1. Finite element methods
László Szirmay-Kalos

2. Representation of functions by finite data
Finite function series: L(p)  Lj bj (p) 1
box 1
tent b1 b1 b2 b2 b3 b3
Piece-wise constant
Piece-wise linear

3. 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)

4. Rendering equation in function space
L*(p) = Lj bj (p)  L L L
+Le b2 b1 L*
Original rendering
equation
Finite element
approximation

5. 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*

6. L* Adjoint base <bi , bj’> = 1 if i=j and 0 otherwise b2 L* b2’
Equality is required in a subspace of adjoint basis functions: b1’, b2’ ,..., bn’
orthogonality:
<bi , bj’> = 1 if i=j and
0 otherwise b2
L*
+Le L*
b2’ b1
projection
b1’

7. Derivation of the projected rendering equation
FEM:
Projecting to an adjoint base: < •, bi’>
L(p)  Lj bj (p)
p=(x,w)
Lj bj (p)  Lje bj (p) + t Lj bj (p)
Li = Lie +  Lj <tbj ,bi’>

8. Projected rendering equation = linear equation for Lj
Rij = <tbj ,bi’>
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)

9. Galerkin’s method <bi ,bi’>=1  bi’ = bi /<bi ,bi>
The base and the adjoint base are the same except for a normalization constant:
<bi ,bi’>=1  bi’ = bi /<bi ,bi>
Error is orthogonal to the original base
Point collocation method
equality is required at finite dot points pi
bi’ (p) = (p - pi)

10. Example: Diffuse case Galerkin+constant basis
<u,v>=Su(x)v(x)dx  <bi,bi> = Ai Aj
bi is 1 on patch i w’
h(x,-w’) ’ Ai x
<tbj,bi’>= 1/Ai Ai bj (h(x,-w’)) fr(x) cos’ dw’dx

11. Solid angle  Area integralAj 
h(x,-w’) = y w’ ’ Ai
dw’= dy cos / |x - y|2 x
<tbj,bi>=1/AiAiAjv(x,y) fr(x) dydx = ai Fij
cos’ cos 
|x - y|2
Patch-patch form factor:
Albedo:
cos’ cos 
ai = fri  Fij=1/Ai AiAj v(x,y) dydx
 |x - y|2

12. Example: Diffuse case Point collocation+linear basisbi
bi’= (x - xi) Aj w’
h(x,-w’) ’ Ai xi
<tbj,bi’>=  bj (h(xi,-w’)) fr(xi) cos’ dw’
cos’ cos 
= Aiv(xi,y) bj (y) fr(xi) dy = ai Fij point-patch
|xi - y|2