function approximation & spherical harmonics
TRANSCRIPT
Function Approximation & Spherical Harmonics
Function approximation
• G(x) ... function to approximate
• B1(x), B2(x), … Bn(x) … basis functions
• G(x) is a linear combination of bases
• Storing a finite number of coefficients ci gives an approximation of G(x)
)()(1
xBcxG i
n
ii∑
=
=
Basis functions
Function approximation
• Linear combination – sum of scaled basis functions
×1c =
=
=
×2c
×3c
Function approximation
• Linear combination – sum of scaled basis functions
( ) =∑=
xBcn
iii
1
Finding the coefficients
• How to find coefficients ci?– Minimize an error measure
• What error measure?– L2 error
2])()([
2 ∫ ∑−=I i
iiL xBcxGE
Approximated function
Original function
Finding the coefficients
• Minimizing EL2 leads to
Where
=
nnnnnn
n
BG
BGBG
c
cc
BBBBBB
BBBBBBBBBB
2
1
2
1
21
2212
12111
∫=I
dxxHxFHF )()(
Finding Coefficients
• Matrix
does not depend on G(x)– Computed just once for a given basis
=
nnnn
n
BBBBBB
BBBBBBBBBB
21
2212
12111
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 approximate1. Compute dot products
2. … (next slide)
[ ]TnBGBGBG 21
Finding the coefficients
2. Compute coefficients as
=
−
nn BG
BGBG
c
cc
2
1
12
1
B
Orthonormal basis
• Orthonormal basis means
• If basis is orthonormal then
≠=
=jiji
BB ji 01
IB =
=
=
10
101
21
2212
12111
nnnn
n
BBBBBB
BBBBBBBBBB
Orthonormal basis
• If the basis is orthonormal, computation of approximation coefficients simplifies to
• We want orthonormal basis functions
=
nn BG
BGBG
c
cc
2
1
2
1
Orthonormal basis
• Projection: How “similar” is the given basis function to the function we’re approximating
∫ ×
×
×
∫∫
1c=
2c=
3c=
Original function Basis functions Coefficients
Another reason for orthonormalbasis 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
0, ϕθϕθ ml
n
l
lm
lmml YcG ∑∑
−
=
=
−=
=
Function approximation with SH
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
∫ ∫∫ ===ππ
ϕθθϕθϕθωωω2
0 0,,,, sin),(),()()( ddYGdYGYGc ml
Smlmlml
Product integral with SH
• Simplified indexing – Yi= Yl,m
– i = l (l+1) + m
• Two functions represented by SH )()(
2
0ωω i
n
iiYfF ∑
=
= )()(2
0ωω i
n
iiYgG ∑
=
=
∑∫=
=2
0)()(
n
iii
S
gfdGF ωωω
SH rotation
• Closed under rotation
• SH rotation matrix
• Fast rotation procedures exist
Representing BRDFs with SH
• For each ωo, the BRDF lobe is a spherical function
• Idea:– Discretize ωo
– For each such ωo, represent BRDF by a set of coeffs
Discretizing the hemisphere
• Want mapping between hemisphere and square – Low distortion (as uniform as possible)
– Continuous
– Fast mapping formulas
• Paraboloid mapping
Paraboloid mapping
s = x / (1-z)t = y / (1-z)