1. Radiosity 1

2. 2 Introduction Ray tracing best with many highly specular surfaces ­Not real scenes Rendering equation describes general shading problem Radiosity solves rendering equation for perfectly diffuse surfaces

3. 3 Terminology Energy ~ light (incident, transmitted) ­Must be conserved Energy flux = luminous flux = power = energy/unit time ­Measured in lumens ­Depends on wavelength so can integrate over spectrum using luminous efficiency curve of sensor Energy density (Φ) = energy flux/unit area

4. 4 Terminology Intensity ~ brightness/lightness ­Brightness/lightness perceptual = flux/area-solid angle = power/unit projected area per solid angle ­Measured in candela Φ = ∫ ∫ I dA dω

5. 5 Rendering Eqn (Kajiya) Consider point on surface N I out (Φ out ) I in (Φ in )

6. 6 Rendering Equation Outgoing light from two sources ­Emission ­Reflection of incoming light Must integrate over all incoming light ­Integrate over hemisphere Must account for foreshortening of incoming light

7. 7 Rendering Equation I out (Φ out ) = E( Φ out ) + ∫ 2π R bd (Φ out, Φ in )I in (Φ in ) cos θ dω bidirectional reflection coefficient angle between normal and Φin emission Note that angle is really two angles in 3D and wavelength is fixed

8. 8 Rendering Equation Rendering equation = energy balance ­Energy in = energy out Integrate over hemisphere Fredholm integral equation ­Cannot solve analytically in general Various approximations of R bd give standard rendering models Should also add occlusion term in front of right side to account for other objects blocking light from reaching surface

9. 9 Another version Consider light at point p arriving from p ’ i(p, p’) = υ(p, p’)(ε(p,p’) + ∫ ρ(p, p’, p’’)i(p’, p’’)dp’’ occlusion = 0 or 1/d 2 emission from p’ to p light reflected at p’ from all points p’’ towards p

10. 10 Radiosity Consider objects to be broken up into flat patches ­may correspond to polygons in model Assume patches = perfectly diffuse reflectors Radiosity = flux = energy/unit area/ unit time leaving patch

11. 11 Notation n patches numbered 1 to n b i = radiosity of patch i a i = area of patch i b i a i = total intensity leaving patch i e i a i = emitted intensity from patch i ρ i = reflectivity of patch i f ij = form factor = fraction of energy leaving patch j that reaches patch i

12. form factor In physics, (usually written e) of material sometimes called emissivity = proportion of energy transmitted by object that can be transferred to another object en.wikipedia.org/wiki/Form_factor_(radiati ve_transfer) en.wikipedia.org/wiki/Form_factor_(radiati ve_transfer) 12

13. 13 Radiosity Equation energy balance b i a i = e i a i + ρ i ∑ f ji b j a j reciprocity f ij a i = f ji a j radiosity equation b i = e i + ρ i ∑ f ij b j emitted reflected

14. 14 Matrix Form b = [b i ] e = [e i ] R = [r ij ] r ij = ρ i if i ≠ jr ii = 0 F = [f ij ]

15. 15 Matrix Form b = e - RFb formal solution b = [I-RF] -1 e Not useful since n usually very large Alternative: use observation that F is sparse Will consider determination of form factors later

16. 16 Solving the Radiosity Equation For sparse matrices, iterative methods usually require only O(n) operations per iteration Jacobi’s method b k+1 = e - RFb k Gauss-Seidel: use immediate updates

17. 17 Series Approximation 1/(1-x) = 1 + x + x 2 + …… b = [I-RF] -1 e = e + RFe + (RF) 2 e +… [I-RF] -1 = I + RF +(RF) 2 +…

18. 18 Rendered Image

19. 19 Patches

20. 20 Computing Form Factors Consider two flat patches

21. 21 Using Differential Patches foreshortening

22. 22 Form Factor Integral f ij = (1/a i ) ∫ ai ∫ ai (o ij cos θ i cos θ j / πr 2 )da i da j occlusion foreshortening of patch i foreshortening of patch j

23. 23 Solving the Intergral Very few cases where integral has (simple) closed form solution ­Occlusion further complicates solution Alternative: use numerical methods Two step process similar to texture mapping ­Hemisphere ­Hemicube

24. 24 Form Factor Examples 1

25. 25 Form Factor Examples 2

26. 26 Form Factor Examples 3

27. 27 Hemisphere Use illuminating hemisphere Center hemisphere on patch with normal pointing up Must shift hemisphere for each point on patch

28. 28 Hemisphere

29. 29 Hemicube Easier to use hemicube instead of hemisphere Rule each side into “pixels” Easier to project on pixels  give delta form factors  can be added up to give desired form factor To get delta form factor need only cast ray through each pixel

30. 30 Hemicube

31. 31 Instant Radiosity Want to use graphics system if possible Suppose make one patch emissive Light from this patch distributed among other patches Shade of other patches ~ form factors Must use multiple OpenGL point sources to approximate uniformly emissive patch