RadiosityWhy Radiosity?Radiosity vs. Ray TracingRay Tracing vs. RadiosityRadiosity IntroductionSolving the rendering equationContinuous Radiosity EquationDiscrete Radiosity EquationThe Radiosity MatrixStandard Solution of the Radiosity MatrixComputing Vertex RadiositiesStages in a Radiosity SolutionProgressive RefinementReordering the Solution for PRProgressive Refinement PseudocodeProgressive Refinement w/out Ambient TermProgressive Refinement with Ambient TermFinite elementsForm Factor DeterminationHemicube AlgorithmForm factor using ray-castingIncreasing the Accuracy of the SolutionAdaptive Subdivision of PatchesAdaptive Subdivision PseudocodeStructure of the SolutionExamplesPowerPoint PresentationLightscape http://www.lightscape.comSlide 29Slide 30Slide 31Discontinuity meshingSlide 33ComparisonHierarchical approachOther basis functionsNext Time: AnimationSlide 1 Lecture 20 6.837 Fall ‘01 RadiosityReferences: Cohen and Wallace, Radiosity and Realistic Image SynthesisSillion and Puech, Radiosity and Global IlluminationThanks to Leonard McMillan for the slidesThanks to François Sillion for imagesAn early application of radiative heat transfer in stables.Slide 2 Lecture 20 6.837 Fall ‘01 Why Radiosity?A powerful demonstration introduced by Goral et al. of the differences between radiosity and traditional ray tracing is provided by a sculpture by John Ferren. The sculpture consists of a series of vertical boards painted white on the faces visible to the viewer. The back faces of the boards are painted 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 before entering the eye. As a result, the colors from the back boards “bleed” onto the white surfaces.eyeSlide 3 Lecture 20 6.837 Fall ‘01 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 20 6.837 Fall ‘01 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 the radiance itself is dependent on the view, which is not the case. The term view-independent refers only to the use of the view to limit the set if locations and directions for which the radiance is computed.Slide 5 Lecture 20 6.837 Fall ‘01 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 in all directions with equal intensity.A formulation for the system of equations is facilitated by dividing the environment into a set of small areas, or patches. The radiosity over a patch is constant.The radiosity, B, of a patch is the total rate of energy leaving a surface and is equal to the sum of the emitted and reflected energies:Radiosity was used for Quake IISlide 6 Lecture 20 6.837 Fall ‘01 Solving the rendering equation L is the radiance from a point on a surface in a given direction ω E is the emitted radiance from a point: E is non-zero only if x’ is emissive V is the visibility term: 1 when the surfaces are unobstructed along the direction ω, 0 otherwise G is the geometry term, which depends on the geometric relationship between the two surfaces x and x’Photon-tracing uses sampling and Monte-Carlo integrationRadiosity uses finite elements: project onto a finite set of basis functions (piecewise constant)Ray tracing computes L [D] S* EPhoton tracing computes L [D | S]* ERadiosity only computes L [D]* E( ) ( ) ( )()( ) ( ), , , ,sL x E x x L x G x x V x x dAw r w� � � � � �= +�r rSlide 7 Lecture 20 6.837 Fall ‘01 Continuous Radiosity Equationxx’xxx’x’BG(x,x’)V(x,x’)EBForm factor•G: geometry term •V: visibility term•No analytical solution, even for simple configurationsFor an environment composed of diffuse surfaces, we have the basic radiosity relationship: reflectivityxSlide 8 Lecture 20 6.837 Fall ‘01 Discrete Radiosity EquationAiAjj=1jijiiiBFEBForm factor• discrete representation• iterative solution• costly geometric/visibility calculationsFor an environment that has been discretized into n patches, over which the radiosity is constant, (i.e. both B and E are constant across a patch), we have the basic radiosity relationship: reflectivitynSlide 9 Lecture 20 6.837 Fall ‘01 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.12nBBB� �� �� �� �� �� �� �M12nEEE� �� �� �� �� �� �� �M=1 11 1 12 1 12 21 2 221111nn n n nnF F FF FF Fr r rr rr r- - -� �� �- -� �� �� �- -� �� �LM OL LSuch an equation exists for each patch, and in a closed environment, a set of n simultaneousequations in n unknown Bi values is obtained:iBiBSlide 10 Lecture 20 6.837 Fall ‘01 Standard Solution of the Radiosity Matrix121211 12i i ii i i i i innn nBBB E BFB EBF FBEB Er r r� � � � � �� �� � � � � �� �� � � � � �� �� � � � � �� �= +� � � � � �� �� � � � � �� �� � � � � �� �� � � � � �� �� �� � � � � �� �� � � � � �MMM M MMLThe radiosity of a single patch i is updated for each iteration by gathering radiosities from all other patches:This method is fundamentally a Gauss-Seidel relaxationSlide 11 Lecture 20 6.837 Fall ‘01 Computing Vertex RadiositiesRecall that radiosity values are constant over the extent of a patch.A standard renderer requires vertex radiosities (intensities). These can be obtained for a vertex by computing the average of the radiosities of patches that contribute to the vertex under
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