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The Rendering Equation

The Rendering Equation. Direct ( local ) illumination Light directly from light sources No shadows Indirect ( global ) illumination Hard and soft shadows Diffuse interreflections (radiosity) Glossy interreflections (caustics). Early Radiosity. Lighting Effects. Hard Shadows.

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The Rendering Equation

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  1. The Rendering Equation • Direct (local) illumination • Light directly from light sources • No shadows • Indirect (global) illumination • Hard and soft shadows • Diffuse interreflections (radiosity) • Glossy interreflections (caustics) University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  2. Early Radiosity University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  3. Lighting Effects Hard Shadows Soft Shadows Caustics Indirect Illumination University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  4. Challenge • To evaluate the reflection equation the incoming radiance must be known • To evaluate the incoming radiance the reflected radiance must be known University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  5. To The Rendering Equation • Questions 1. How is light measured? 2. How is the spatial distribution of light energy described? 3. How is reflection from a surface characterized? 4. What are the conditions for equilibrium flow of light in an environment? University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  6. The Grand Scheme Light and Radiometry Energy Balance Surface Rendering Equation Volume Rendering Equation Radiosity Equation University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  7. Balance Equation • Accountability • [outgoing] - [incoming] = [emitted] - [absorbed] • Macro level • The total light energy put into the system must equal the energy leaving the system (usually, via heat). • Micro level • The energy flowing into a small region of phase space must equal the energy flowing out. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  8. Surface Balance Equation • [outgoing] = [emitted] + [reflected] University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  9. Direction Conventions Surface vs. Field Radiance BRDF University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  10. Surface Balance Equation [outgoing] = [emitted] + [reflected] + [transmitted] BTDF University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  11. Two-Point Geometry Ray Tracing University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  12. Coupling Equations Invariance of radiance University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  13. The Rendering Equation • Directional form Transport operator i.e. ray tracing Integrate over hemisphere of directions University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  14. The Rendering Equation • Surface form Geometry term Integrate over all surfaces Visibility term University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  15. The Radiosity Equation • Assume diffuse reflection 1. 2. University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  16. Integral Equations • Integral equations of the 1st kind • Integral equations of the 2nd kind University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  17. Linear Operators • Linear operators act on functions like matrices act on vectors • They are linear in that • Types of linear operators University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  18. Rendering Operators • Scattering operator • Transport operator University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  19. Solving the Rendering Equation • Rendering Equation • Solution University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  20. Formal Solution • Neumann series • Verify University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  21. Successive Approximations • Successive approximations • Converged University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  22. Successive Approximation University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  23. Light Path University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  24. Light Path University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  25. Light Paths University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  26. Light Transport • Integrate over all paths of all lengths • Question: • How to sample space of paths? University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  27. Classic Ray Tracing • Forward (from eye): E S* (D|G) L From Heckbert University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  28. Photon Paths Radiosity Caustics From Heckbert University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

  29. How to Solve It? • Finite element methods • Classic radiosity • Mesh surfaces • Piecewise constant basis functions • Solve matrix equation • Not practical for rendering equation • Monte Carlo methods • Path tracing (distributed ray tracing) • Randomly trace ray from the eye • Bidirectional ray tracing • Photon mapping University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

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