1Global IlluminationTom FunkhouserPrinceton UniversityCOS 526, Spring 2006Overview• Rendering is integration Direct illumination Global illumination Rendering equation• Path types L(S|D)*E• Solution methods OpenGL Ray tracing Path tracing RadiosityOpenGLAssumedirect illuminationfrom point lightsand ignore visibilityAssumedirect illuminationfrom point lightsand ignore visibilityΩ•+=ωωωωωωωdnxLxfxLxLireo))(,'()',,'()','()','(n'xω'ωLightCameraΩ=•+=nlightsiireonxLxfxLxL1))(,'()',,'()','()','(ωωωωωωDirect Illuminationn'xω'ωωdLight CameraΩΩ•+=LdnxLxfxLxLireoωωωωωωω))(,'()',,'()','()','(Ray TracingAssume only significantindirect illumination due to perfect specular reflection and refractionAssume only significantindirect illumination due to perfect specular reflection and refractionn'xω'ωCameraΩSurface LightspecularnxLxfxLxLnlightsiireo+•+==1))(,'()',,'()','()','(ωωωωωωΩ•+=ωωωωωωωdnxLxfxLxLireo))(,'()',,'()','()','(Global Illuminationn'xω'ωωdΩΩ•+=ωωωωωωωdnxLxfxLxLireo))(,'()',,'()','()','(SurfaceSurfaceLight2Monte Carlo Path TracingΩ•+=ωωωωωωωdnxLxfxLxLireo))(,'()',,'()','()','(n'xω'ωωdΩSurfaceSurfaceLightEstimate integral for each pixel by random samplingEstimate integral for each pixel by random samplingRendering Equationn'xω'ωωdΩΩ•+=ωωωωωωωdnxLxfxLxLireo))(,'()',,'()','()','(SurfaceSurfaceKajiya 1986Rendering Equation (2)n'xx"xiΘ′ω'ωdA'dAoΘ→→→+→=→SredAxxGxxVxxLxxxfxxLxxL )',()',()'()"'()"'()"'(2'coscos)',(xxxxGoi−ΘΘ′=Kajiya 1986Overview• Rendering is integration Direct illumination Global illumination Rendering equation• Path types L(S|D)*E• Solution methods OpenGL Ray tracing Path tracing RadiosityPath Types Path Types?Henrik Wann Jensen3Path Types?HerfPath Types?Henrik Wann JensenPath Types?Henrik Wann JensenPath Types?RenderParkPath Types?Paul DebevecPath Types?4Path TypesJensenRay tracingHenrik Wann JensenPath TypesJensen+ soft shadowsHenrik Wann JensenPath TypesJensen+ causticsHenrik Wann JensenPath TypesJensen+ indirect diffuse illuminationHenrik Wann JensenPath Types• OpenGL LDE• Ray tracing LDS*E• Path tracing L(D|S)*E• Radiosity LD*EJohn HartOverview• Rendering is integration Direct illumination Global illumination Rendering equation• Path types L(S|D)*E• Solution methods OpenGL Ray tracing Path tracing Radiosity5Indirect Diffuse Illumination Rendering Equationn'xx"xiΘ′ω'ωdA'dAoΘ→→→+→=→SredAxxGxxVxxLxxxfxxLxxL )',()',()'()"'()"'()"'(2'coscos)',(xxxxGoi−ΘΘ′=Kajiya 1986Radiosity Equation→→→+→=→SredAxxGxxVxxLxxxfxxLxxL )',()',()'()"'()"'()"'(Assume everything is Lambertianπρ)"'()'( xxxfxr→→=+=SedAxxGxxVxLxxLxL )',()',()()'()'()'(πρΩ=ωθdLBocosπBL =+=SedAxxGxxVxBxxBxB )',()',()()'()'()'(πρConvert toRadiositiesRadiosity Approximation+=SedAxxGxxVxBxxBxB )',()',()()'()'()'(πρ=+=NjijjiiiFBEB1ρwhereijA AoiijiijdAdArVAFi j ΘΘ′=2coscos1πDiscretize the surfacesinto “elements”iΘ′jAiAoΘrSystem of Equations=+=NjijjiiiFBEB1ρ = −−−−−−−−−−−−−nnnnnnnnnnnnnnEEEBBBFFFFFFFFF......1..................1...12121,1,,111,11,222,221,22,111,11ρρρρρρρρρ=−=NjijjiiiFBBE1ρiNjijjiiEFBB =−=1ρijNjijiiNjiiiEBFBF =−−== 11)1(ρρThis is an energy balanceequation=+=NjjjjiiiiiiABFAEAB1ρRadiosity Intuition6Radiosity• Issues Computing form factors Selecting basis functions for radiosities Solving linear system of equations Meshing surfaces into elements Rendering images Form Factor• Fraction of energy leaving element i that arrives at element jijA AoiijiijdAdArVAFi j ΘΘ′=2coscos1πiΘ′jAiAoΘrForm Factor Intuition2coscos1rVAFjiijidjdiπΘΘ=−iΘ′jdAidAoΘr2cosrAiiΘProjection tohemispherejΘcosProjection to diskπ1 Divide byarea of diskComputing Form FactorsCohen & WallaceSolving the System of Equations• Challenges: Size of matrix Cost of computing form factors Computational complexity = −−−−−−−−−−−−−nnnnnnnnnnnnnnEEEBBBFFFFFFFFF......1..................1...12121,1,,111,11,222,221,22,111,11ρρρρρρρρρA x = bSolving the System of Equations• Solution methods: Invert the matrix – O(n3) Iterative methods – O(n2) Hierarchical methods – O(n) = −−−−−−−−−−−−−nnnnnnnnnnnnnnEEEBBBFFFFFFFFF......1..................1...12121,1,,111,11,222,221,22,111,11ρρρρρρρρρA x = b7Gauss-Seidel Iteration Gauss-Seidel Iteration• Two interpretations: Iteratively relax rows of linear system Iteratively gather radiosity to elements = −−−−−−−−−−−−−nnnnnnnnnnnnnnEEEBBBFFFFFFFFF......1..................1...12121,1,,111,11,222,221,22,111,11ρρρρρρρρρGauss-Seidel Iteration• Two interpretations: Iteratively relax rows of linear system Iteratively gather radiosity to elementsProgressive Radiosity• Interpretation: Iteratively shoot “unshot” radiosity from elements Select shooters in order of unshot radiosityProgressive Radiosity• Adaptive refinementYeapProgressive Radiosity8Progressive Radiosity Surface Meshing• Store radiosity across surface Few elements Represents function well Few visible artifactsCohen & WallaceArtifacts of Bad Surface Meshing Cohen & WallaceAdaptive Meshing• Refine mesh in areas of high residualYeapAdaptive MeshingCohen & WallaceUniform mesh Adaptive meshError ComparisonCohen & Wallace9Adaptive Meshing
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