Radiosity 김 성 남

Contents Definition/Goal Basic Radiosity Method Progressive Radiosity Method Mesh substructuring Hierarchical Radiosity Ray Tracing and Radiosity

Definition Radiosity - 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 Definition

Goal Simulate diffuse inter-object reflections and shadows

Basic Idea Treat every polygon as light source Advantages - Physically models shadows and indirect diffuse illumination - Independent of any viewpoint

Radiosity Equation - Equation Formulation B k : Radiosity of patch k E k : Emission of patch k ρ k : Reflectivity of patch k F jk : Form Factor between patch j and k Basic Radiosity Model B k = E k + ρ k B j F jk n ∑ j=1

Background - radian θ = s/r - solid angle ω = A/r 2 ω : steradian Basic Radiosity Model s r θ r ω A

Radiosity of patch : B B k : Radiosity of patch k - total diffuse reflection rate of energy leaving surface k per unit area dB : differential amount of radiant energy (joules/secmeter 2, watts/meter 2 ) Basic Radiosity Model x y z θ Φ N dB dωdω

I : Intensity or luminance - radiant energy per unit time per unit projected area per unit solid angle (watts/meter 2 steradians) I = dB / dωcosΦ assuming the surface is ideal diffuse reflector, I = constant B = ∫ hemi dB B = I ∫ hemi cosΦdω Basic Radiosity Model Φ Φ dA dAcosΦ

dω = dS / r 2 = sinΦdΦdθ B = I ∫ o ∫ o cosΦsinΦdΦdθ = Iπ Basic Radiosity Model x y z θ Φ N dB dωdω dS 2π2π π/2

Enclose of surfaces for the radiosity model H k = B j F jk H k : Insident energy F jk : Form factor for surfaces j and k (fractional amount of surfaces j k) Basic Radiosity Model ∑j∑j HkHk BkBk Surface k

B k = E k + ρ k H k Basic Radiosity Model B k = E k + ρ k B j F jk n ∑ j=1

Emission of patch : E if surface k <> light source E = 0 else rate of energy emitted from surface per unit area(watts/meter 2 ) Reflectivity factor of patch : ρ percent of incident light reflected in all directions Basic Radiosity Model

Form factor : F energy transfer from surface j to surface k dB j dA j = (I j cosΦ j dω)dA j solid angle dω : the projection of area element dA k perpendicular to the direction dB j dω = dA / r 2 = cosΦ k dA k / r 2 Basic Radiosity Model

dω = dA / r 2 = cosΦ k dA k / r 2 dB j dA j = I j cosΦ j cosΦ k dA j dA k / r 2 B j = πI j 이므로 Basic Radiosity Model ΦjΦj NjNj dB j dωdω Surface j Surface k NkNk ΦkΦk dA j dA k F dAj,dAk = Total energy leaving dA j Energy incident on dA k = I j cosΦ j cosΦ k dA j dA k r2r2 B j dA j 1 F dAj,dAk = πr2πr2 cosΦ j cosΦ k dA k

- condition F jk = 1, for all k(conservation of energy) A j F jk = A k F kj (uniform light reflection) F jj = 0, for all j ( assuming only plane or convex surface patches) Basic Radiosity Model πr2πr2 cosΦ j cosΦ k F dAj,Ak = dA k ∫ surf j F jk = A 1 ∫ surf j ∫ surf k πr2πr2 cosΦ j cosΦ k dA k dA j n ∑ j=1

Form factor 의 계산속도 향상을 위해서 HemiSphere HemiCube (spherical surface) (linear(plane) surface) Basic Radiosity Model

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

= (1 – ρ k F k )B k - ρ k B j F jk = E k, k =1,2,3,…,n Matrix Formulation → Gauss-Seidel Iteration : O(n 2 ) 1 – ρ 1 F 11 – ρ 1 F 12 …. – ρ 1 F 1n B 1 E 1 1 – ρ 2 F 21 1 – ρ 2 F 22 …. – ρ 2 F 2n B 2 E 2 1 – ρ n F n1 – ρ n F n2 …. 1 – ρ n F nn B n E n B k = E k + ρ k B j F jk n ∑ j=1 Basic Radiosity Model

Matrix Solution Methods - Gaussian elimination : O(n 3 ) - LU Decomposition decomposing an N x N matrix A a lower Triangular Matrix L a upper Triangular Matrix U LU = A Basic Radiosity Model L U11 U12 U13 A11 A12 A13 L21 L U22 U23 A21 A22 A23 L31 L32 L U33 A31 A32 A33 =

Progressive Refinement Goal - Speep up the calculation time - Reduce storage requirement Method - initially, set B k = E k 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(n 2 ) - but get pretty good solutions more quickly Progressive Refinement

Example Iteration 1 Iteration 2 Iteration 24 Iteration 100

Mesh Substructuring Adaptive subdivision - To improve computation time for n 2 form factors

partition each patch into multiple subpatch along gradients of radiosity using the hemicube replace more accurate area-weighted average of the form factors from its subpatches until the differences reach an acceptable level Speed : O(np) - p : partition Mesh 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 Radiosity

Substructure & Hierarchical Example : using hemicube to compute form factors Hierarchical radiosity Mesh substructuring

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

Diffuse radiosityDiffuse first pass & ray-tracing second pass Diffuse first pass with extended from form factors & ray tracing second pass Example