Finite element methods

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Finite element methods
László Szirmay-Kalos
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Representation of functions by finite data
Finite function series L(p) ? ??Lj bj (p)
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1
box
tent
b1
b1
b2
b2
b3
b3
Piece-wise constant
Piece-wise linear
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Representation of the radiance

• Finite elements L(p) ? ??Lj bj (p)
• bj total function system
• box, tent, harmonic, Chebishev, etc.
• diffuse radiosity piece-wise constant
• non-diffuse case
• partitioned hemisphere (piece-wise constant),
• directional distributions (spherical harmonics)
• illumination networks (links)

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Rendering equation in function space
L(p) ??Lj bj (p)
?
?L
L
L
Le
b2
b1
L
Original rendering equation
Finite element approximation
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Projected rendering equation
?L
L(p) ??Lj bj (p)
Basis functions
b2
Le
b1
L
b2
?F L
b1
Adjoint base
Le
L Le ?F L
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Adjoint base

• Equality is required in a subspace of adjoint
  basis functions b1, b2 ,..., bn
• orthogonality

ltbi , bjgt 1 if ij and 0
otherwise
b2
?L
Le
L
b2
b1
projection
b1
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Derivation of the projected rendering equation

• FEM
• Projecting to an adjoint base lt , bigt

L(p) ? ??Lj bj (p)
p(x,w)
??Lj bj (p) ? ??Lje bj (p) t ??Lj bj (p)
Li Lie ?? Lj lttbj ,bigt
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Projected rendering equation linear equation
for Lj
Rij lttbj ,bigt
L Le R L
FEM 1. define basis functions and adjoint
basis function tesselation, function shape 2.
Evaluate Rij 3. Solve the linear equation for
L1, L2 ,, Ln 4. For any p L(p) ? ??Lj bj (p)
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Galerkins method

• The base and the adjoint base are the same except
  for a normalization constant
• ltbi ,bigt1 ? bi bi /ltbi ,bigt
• Error is orthogonal to the original base
• Point collocation method
• equality is required at finite dot points pi
• bi (p) ?(p - pi)

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Example Diffuse caseGalerkinconstant basis
ltu,vgt?Su(x)v(x)dx ? ltbi,bigt Ai
Aj
bi is 1 on patch i
w
h(x,-w)
?
Ai
x
lttbj,bigt 1/Ai ?Ai?? bj (h(x,-w)) fr(x) cos?
dwdx
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Solid angle ? Area integral
Aj
?
h(x,-w) y
w
?
Ai
dw dy cos ?/ x - y2
x
lttbj,bigt1/Ai?Ai?Ajv(x,y) fr(x)
dydx ai Fij
cos? cos ?
x - y2
Patch-patch form factor
Albedo
cos? cos ?
ai fri ? Fij1/Ai ?Ai?Aj v(x,y)
dydx
? x - y2
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Example Diffuse casePoint collocationlinear
basis
bi
bi ?(x - xi)
Aj
w
h(x,-w)
?
Ai
xi
lttbj,bigt ?? bj (h(xi,-w)) fr(xi) cos? dw
cos? cos ?
?Aiv(xi,y) bj (y) fr(xi) dy
ai Fij point-patch
xi - y2