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Function Representation & Spherical Harmonics

Function 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

x B c x G

i n i i

∑

=

=

Examples of basis functions

Tent function (linear interpolation) Associated Legendre polynomials

Function approximation

• Linear combination

– sum of scaled basis functions

×

1

c = = = ×

2

c ×

3

c

Function approximation

• Linear combination

– sum of scaled basis functions

( )=

∑

=

x B c

n i i i 1

Finding the coefficients

• How to find coefficients ci?

– Minimize an error measure

• What error measure?

– L2 error

2

] ) ( ) ( [

2 ∫

∑

− =

I i i i L

x B c x G E

Approximated function Original function

Finding the coefficients

• Minimizing EL2 leads to

Where

              =                          

n n n n n n n

B G B G B G c c c B B B B B B B B B B B B B B B B        

2 1 2 1 2 1 2 2 1 2 1 2 1 1 1

∫

=

I

dx x H x F H F ) ( ) (

Finding the coefficients

• Matrix

does not depend on G(x)

– Computed just once for a given basis

              =

n n n n n

B B B B B B B B B B B B B B B B      

2 1 2 2 1 2 1 2 1 1 1

B

Finding 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)

[ ]

T n

B G B G B G 

2 1

Finding the coefficients

• 2. Compute coefficients as

              =            

− n n

B G B G B G c c c  

2 1 1 2 1

B

Orthonormal basis

• Orthonormal basis means
• If basis is orthonormal then

   ≠ = = j i j i B B

j i

1

I B =             =               = 1 1 1

2 1 2 2 1 2 1 2 1 1 1

      

n n n n n

B B B B B B B B B B B B B B B B

Orthonormal basis

• If the basis is orthonormal, computation of

approximation coefficients simplifies to

• We want orthonormal basis functions

              =            

n n

B G B G B G c c c  

2 1 2 1

Orthonormal basis

• Projection: How “similar” is the given basis

function to the function we’re approximating

∫

× × ×

∫ ∫

1

c =

2

c =

3

c =

Original function Basis functions Coefficients

Another 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 GI

• Illumination integral

Lo = ∫ Li(ωi) BRDF (ωi) cosθi dωi

Spherical Harmonics

Spherical 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 Functions

Spherical 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 SH

• n…approximation order
• There are n2 harmonics for order n

) , ( ) , (

, 1 ,

ϕ θ ϕ θ

m l n l l m l m m l Y

c G

∑ ∑

− = = − =

=

Function 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

∫∫ ∫

= = =

π π

ϕ θ θ ϕ θ ϕ θ ω ω ω

2 0 0 , , , ,

sin ) , ( ) , ( ) ( ) ( d d Y G d Y G Y G c

m l S m l m l m l

Function approximation with SH

Product integral with SH

• Simplified indexing

– Yi= Yl,m – i = l (l+1) + m

• Two functions

represented by SH

) ( ) (

2

ω ω

i n i iY

f F

∑

=

= ) ( ) (

2

ω ω

i n i iY

g G

∑

=

=

∑ ∫

=

=

2

) ( ) (

n i i i S

g f d G F ω ω ω