The Rendering Equation

What is Global Illumination?

• Approximation of Physically Based Illumination
• Attempts to compute the light transfer between all objects in a scene, including direct and indirect illumination.
• Types of light interactions between objects
  • Diffuse to Diffuse
  • Specular to Diffuse
  • Diffuse to Specular
  • Specular to Specular
    
• Local Illumination Model : L (D|S) E
  • Each path is terminated by the eye and a light
    E - the eye
    L - the light
  • Each bounce involves an interaction with a surface
    D - diffuse reflection or transmission
    S - specular reflection or refraction
    

The Rendering Equation

• General formulation of the global illumination problem: it describes how the radiance from surface x reflects from the surface x':
  

  Lightout = Lightemitted + Lightin

  or

  • L is the radiance from a point on a surface in a given direction ω
  • E is the emitted radiance from a point: E is non-zero only if x' is emissive
  • V is the visibility term: 1 when surfaces are unobstructed along the direction ω, 0 otherwise
  • G is the geometry term, depends on geometric relationship between two surfaces x and x'
  • S represents all points on all surfaces in the scene
   
• Physically, a measure of the amount light in the scene, i.e., determine what sensors, if placed in the scene, would sense if they respond to incident radiance. Then "amount of light" should mean something like "how many photons" but no photons go through a single point. Instead, represent photon density.
• Note, this equation is recursive!
   

Approximation Required

• Typically cannot solve integral directly for discrete image
• Practically, a "sampled" representation of such a function, i.e., a function from a finite grid to colors. At each point of the grid, the function should indicate the amount of light coming through that point towards your eye.
   

• Current Assumptions
  For essentially all work in graphics, assume light is emitted, scattered, and absorbed at surface (i.e., between surfaces, it travels in straight lines).
  • No participating media (clouds, smoke)
  • No varying index of refraction (heated air or water)
  • No inter-light-beam effects (lasers)
  • No wave- or quantum-level effects (diffraction, self-shadows, etc.)

Views of the Rendering Equation

Graphics pipeline (OpenGL): LE

• Computes only direct lighting based on angle of incidence between light and geometry
• Approximates geometry through small polygons, each object is processed one at a time
• Mostly view independent

Ray Tracing: L S* E

• Examples

• View of rendering equation: approximate by sampling along rays where the integrand is likely to be large
• Takes place in image space: try to guess where the photons that hit the screen come from
• Algorithm: cast a ray from camera through each pixel in final image into scene, see what it hits and hits and ...
• View-dependent: must be recalculated when camera is moved
• Provides improved illumination model, but does not handle
  • Specular-to-diffuse
  • Diffuse-to-diffuse
  • Diffuse-to-specular
  • Caustics (focused light)

Radiosity: L D* E

• Examples

• View of rendering equation: approximate by iterating over diffuse scattering of radience
• Takes place in object space: try to guess how much light is emitted from each object in scene
• Algorithm: divide objects into patches, approximate light gathered and reflected from patch, refine and refine and ...
• View independent

Two Pass Algorithms

• Ray Tracing and Radiosity can be combined
  • Pass 1: enhanced radiosity to account for diffuse to diffuse and specular to diffuse
  • Pass 2: diffuse to specular and specular to specular using enhanced ray tracing
• Why two passes?
  • view-independent specularity and view-dependent diffuse computations are too expensive
  • view-independent diffuse computation followed by view-dependent specular computation is doable

Remaining Hard Problems

• Reflective Diffraction Effects
  • thin films
  • feathers of a blue jay, butterflies
  • oil on water
  • CDs

• Anisotropy
  • hair
  • brushed metals
  • strands pulled materials
  • satin and velvet cloths

References

• Notes on Ray Tracing
• Notes on Radiosity
• Radiosity Form Factor Applet
• Radiosity Shooting vs Gathering Applet
• The Rendering Equation by Jim Kajiya