1Measures of IlluminationThe Radiosity EquationForm FactorsRadiosity AlgorithmsMeasures of IlluminationThe Radiosity EquationForm FactorsRadiosity AlgorithmsRadiosityRadiosityAlternative NotesAlternative Notes• SIGGRAPH 1993 Education Slide Set –Radiosity Overview, by Stephen Spencerwww.siggraph.org/education/materials/HyperGraph/radiosity/overview_1.htm2Limitations of Ray TracingLimitations of Ray TracingLocal vs. Global IlluminationLocal vs. Global Illumination• Local illumination: Phong model (OpenGL)– Light to surface to viewer– No shadows, interreflections– Fast enough for interactive graphics• Global illumination: Ray tracing– Multiple specular reflections and transmissions– Only one step of diffuse reflection• Global illumination: Radiosity– All diffuse interreflections; shadows– Advanced: combine with specular reflection3Image vs. Object SpaceImage vs. Object Space• Image space: Ray tracing– Trace backwards from viewer– View-dependent calculation– Result: rasterized image (pixel by pixel)• Object space: Radiosity– Assume only diffuse-diffuse interactions– View-independent calculation– Result: 3D model, color for each surface patch– Can render with OpenGLClassical Radiosity MethodClassical Radiosity Method• Divide surfaces into patches (elements)• Model light transfer between patches as system of linear equations• Important assumptions:– Reflection and emission are diffuse• Recall: diffuse reflection is equal in all directions• So radiance is independent of direction– No participating media (no fog)– No transmission (only opaque surfaces)– Radiosity is constant across each element– Solve for R, G, B separately4Balance of EnergyBalance of Energy• Lambertian surfaces (ideal diffuse reflector)• Divided into n elements• Variables– AiArea of element i (computable)– BiRadiosity of element i (unknown)– EiRadiant emitted flux density of element i (given)– ρiReflectance of element i (given)– Fj iForm factor from j to i (computable)Form FactorsForm Factors• Form factor Fi j: Fraction of light leaving element i arriving at element j• Depends on– Shape of patches i and j– Relative orientation of both patches– Distance between patches– Occlusion by other patches5Form Factor EquationForm Factor Equation• Polar angles θ and θ’ between normals and ray between x and y• Visibility function v(x,y) = 0 if ray from x to y is occluded, v(x,y) = 1 otherwise• Distance r between x and yReciprocityReciprocity• Symmetry of form factor• Divide earlier radiosity equationby Ai6Radiosity as a Linear SystemRadiosity as a Linear System• Restate radiosity equation• In matrix form• Known: reflectances ρi, form factors Fi, emissions Ei• Unknown: Radiosities Bi• n linear equations in n unknownsRadiosity “Pipeline”Radiosity “Pipeline”Form factorcalculationSolution ofRadiosity EqVisualizationScene Geometry Reflectance PropertiesViewing Conditions(camera setup)RadiosityImage7VisualizationVisualization• Radiosity solution is viewer independent• Can exploit graphics hardware to obtain image• Convert color on patch to vertex color• Easy part of radiosity methodComputing Form FactorsComputing Form Factors• Visibility critical• Two principal methods– Hemicube: exploit z-buffer hardware– Ray casting (can be slow)– Both exhibit aliasing effects• For inter-visible elements– Many special cases can be solved analytically– Avoid full numeric approximation of double integral8Hemicube AlgorithmHemicube Algorithm• Render model onto a hemicube as seen from the center of a patch• Store patch identifiers j instead of color• Use z-buffer to resolve visibility• Efficiently implementable in hardware• See Cohen and Greenberg, SIGGRAPH ‘85Resolution IssuesResolution Issues• Each patch is a constant color .. how can we obtain smooth, visually pleasing images?9[Chandran et al]WireframeNo Intensity Interpolation10WireframeResolution 30011Resolution 1200Resolution 250012Resolution 2500, InterpolatedSolving the Radiosity EquationSolving the Radiosity Equation• Direct form• As matrix equation• Unknown: radiosity Bi • Known: emission Ei, form factor Fi j, reflect. ρi13Classical Radiosity AlgorithmsClassical Radiosity Algorithms• Matrix Radiosity (Gathering)– Diagonally dominant matrix– Use Jacobi’s method (iterative solution)– Time and space complexity is O(n2) for n elements– Memory cost excessive• Progressive Refinement Radiosity (Shooting)– Solve equations incrementally with form factors– Time complexity is O(n s) for s iterations– Used more commonly (space complexity O(n))Matrix RadiosityMatrix Radiosity• Compute all form factors Fi j• Make initial approximation to radiosity– Emitting elements Bi= Ei– Other elements Bi= 0• Apply equation to get next approximation• Iterate with new approximation• Intuitively– Gather incoming light for each element i– Base new estimate on previous estimate14Radiosity SummaryRadiosity Summary• Assumptions– Opaque Lambertian surfaces (ideal diffuse)– Radiosity constant across each element• Radiosity computation structure– Break scene into patches– Compute form factors between patches• Lighting independent– Solve linear radiosity equation• Viewer independent– Render using standard hardware15Solid AngleSolid Angle• 2D angle subtended by object O from point x:– Length of projection onto unit circle at x– Measured in radians (0 to 2π)• 3D solid angle subtended by O from point x:– Area of of projection onto unit sphere at x– Measured in steradians (0 to 4π)J. StewartRadiant Power and RadiosityRadiant Power and Radiosity• Radiant power P– Rate at which light energy is transmitted– Dimension: power = energy / time• Flux density Φ– Radiant power per unit area of the surface– Dimension: power / area• Irradiance E: incident flux density of surface• Radiosity B: exitant flux density of surface– Dimension: power / area• Flux density at a point Φ(x) = dP/dA (or dP/dx)16Power at Point in a DirectionPower at Point in a Direction• Radiant intensity I– Power radiated per unit solid angle by point source– Dimension: power / solid angle• Radiant intensity in direction ω– I(ω) = dP/dω• Radiance L(x, ω)– Flux density at point x in direction ω– Dimension: power / (area solid angle)RadianceRadiance• Measured across surface in direction
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