1. Function Representation & Spherical Harmonics

2. Function approximation G(x)... function to represent B 1 (x), B 2 (x), … B n (x) … basis functions G(x) is a linear combination of basis functions Storing a finite number of coefficients c i gives 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 c i ? – Minimize an error measure What error measure? – L 2 error Approximated function Original function

7. Finding the coefficients Minimizing E L 2 leads to Where

8. Finding the coefficients Matrix does not depend on G(x) – Computed just once for a given basis

9. Finding the coefficients Given a basis {B i (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)

10. Finding the coefficients 2. 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 functionBasis functions Coefficients

14. Another reason for orthonormal basis functions f(x) = fifi B i (x) g(x) = gigi B i (x)  f(x)g(x)dx = fifi gigi Intergral of product = dot product of coefficients

15. Application to GI Illumination integral L o = ∫ L i (  i ) BRDF (  i ) cos  i d  i

16. Spherical Harmonics

17. Spherical harmonics Spherical function approximation Domain I = unit sphere S – directions in 3D Approximated function: G(θ,φ) Basis functions: Y i (θ,φ)= Y l,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: Y l,m (θ,φ) = K. Ψ(φ). P l,m (cos θ) – Y l,m (θ,φ) is separable in θ and φ

20. Function approximation with SH n…approximation order There are n 2 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 – Y i = Y l,m – i = l (l+1) + m Two functions represented by SH