Function Representation

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Function Representation Spherical Harmonics
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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 ci gives
  an approximation of G(x)

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Examples of basis functions
Tent function (linear interpolation)
Associated Legendre polynomials
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Function approximation

• Linear combination
• sum of scaled basis functions

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Function approximation

• Linear combination
• sum of scaled basis functions

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Finding the coefficients

• How to find coefficients ci?
• Minimize an error measure
• What error measure?
• L2 error

Approximated function
Original function
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Finding the coefficients

• Minimizing EL2 leads toWhere

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Finding the coefficients

• Matrixdoes not depend on G(x)
• Computed just once for a given basis

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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)

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Finding the coefficients

• Compute coefficients as

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Orthonormal basis

• Orthonormal basis means
• If basis is orthonormal then

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Orthonormal basis

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

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Orthonormal basis

• Projection How similar is the given basis
  function to the function were approximating

Original function
Basis functions
Coefficients
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Another reason for orthonormal basis functions

• Intergral of product dot product of coefficients

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Application to GI

• Illumination integral
• Lo ? Li(wi) BRDF (wi) cosqi dwi

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Spherical Harmonics
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Spherical harmonics

• Spherical function approximation
• Domain I unit sphere S
• directions in 3D
• Approximated function G(?,f)
• Basis functions Yi(?,f) Yl,m(?,f)
• indexing i l (l1) m

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The SH Functions
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Spherical harmonics

• K normalization constant
• P Associated Legendre polynomial
• Orthonormal polynomial basis on (0,1)
• In general Yl,m(?,f) K . ?(f) . Pl,m(cos ?)
• Yl,m(?,f) is separable in ? and f

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Function approximation with SH

• napproximation order
• There are n2 harmonics for order n

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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

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Function approximation with SH
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Product integral with SH

• Simplified indexing
• Yi Yl,m
• i l (l1) m
• Two functions represented by SH