Global Illumination Jian Huang CS 594 Fall 2002 This set of slides reference text book and the course note of Dutre et al on SIGGRAPH 2001 Looking 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 srm2 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 quantity Irradiance 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 well Relationships among the Radiometric Units Path 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 LSSDE BRDF 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 angle BRDF The geometry of BRDF BRDF 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 subsurface scattering reflectance distribution function The transparent side can be diffuse specular or glossy Reflectance 3 forms The 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 used The 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 surfaces Linear 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 somewhere Examples 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 raytracing 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 integral Basic 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 methods When to Use Monte Carlo High dimension integration the sample points needed in deterministic approach exponential increase Complex integrand practically can t tell the error bound for deterministic approaches Monte Carlo is always un biased and for rendering purpose it converts errors into noise Two Types of Monte Carlo Monte Carlo integration methods can roughly be subdivided in two categories those that have no information about the function to be integrated blind Monte Carlo those that do have some kind of information available about the function informed Monte Carlo Intuitively one expects that informed Monte Carlo methods to produce more accurate results as opposed to blind Monte Carlo methods The basic Monte Carlo integration is a blind Monte Carlo method Importance Sampling An informed Monte Carlo Importance sampling uses a non uniform probability function pdf x for generating samples By choosing the probability function pdf x wisely on the basis of some knowledge of the function to be integrated we can often reduce the variance Can prove if can get
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