Global Illumination Greg Humphreys CS445 Intro Graphics University of Virginia Fall 2003 Lighting Simulation The Rendering Equation Given a scene consisting of geometric primitives with material properties and a set of light sources compute the illumination at each point on each surface Challenges Primitives complex lights materials shapes Exponential number of paths dense coupling How to solve it Radiosity Ray Tracing Finite element Monte Carlo Lighting Example Cornell Box Hard Shadows Caustics Indirect Illumination Surface Color Lighting Example Diffuse Reflection Surface Color Diffuse Shading Lighting Example Shadows No Shadows Shadows Lighting Example Soft Shadows Hard Shadows Point Light Source Soft Shadows Area Light Source Radiosity Cornell Experment Measured Simulated Program of Computer Graphics Cornell University Early Radiosity Very Early Radiosity ry Moon and Domina Spencer Lighting Design 1948 MIT Lighting Effects Hard Shadows Caustics Soft Shadows Indirect Illumination Caustics Henrik Wann Jensen 1995 Henrik Wann Jensen 1995 Complex Indirect Illumination Courtyard House with Curved Elements Mies van der Rohe Modeling Stephen Duck Rendering Henrik Wann Jensen Measuring things Flux Power rate at which light energy is emitted Measured in Solid angle 3D generalization of angle Measured in 1 l l Intensity Flux per solid angle Measured in 1 s s Radiance Radiance intensity per unit foreshortened area Foreshortened area found by multiplying the area by cos Think of the projection of the area onto the plane perpendicular to the direction of radiation Properties of radiance Remains constant along a ray The response of a sensor is proportional to the incident radiance Irradiance Irradiance flux per unit area E Li cos qi d wi W Example Point Light Sources Energy distribution has an irritating singularity The flux in some small differential solid angle is d I d Assume isotropic light source Then irradiance at a point on a unit sphere is I 4 Point Source Irradiance Irradiance on a small surface from a point light xs N x d cos E I 2 dA 4 x xs BRDF s Bidirectional Reflection Distribution Function How much light is reflected in direction o from direction i f r i i o o f r wi wo Two main properties Reciprocity f r wi wo f r wo wi Energy preservation f w w dw 1 W r i o The Reflection Equation How much light reflects in some given direction Take light coming from all incoming directions multiply it by the BRDF multiply by cos r r Lr wr f r wi wr Li wi cos qi dwi W Aside Delta Functions The Dirac delta function is defined as follows d x 0 if x 0 d x dx 1 d x y f x dx f y BRDF Example Mirror Mirror reflection so r i r i Also since no light is absorbed Lr r r Li r r Mirror s BRDF uses delta functions to enforce this cos i cos r fr i r cos i Diffuse Reflection Light is equally likely to be scattered in any direction regardless of the incident direction The BRDF is a constant r r Lr wr f r Li wi cos qi dwi W r fr Li wi cos qi d wi W fr E Indirect Illumination Radiance is invariant along a ray The radiance at x due to the radiance from x is Li x wi Lo x wo V x x V x x is a boolean visibility function So Close Hemispherical integral bad Surface integral good Relationship between solid angle and projected surface area dw So define G G x x cos qo dA 2 x x cos qi cos qo x x 2 V x x Change variables in the reflection equation L x w f r x L x w G x x dA S The Rendering Equation Incorporate emission r L x w w Le x f r x L x w G x x dA S This completely captures all light transport in a scene Is this true Global illumination solve the rendering equation But it s too hard The Radiosity Equation Assume all surfaces are diffuse BRDF is a constant we can pull it out of the integral B x E x f r x B x G x x dA S Solving the radiosity equation Radiosity solutions are view independent This is actually a tractable problem Bounce light around in the scene absorbing some and reflecting some until everything settles down What s that called The Radiosity Equation We can t compute integrals but that s OK Cut up the scene into little patches Sum the light contribution over all patches N Aj j 1 Ai Bi Ei i B j F j i Fi j the fraction of energy leaving patch j that arrives at patch i Called the form factor between the two patches Form Factor Facts Reciprocity relationship between form factors Ai Fi j A j F j i Simplify the summation this is pretty simple considering the rendering equation N Bi Ei i B j Fi j j 1 Rearrange terms N Bi i B j Fi j Ei j 1 Wait A Minute This looks suspiciously like a system of equations N Bi i B j Fi j Ei j 1 1 1 F1 1 F 2 2 1 N FN 1 1 F1 2 1 2 F2 2 N FN 2 1 F1 N B1 E1 2 F2 N B2 E2 1 N FN N BN E N This is Ye Olde Huge Matrixe Solve it using numerical techniques Finding Form Factors The form factor between two tiny surface patches is cos i cos j dFdi dj Vij dA j 2 r Vij is the binary visibility function So the true form factor is Fi j 1 Ai Ai Aj cos i cos j r 2 Vij dA j dAi Nusselt s Method Project the visible areas of Aj onto a unit hemisphere centered at dAi and then onto the base The ratio of this projected area to the area of the base circle is the form factor Hemicubes Approximate the hemisphere Hemicubes Each small hemicube cell has a precomputed delta form factor Fp cos i cos p r 2 A We can render the scene using normal Z buffer scan conversion onto the faces of the hemicube Progressive Refinement Radiosity solving is really slow Display a reasonable picture while solving The approach so far estimate the radiosity of patch I based on the estimates of all other patch radiosities Bi due to B j i B j Fi This is called gathering Poor intermediate results Why j Shooting Instead of gathering light we can shoot the light energy stored at each patch to every other patch B j due to Bi j Bi F j i This requires knowing all the form factors at once That s bad Rewrite the equation as B j due to Bi j Bi Fi j Ai Aj Choose the patch with maximum stored energy Starting with the light sources Intermediate Results Display the latest radiosity values at each patch Use ambient to make up the difference Set initial radiosities to the emission Compute an average diffuse reflectivity for the scene N A i …
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