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Function Representation & Spherical Harmonics

Function Representation & Spherical Harmonics. 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).

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Function Representation & Spherical Harmonics

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  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

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