Function Representation & Spherical Harmonics
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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