Global IlluminationLooking BackRadianceIrradiance and RadiosityRelationships among the Radiometric UnitsPath NotationBRDFSlide 8BRDF propertiesBRDF ExamplesReflectanceThe Rendering EquationSlide 13Linear OperatorNeumann Series SolutionExamplesImplicationsMonte Carlo Techniques in Global IlluminationBasic Monte Carlo IntegrationSlide 20When to Use Monte Carlo?Two Types of Monte CarloImportance SamplingStratified SamplingMore On Ray-TracingCone Tracing (1984)Distributed Ray-TracingSampling Other DimensionsPath TracingPathIndirect paths - surface samplingIndirect paths - source shootingIndirect paths - receiver gatheringIndirect pathsComplex path generatorsBidirectional ray tracingClassic ray tracing?Global IlluminationJian Huang, CS 594, Fall 2002This set of slides reference text book and the course note of Dutre et. al on SIGGRAPH 2001Looking Back•Ray-tracing and radiosity both computes global illumination•Is there a more general methodology?•It’s a game of light transport.Radiance•Radiance (L): for a point in 3D space, L is the light flux per unit projected area per unit solid angle, measured in W/(sr-m2)–sr – steradian: unit of solid angle•A cone that covers r2 area on the radius-r hemisphere•A total of 2 sr on a hemisphere .–power density/solid angel–The fundamental radiometric quantityIrradiance and Radiosity•Irradiance (E)–Integration of incoming radiance over all directions, measured in W/m2–Incident radiant power (Watt) on per unit projected surface area•Radiance distribution is generally discontinuous, irradiance distribution is generally continuous, due to the integration–‘shooting’, distribute radiance from a surface–‘gathering’, integrating irradiance and accumulate light flux on surface•Radiosity (B) is–Exitant radiant power (Watt) on per unit projected surface area, measured in W/m2 as wellRelationships among the Radiometric UnitsPath Notation•A non-mathematical way to categorize the behavior of global illumination algorithm–Diffuse to diffuse transfer–Specular to diffuse transfer–Diffuse to specular transfer–Specular to specular transfer•Heckbert’s string notation (1990): as light ray travels from source (L) to eye (E):–LDDE, LDSE+LDDE, LSSE+LDSE, LSDE, LSSDEBRDF•Materials interact with light in different ways, and different materials have different appearances given the same lighting conditions. •The reflectance properties of a surface are described by a reflectance function, which models the interaction of light reflecting at a surface.•The bi-directional reflectance distribution function (BRDF) is the most general expression of reflectance of a material•The BRDF is defined as the ratio between differential radiance reflected in an exitant direction, and incident irradiance through a differential solid angleBRDF•The geometry of BRDFBRDF properties•Positive, and variable in regard to wave-length•Reciprocity: the value of the BRDF will remain unchanged if the incident and exitant directions are interchanged.•Generally, the BRDF is anisotropic. •BRDF behaves as a linear function with respect to all incident directions.BRDF Examples•Diffuse surface (Lambertian)•Perfect specular surface–BRDF is non-zero in only one exitant direction•Glossy surfaces (non ideally specular)–Difficult to model analytically•Transparent surfaces–Need to model the full sphere (hemi-sphere is not enough)–BRDF is not usually enough, need BSSRDF (bi-directional sub-surface scattering reflectance distribution function)–The transparent side can be diffuse, specular or glossyReflectance•3 formsThe Rendering Equation•Proposed by Jim Kajiya in his SIGGRAPH’1986 paper–Light transport equation in a general form–Describes not only diffuse surfaces, but also ones with complex reflective properties–Goal of computer graphics: solution of the rendering equation!–Looks simple and natural, but really is too complex to be solved exactly; various techniques to nd approximate solutions are usedThe Rendering Equation•I(x,x’) = intensity passing from x’ to x•g(x,x’) = geometry term (1, or 1/r2, if x visible from x’, 0 otherwise)• (x,x’) = intensity emitted from x’ in the direction of x• (x,x’,x’’) = scattering term for x’ (fraction of intensity arriving at x’ from the direction of x’’ scattered in the direction of x)•S = union of all surfacesLinear Operator•Define a linear operator, M.•The rendering equation:•How to solve it?Neumann Series Solution•Start with an initial guess I0•Compute a better solution•Computer an even better solution•Then,•In practice one needs to truncate it somewhereExamples•No shading/illumination, just draw surfaces as emitting themselves:•Direct illumination, no shadows:•Direct illumination with shadows:Implications•How successful is a global illumination algorithm?–The first term is simple, just visibility–How an algorithm handles the remaining terms and the recursion?–How does it handle the combinations of diffuse and specular reflectivity•The rendering equation is a view-independent statement of the problem•How are the radiosity algorithm and the ray-tracing algorithm?Monte Carlo Techniques in Global Illumination•Monte Carlo is a general class of estimation method based on statistical sampling–The most famous example: to estimate •Monte Carlo techniques are commonly used to solve integrals with no analytical or numerical solution–The rendering equation has one such integralBasic Monte Carlo Integration•Suppose we want to numerically integrate a function over an integration domain D (of dimension d), i.e., we want to compute the value of the integral I:•Common deterministic approach: construct a number of sample points, and use the function values at those points to compute an estimate of I. •Monte Carlo integration basically uses the same approach, but uses a stochastic process to generate the sample points. And would like to generate N sample points distributed uniformly over D.Basic Monte Carlo Integration•The mean of the evaluated function values at each randomly generated sample point multiplied by the area of the integration domain, provides an unbiased estimator for I:•Monte Carlo methods provides an un-biased estimator•The variance reduces as N increases•Usually, given the same N, deterministic approach produces less error than Monte Carlo methodsWhen to Use Monte
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