basisOrthonormal basis

• Orthonormal basis means

• If basis is orthonormal then

ji

jiBB ji 0

1

IB

10

1

01

21

2212

12111

nnnn

n

BBBBBB

BBBB

BBBBBB

Orthonormal basisOrthonormal basis

• If the basis is orthonormal, computation of approximation coefficients simplifies to

• We want orthonormal basis functions

nn BG

BG

BG

c

c

c

2

1

2

1

Orthonormal basisOrthonormal basis

• Projection: How “similar” is the given basis function to the function we’re approximating

1c

2c

3cOriginal function Basis functions Coefficients

Another reason for orthonormal basis functionsAnother reason for orthonormal basis functions

f(x) = fi Bi(x)

g(x) = gi Bi(x)

f(x)g(x)dx = fi gi

• Intergral of product = dot product of coefficients

Application to GIApplication to GI

• Illumination integral

Lo = ∫ Li(i) BRDF (i) cosi di

Spherical HarmonicsSpherical Harmonics

Spherical harmonicsSpherical harmonics

• Spherical function approximation

• Domain I = unit sphere S

– directions in 3D

• Approximated function: G(θ,φ)

• Basis functions: Yi(θ,φ)= Yl,m(θ,φ)

– indexing: i = l (l+1) + m

The SH FunctionsThe SH Functions

Spherical harmonicsSpherical harmonics

• K … normalization constant

• P … Associated Legendre polynomial

– Orthonormal polynomial basis on (0,1)

• In general: Yl,m(θ,φ) = K . Ψ(φ) . Pl,m(cos θ)

– Yl,m(θ,φ) is separable in θ and φ

Function approximation with SHFunction approximation with SH

• n…approximation order

• There are n2 harmonics for order n

),(),( ,

1

0, ml

n

l

lm

lmml YcG

Function approximation with SHFunction approximation with SH

• Spherical harmonics are orthonormal

• Function projection

– Computing the SH coefficients

– Usually evaluated by numerical integration

• Low number of coefficients

low-frequency signal

2

0 0

,,,, sin),(),()()( ddYGdYGYGc ml

S

mlmlml

Function approximation with SHFunction approximation with SH

Product integral with SHProduct integral with SH

• Simplified indexing

– Yi= Yl,m

– i = l (l+1) + m

• Two functions represented by SH )()(

2

0

i

n

iiYfF

)()(2

0

i

n

iiYgG

2

0

)()(n

iii

S

gfdGF