Introduction to Radiosity

Transcription

1 Introduction to Radiosity Produce photorealistic pictures using global illumination Mathematical basis from the theory of heat transfer Enables color bleeding Provides view independent representation Unfortunately, expensive Page 1 of 7

2 Introduction to Radiosity Produce photorealistic pictures using global illumination Mathematical basis from the theory of heat transfer Enables color bleeding Provides view independent representation Unfortunately, expensive Page 1 of 7 Flat shading is improved by discretizing environment into patches

3 Introduction to Radiosity Produce photorealistic pictures using global illumination Mathematical basis from the theory of heat transfer Enables color bleeding Provides view independent representation Unfortunately, expensive Page 1 of 7 Flat shading is improved by discretizing environment into patches Patches have an associated brightness or radiosity.

4 Introduction to Radiosity Produce photorealistic pictures using global illumination Mathematical basis from the theory of heat transfer Enables color bleeding Provides view independent representation Unfortunately, expensive Page 1 of 7 Flat shading is improved by discretizing environment into patches Patches have an associated brightness or radiosity. Radiosity constant over a patch The Cornell room painted by the hierarchical radiosity algorithm using interpolated shading.

5 Radiosity Basics The radiosity, B(x) is the total power leaving a point x per unit area The power emitted by a patch i is the sum of its intrinsic emitted power and fractions of the the power received from other patches in the environment. Only sources of light have intrinsic emitted power, all other patches appear bright because they receive light from other patches Page 2 of 7

6 Page 2 of 7 Radiosity Basics The radiosity, B(x) is the total power leaving a point x per unit area The power emitted by a patch i is the sum of its intrinsic emitted power and fractions of the the power received from other patches in the environment. Only sources of light have intrinsic emitted power, all other patches appear bright because they receive light from other patches Define form factor F ji : reaching patch i. Therefore A i B i = A i E i + ρ i n j=1 F ji B j A j It turns out that F ji A j = F i j A i fraction of energy leaving patch j and The net result is a system of equations of the form MB = E

7 Radiosity Basics 1 ρ 1 F 11 ρ 1 F 12 ρ 1 F 1n ρ 2 F 21 1 ρ 2 F 22 ρ 2 F 2n... ρ n F n1 ρ 1 F n2 1 ρ n F nn B 1 B 2. B n = E 1 E 2. E n Page 3 of 7 F ii is zero if the patch is flat F i j depends only on the geometry, and can be calculated independent of the light source

8 Radiosity Basics 1 ρ 1 F 11 ρ 1 F 12 ρ 1 F 1n ρ 2 F 21 1 ρ 2 F 22 ρ 2 F 2n... ρ n F n1 ρ 1 F n2 1 ρ n F nn B 1 B 2. B n = E 1 E 2. E n Page 3 of 7 F ii is zero if the patch is flat F i j depends only on the geometry, and can be calculated independent of the light source Currently we have considered the unoccluded case, that is we have not explicitly modeled the visibility between patches

9 Solution method Change the geometry of the scene Discretized environment Page 4 of 7 Change the wavelength dependent properties Change view Form factor calculations Full matrix solution Standard renderer View independent solution Specific view of the environment

10 So What s The Form Factor Page 5 of 7 If we consider a differential area da, the amount of power it sends out in a (small) cone is proportional to dacosθ dω, where dω is the solid angle subtended by any area in the cone The total power transmitted by the differential area in the hemisphere is da Ω cosθ dω = π da (why?) So the differential formfactor between areas da i and da j F dai da j = cosθ i cosθ j da j πr 2 z Surface da θ y N i θ i φ θ j da j Nj d ω x I( θ, φ ) j (area A j ) da i i (area A i )

11 So What s The Form Factor Page 6 of 7 Differential area da j to patch A i cosθ i cosθ j da j F dai A j = A j πr 2 Patch to patch form factor = Area average of F da j da i F i j = 1 A i A i cosθ i cosθ j da j da i A j πr 2 Notice that F i j A i = F ji A j

12 So What s The Form Factor Page 6 of 7 Differential area da j to patch A i cosθ i cosθ j da j F dai A j = A j πr 2 Patch to patch form factor = Area average of F da j da i F i j = 1 A i A i cosθ i cosθ j da j da i A j πr 2 Notice that F i j A i = F ji A j Notice also that the differential formfactor F xa j = cosθ x dω where ω is the solid angle subtended by patch A j and θ x is the angle made by the normal at a point x

13 Form Factor Computation: Hemicube PATCH j PATCH i Hemicube constructed over the center of patch. Divided into pixels each of which have form factors. Page 7 of 7 Form factor of patch j is the sum of form factors of pixels on to which it projects. The delta form factor for a pixel on the top face is the normalized value 1 π(1+x 2 +y 2 ) 2 Assumption patches are distant, and cube of size 2 units If two or more patches project on to a pixel, the nearest patch gets the form factor of the pixel