1、Radisoity,Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts Director, Arts Technology Center University of New Mexico,2,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Introduction,Ray tracing is best with many highly specular sufaces Not charact

2、eristic of real scenes Rendering equation describes general shading problem Radiosity solves rendering equation for perfectly diffuse surfaces,3,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Terminology,Energy light (incident, transmitted) Must be conserved Energy flux = luminous flux

3、= power = energy/unit time Measured in lumens Depends on wavelength so we can integrate over spectrum using luminous efficiency curve of sensor Energy density () = energy flux/unit area,4,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Terminology,Intensity brightness Brightness is perce

4、ptual = flux/area-solid angle = power/unit projected area per solid angle Measured in candela = I dA d,5,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Rendering Eqn (Kajiya),Consider a point on a surface,N,Iout(out),Iin(in),6,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,

5、Rendering Equation,Outgoing light is 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,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Rendering Equation,Iout(out) =

6、E(out) + 2Rbd(out, in )Iin(in) cos d,bidirectional reflection coefficient,angle between normal and in,emission,Note that angle is really two angles in 3D and wavelength is fixed,8,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Rendering Equation,Rendering equation is an energy balance E

7、nergy in = energy out Integrate over hemisphere Fredholm integral equation Cannot be solved analytically in general Various approximations of Rbd give standard rendering models Should also add an occlusion term in front of right side to account for other objects blocking light from reaching surface,

8、9,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Another version,Consider light at a point p arriving from p,i(p, p) = (p, p)(p,p)+ (p, p, p)i(p, p)dp,occlusion = 0 or 1/d2,emission from p to p,light reflected at p from all points p towards p,10,Angel: Interactive Computer Graphics 4E A

9、ddison-Wesley 2005,Radiosity,Consider objects to be broken up into flat patches (which may correspond to the polygons in the model) Assume that patches are perfectly diffuse reflectors Radiosity = flux = energy/unit area/ unit time leaving patch,11,Angel: Interactive Computer Graphics 4E Addison-Wes

10、ley 2005,Notation,n patches numbered 1 to n bi = radiosity of patch I ai = area patch I total intensity leaving patch i = bi ai ei ai = emitted intensity from patch I i = reflectivity of patch I fij = form factor = fraction of energy leaving patch j that reaches patch i,12,Angel: Interactive Compute

11、r Graphics 4E Addison-Wesley 2005,Radiosity Equation,energy balance,biai = eiai + i fjibjaj,reciprocity,fijai = fjiaj,radiosity equation,bi = ei + i fijbj,13,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Matrix Form,b = bi,e = ei,R = rij,rij = i if i j,rii = 0,F = fij,14,Angel: Interac

12、tive Computer Graphics 4E Addison-Wesley 2005,Matrix Form,b = e - RFb,formal solution,b = I-RF-1e,Not useful since n is usually very large Alternative: use observation that F is sparse We will consider determination of form factors later,15,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005

13、,Solving the Radiosity Equation,For sparse matrices, iterative methods usually require only O(n) operations per iteration Jacobis method,bk+1 = e - RFbk,Gauss-Seidel: use immediate updates,16,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Series Approximation,1/(1-x) = 1 + x + x2+ ,b =

14、I-RF-1e = e + RFe + (RF)2e +,I-RF-1 = I + RF +(RF)2+,17,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Rendered Image,18,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Patches,19,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Computing Form Factors,Consider two

15、 flat patches,20,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Using Differential Patches,foreshortening,21,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Form Factor Integral,fij = (1/ai) ai ai (oij cos i cos j / r2 )dai daj,occlusion,foreshortening of patch i,foreshorten

16、ing of patch j,22,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Solving the Intergral,There are very few cases where the integral has a (simple) closed form solution Occlusion further complicates solution Alternative is to use numerical methods Two step process similar to texture mappi

17、ng Hemisphere Hemicube,23,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Form Factor Examples 1,24,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Form Factor Examples 2,25,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Form Factor Examples 3,26,Angel: Interacti

18、ve Computer Graphics 4E Addison-Wesley 2005,Hemisphere,Use illuminating hemisphere Center hemisphere on patch with normal pointing up Must shift hemisphere for each point on patch,27,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Hemisphere,28,Angel: Interactive Computer Graphics 4E Addison-Wesley 2005,Hemicube,Easier to use a hemicube instead of a hemisphere Rule ea