radiosity 2001.2.21 김 성 남. contents definition/goal basic radiosity method progressive...

RadiosityRadiosity

2001.2.21

김 성 남

ContentsContents

Definition/GoalBasic Radiosity MethodProgressive Radiosity MethodMesh substructuringHierarchical RadiosityRay Tracing and Radiosity

DefinitionDefinitionRadiosity

- method for describing diffuse reflection- We can accurately model diffuse reflecitons from a surface by considering the radiant energy transfers between surfaces → conservation of energy laws- independent on viewer

Ray Tracing Specular reflection- dependent on viewer

Radiosity- object 의 diffuse reflection 특성을 이용- light source 가 object surface 를 가질수 있음 .

Ray Tracing- object 의 specular reflection, transparency 특성을 이용- light, object surfaces 항상 분리

DefinitionDefinition

GoalGoalSimulate diffuse inter-object reflections

and shadows

Basic IdeaBasic IdeaTreat every polygon as light source

Advantages- Physically models shadows and indirect diffuse illumination- Independent of any viewpoint

Radiosity Equation- Equation Formulation

Bk : Radiosity of patch kEk : Emission of patch kρk : Reflectivity of patch kFjk : Form Factor between patch j and k

Basic Radiosity ModelBasic Radiosity Model

Bk = Ek + ρk BjFjk

n

∑j=1

Background- radian θ = s/r

- solid angle ω = A/r2

ω : steradian


s

r

θ

rω

A

Radiosity of patch : BBk : Radiosity of patch k

- total diffuse reflection rate of energy leaving surface k per unit area

dB : differential amount of radiant energy (joules/sec•meter2, watts/meter2)


xy

z

θ

ΦN

dB

dω

I : Intensity or luminance- radiant energy per unit time per unit projected area per unit solid angle (watts/meter2•steradians)

I = dB / dωcosΦassuming the surface is ideal diffuse reflector,I = constant

B = ∫hemi dB B = I ∫hemi cosΦdω

Basic Radiosity ModelBasic Radiosity ModelΦ

Φ

dA

dAcosΦ

dω = dS / r2 = sinΦdΦdθ

B = I ∫o ∫o cosΦsinΦdΦdθ = Iπ


xy

z

θ

ΦN

dB

dωdS

2π π/2

Enclose of surfaces for the radiosity model

Hk = BjFjk

Hk : Insident energyFjk : Form factor for surfaces j and k(fractional amount of surfaces j k)


∑j

Hk

Bk

Surface k

Bk = Ek + ρkHk


Bk = Ek + ρk BjFjk

n

∑j=1

Emission of patch : Eif surface k <> light source E = 0else rate of energy emitted from surface per unit area(watts/meter2)

Reflectivity factor of patch : ρpercent of incident light reflected in all directions


Form factor : Fenergy transfer from surface j to surface k

dBjdAj = (IjcosΦjdω)dAj

solid angle dω : the projection of area element dAk

perpendicular to the direction dBj

dω = dA / r2 = cosΦkdAk / r2


dω = dA / r2 = cosΦkdAk / r2

dBjdAj = IjcosΦjcosΦkdAjdAk / r2

Bj = πIj 이므로


ΦjNj

dBj

dω

Surface j

Surface k

Nk

Φk

dAj

dAkFdAj,dAk = Total energy leaving dAj

Energy incident on dAk

=IjcosΦjcosΦkdAjdAk

r2•

BjdAj

1

FdAj,dAk = πr2

cosΦjcosΦkdAk

- condition

Fjk = 1, for all k(conservation of energy)

AjFjk = AkFkj (uniform light reflection)

Fjj = 0, for all j ( assuming only plane or convex surface patches)


πr2

cosΦjcosΦkFdAj,Ak = dAk∫surf j

Fjk = A1 ∫surf j∫surf k πr2

cosΦjcosΦk dAkdAj

n

∑j=1

Form factor 의 계산속도 향상을 위해서

HemiSphere HemiCube(spherical surface) (linear(plane) surface)


HemiCube Method- project scene onto hemicube positioned at centroid of patch i- count pixel coverage to determine form factors Fi - can use H/W z-buffer → speed up → use Image precision, prone to aliasing


.

.

....

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.

....

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.

.=

(1 – ρkFk)Bk - ρk BjFjk = Ek, k =1,2,3,…,n

Matrix Formulation → Gauss-Seidel Iteration : O(n2) 1 – ρ1F11 – ρ1F12 …. – ρ1F1n B1 E1 1 – ρ2F21 1 – ρ2F22 …. – ρ2F2n B2 E2

1 – ρnFn1 – ρnFn2 …. 1 – ρnFnn Bn En

Bk = Ek + ρk BjFjk

n

∑j=1


Matrix Solution Methods- Gaussian elimination : O(n3)

- LU Decomposition decomposing an N x N matrix A a lower Triangular Matrix L a upper Triangular Matrix U LU = A


L11 0 0 U11 U12 U13 A11 A12 A13L21 L22 0 0 U22 U23 A21 A22 A23L31 L32 L33 0 0 U33 A31 A32 A33

=

Progressive RefinementProgressive RefinementGoal

- Speep up the calculation time- Reduce storage requirement

Method- initially, set Bk = Ek for all surface patches- select the patch with the highest values • store one hemicube • light source are chosen first - calculate radiosity- discard this value- next step, repeat.

Display- from dark scene to a fully illuminated one- to produce more useful initial view, can set ambient light- at each stage of iteration, reduce ambient light

Speed : O(n2)- but get pretty good solutions more quickly

Progressive RefinementProgressive Refinement

Progressive RefinementProgressive RefinementExample

Iteration 1 Iteration 2

Iteration 24 Iteration 100

Mesh SubstructuringMesh SubstructuringAdaptive subdivision

- To improve computation time for n2 form factors

partition each patch into multiple subpatch along gradients of radiosity

using the hemicubereplace more accurate area-weighted average

of the form factors from its subpatchesuntil the differences reach an acceptable levelSpeed : O(np)

- p : partition

Mesh SubstructuringMesh Substructuring

Multiresolution Computation- substructure patches into quad-tree- Transfer energy using lower resolution mesh elements if can do so within error tolerance

Speed : O(n)

Hierarchical RadiosityHierarchical Radiosity

Substructure & HierarchicalSubstructure & HierarchicalExample : using hemicube to compute form factors

Hierarchical radiosity Mesh substructuring

Dilemma- Radiosity : diffuse reflection- Ray Tracing : specular reflection

Combine

Ray Tracing and RadiosityRay Tracing and Radiosity

Ray Tracing and RadiosityRay Tracing and Radiosity

Diffuse radiosity Diffuse first pass & ray-tracing second pass

Diffuse first passwith extended

from form factors & ray tracing second pass

Example

radiosity 2001.2.21 김 성 남. contents definition/goal basic radiosity method progressive...

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