Slide 1 Lecture 23 6.837 Fall ‘00 RadiosityReferences:Cohen and Wallace,Radiosity and RealisticImage SynthesisSillion and Puech,Radiosity and GlobalIlluminationThanks to François Sillion forimagesAn early application of radiative heat transfer in stables.Slide 2 Lecture 23 6.837 Fall ‘00 Why Radiosity?A powerful demonstration introduced by Goral et al. of the differences between radiosity andtraditional ray tracing is provided by a sculpture by John Ferren. The sculpture consists of a series ofvertical boards painted white on the faces visible to the viewer. The back faces of the boards arepainted bright colors. The sculpture is illuminated by light entering a window behind the sculpture,so light reaching the viewer first reflects off the colored surfaces, then off the white surfaces beforeentering the eye. As a result, the colors from the back boards “bleed” onto the white surfaces.eyeSlide 3 Lecture 23 6.837 Fall ‘00 Radiosity vs. Ray TracingOriginal sculpture litby daylight from the rear.Image rendered with radiosity. note color bleeding effects.Ray traced image. A standardRay tracer cannot simulate theinterreflection of light between diffuse Surfaces.Slide 4 Lecture 23 6.837 Fall ‘00 Ray Tracing vs. RadiosityRay tracing is an image-space algorithm, while radiosity is computed in object-space.Because the solution is limited by the view, ray tracing is often said to provide a view-dependent solution, although this is somewhat misleading in that it implies that theradiance itself is dependent on the view, which is not the case. The term view-independentrefers only to the use of the view to limit the set if locations and directions for which theradiance is computed.Slide 5 Lecture 23 6.837 Fall ‘00 Radiosity IntroductionThe radiosity approach to rendering has its basis in the theory of heat transfer.This theory was applied to computer graphics in 1984 by Goral et al.Surfaces in the environment are assumed to be perfect (or Lambertian) diffusers,reflectors, or emitters. Such surfaces are assumed to reflect incident light inall directions with equal intensity.A formulation for the system of equations is facilitated by dividing theenvironment into a set of small areas, or patches. The radiosity over a patch isconstant.The radiosity, B, of a patch is the total rate of energy leaving a surface and isequal to the sum of the emitted and reflected energies:Slide 6 Lecture 23 6.837 Fall ‘00 Interchange Between PatchesWe can set up an equation that relates the energy reflected from a patch to anyself-emitted energy plus the energy incoming from all other patches as follows:rdAidAjθjθijiii ii i j jdAdAjBdA EdA B dA Fρ=+òRadiosity x area = emitted energy + reflected energySlide 7 Lecture 23 6.837 Fall ‘00 Radiosity EquationAiAjå+=j=1jijiiiBFEBρForm factor• discrete representation• iterative solution• costly geometric/visibility calculationsFor an environment that has been discretized into n patches, over which theradiosity is constant, (i.e. both B and E are constant across a patch), we have thebasic radiosity relationship:reflectivitynSlide 8 Lecture 23 6.837 Fall ‘00 The Radiosity MatrixA solution yields a single radiosity value Bi for each patch in the environment – a view-independent solution. The Bi values can be used in a standard renderer and a particularview of the environment constructed from the radiosity solution.12nBBBéùêúêúêúêúêúëûM12nEEEéùêúêúêúêúêúëûM=111 112 11221 2221111nnn nnnFF FFFFFρρ ρρρρρ−− −éùêú−−êúêúêú−−êúëûLMOLLSuch an equation exists for each patch, and in a closed environment, a set of n simultaneousequations in n unknown Bi values is obtained:iBiBSlide 9 Lecture 23 6.837 Fall ‘00 Standard Solution of the Radiosity Matrix12121112ii iii ii iinnnnBBBEBFBEBFFBEBEρρ ρéùéù éùéùêúêú êúêúêúêú êúêúêúêú êúêú=+êúêú êúêúêúêú êúêúêúêú êúêúêúêú êúêúêúêúêú êúëûëûëû ëûMMMM MMLThe radiosity of a single patch i is updated for each iteration by gathering radiosities from all other patches:Slide 10 Lecture 23 6.837 Fall ‘00 Computing Vertex Radiosities✔Recall that radiosity values areconstant over the extent of a patch.✔A standard renderer requiresvertex radiosities (intensities).These can be obtained for a vertexby computing the average of theradiosities of patches that contributeto the vertex under consideration.✔Vertices on the edge of a surfacecan be allocated values byextrapolation through interior vertexvalues, as shown on the right:Slide 11 Lecture 23 6.837 Fall ‘00 Stages in a Radiosity SolutionInput of scene geometryInput of reflectance propertiesViewing conditionsVisualizationSolution to the systemof equationsRadiosity solutionForm factorcalculationRadiosity imageSlide 12 Lecture 23 6.837 Fall ‘00 Progressive Refinement✔ The idea of progressive refinement is to provide a quickly rendered image tothe user that is then gracefully refined toward a more accurate solution. Theradiosity method is especially amenable to this approach.✔ The two major practical problems of the radiosity method are the storage costsand the calculation of the form factors.✔ The requirements of progressive refinement and the elimination ofprecalculation and storage of the form factors are met by a restructuring of theradiosity algorithm.✔ The key idea is that the entire image is updated at every iteration, rather than asingle patch.Slide 13 Lecture 23 6.837 Fall ‘00 Reordering the Solution for PRShooting: the radiosity of all patches is updated for each iteration:11 1122 22iiinn nniBB FBB FBBB Fρρρéùéùé ùéùêúêúê úêúêúêúê úêúêúêúê úêú=+êúêúê úêúêúêúê úêúêúêúê úêúêúêúê úêúêúêúêúê úëûëûëûë ûMLLLMMMLLMMSlide 14 Lecture 23 6.837 Fall ‘00 Progressive Refinement Pseudocode}i;element ofintensity the as B using imagedisplay 0B}rad;BB rad;BB;B rad{ element)(every for ;largest is Bthat such i, pick{ converged)(not whileiijjjjii=∆∆+=∆+∆=∆ρ∗∆=∆∗∆jijiFASlide 15 Lecture 23 6.837 Fall ‘00 Progressive Refinement w/out Ambient TermSlide 16 Lecture 23 6.837 Fall ‘00 Progressive Refinement with Ambient TermSlide 17 Lecture 23 6.837 Fall ‘00 Form Factor DeterminationAjAjr = 1FdAi,AjdAiThe Nusselt analog: the form factor of a patch is equivalent to the faction of thethe unit
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