1、Radisoity1Angel: Interactive Computer Graphics 5E Addison-Wesley 2009原著Ed AngelProfessor of Computer Science, Electrical and Computer Engineering, and Media ArtsUniversity of New Mexico編輯 武漢大學(xué)計(jì)算機(jī)學(xué)院圖形學(xué)課程組2Angel: Interactive Computer Graphics 5E Addison-Wesley 2009IntroductionRay tracing is best with

2、many highly specular sufacesNot characteristic of real scenesRendering equation describes general shading problemRadiosity solves rendering equation for perfectly diffuse surfaces3Angel: Interactive Computer Graphics 5E Addison-Wesley 2009TerminologyEnergy light (incident, transmitted)Must be conser

3、vedEnergy flux = luminous flux = power = energy/unit timeMeasured in lumensDepends on wavelength so we can integrate over spectrum using luminous efficiency curve of sensorEnergy density () = energy flux/unit area4Angel: Interactive Computer Graphics 5E Addison-Wesley 2009TerminologyIntensity bright

4、ness Brightness is perceptual= flux/area-solid angle = power/unit projected area per solid angleMeasured in candela = I dA d5Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Rendering Eqn (Kajiya)Consider a point on a surface NIout(out)Iin(in)6Angel: Interactive Computer Graphics 5E Addiso

5、n-Wesley 2009Rendering EquationOutgoing light is from two sourcesEmissionReflection of ing lightMust integrate over all ing lightIntegrate over hemisphereMust account for foreshortening of ing light7Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Rendering EquationIout(out) = E(out) + 2Rb

6、d(out, in )Iin(in) cos d bidirectional reflection coefficientangle between normal and inemissionNote that angle is really two angles in 3D and wavelength is fixed8Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Rendering EquationRendering equation is an energy balanceEnergy in = energy ou

7、tIntegrate over hemisphereFredholm integral equationCannot be solved analytically in generalVarious approximations of Rbd give standard rendering modelsShould also add an occlusion term in front of right side to account for other objects blocking light from reaching surface9Angel: Interactive Comput

8、er Graphics 5E Addison-Wesley 2009Another versionConsider light at a point p arriving from pi(p, p) = (p, p)(p,p)+ (p, p, p)i(p, p)dpocclusion = 0 or 1/d2emission from p to plight reflected at p from all points p towards p10Angel: Interactive Computer Graphics 5E Addison-Wesley 2009RadiosityConsider

9、 objects to be broken up into flat patches (which may correspond to the polygons in the model)Assume that patches are perfectly diffuse reflectorsRadiosity = flux = energy/unit area/ unit time leaving patch11Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Notationn patches numbered 1 to n

10、bi = radiosity of patch Iai = area patch Itotal intensity leaving patch i = bi aiei ai = emitted intensity from patch Ii = reflectivity of patch Ifij = form factor = fraction of energy leaving patch j that reaches patch i12Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Radiosity Equation

11、energy balancebiai = eiai + i fjibjajreciprocityfijai = fjiaj radiosity equationbi = ei + i fijbj13Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Matrix Formb = bie = eiR = rijrij = i if i j rii = 0 F = fij14Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Matrix Formb = e - RF

12、bformal solutionb = I-RF-1eNot useful since n is usually very largeAlternative: use observation that F is sparseWe will consider determination of form factors later15Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Solving the Radiosity EquationFor sparse matrices, iterative methods usuall

13、yrequire only O(n) operations per iterationJacobis method bk+1 = e - RFbkGauss-Seidel: use immediate updates16Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Series Approximation1/(1-x) = 1 + x + x2+ b = I-RF-1e = e + RFe + (RF)2e + I-RF-1 = I + RF +(RF)2+17Angel: Interactive Computer Gra

14、phics 5E Addison-Wesley 2009Rendered Image18Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Patches19Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Computing Form FactorsConsider two flat patches20Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Using Differential Pa

15、tchesforeshortening21Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Form Factor Integralfij = (1/ai) ai ai (oij cos i cos j / r2 )dai dajocclusionforeshortening of patch iforeshortening of patch j22Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Solving the IntergralThere are

16、very few cases where the integral has a (simple) closed form solutionOcclusion further complicates solutionAlternative is to use numerical methodsTwo step process similar to texture mapping HemisphereHemicube23Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Form Factor Examples 124Angel:

17、Interactive Computer Graphics 5E Addison-Wesley 2009Form Factor Examples 225Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Form Factor Examples 326Angel: Interactive Computer Graphics 5E Addison-Wesley 2009HemisphereUse illuminating hemisphereCenter hemisphere on patch with normal pointi

18、ng upMust shift hemisphere for each point on patch27Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Hemisphere28Angel: Interactive Computer Graphics 5E Addison-Wesley 2009HemicubeEasier to use a hemicube instead of a hemisphereRule each side into “pixels”Easier to project on pixels which give delta form factors that can be added up to give desired from factorTo get a delta form factor we need only cast a ray through each pixel29Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Hemicube30Angel: Interactive Computer Gra