Function Representation & Spherical Harmonics

1. Function Representation & Spherical Harmonics

2. 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 cigives an approximation of G(x)

3. Examples of basis functions Tent function (linear interpolation) Associated Legendre polynomials

4. Function approximation • Linear combination • sum of scaled basis functions

5. Function approximation • Linear combination • sum of scaled basis functions

6. Finding the coefficients • How to find coefficients ci? • Minimize an error measure • What error measure? • L2 error Original function Approximated function

7. Finding the coefficients • Minimizing EL2 leads toWhere

8. Finding the coefficients • Matrixdoes not depend on G(x) • Computed just once for a given basis

9. Finding the coefficients • Given a basis {Bi(x)} • Compute matrix B • Compute its inverse B-1 • Given a function G(x) to approximate • Compute dot products • … (next slide)

10. Finding the coefficients • Compute coefficients as

11. Orthonormal basis • Orthonormal basis means • If basis is orthonormal then

12. Orthonormal basis • If the basis is orthonormal, computation of approximation coefficients simplifies to • We want orthonormal basis functions

13. Orthonormal basis • Projection: How “similar” is the given basis function to the function we’re approximating Original function Basis functions Coefficients

14. fi Bi(x) f(x) = fi gi gi Bi(x)  g(x) = f(x)g(x)dx = Another reason for orthonormal basis functions • Intergral of product = dot product of coefficients

15. Application to GI • Illumination integral • Lo= ∫Li(wi)BRDF(wi) cosqidwi

16. Spherical Harmonics

17. 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

18. The SH Functions

19. 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

20. Function approximation with SH • n…approximation order • There are n2 harmonics for order n

21. 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

22. Function approximation with SH

23. Product integral with SH • Simplified indexing • Yi= Yl,m • i = l(l+1)+m • Two functions represented by SH