Paper Summaries Any takers? The Rendering Equation Assignments - PDF document

Paper Summaries • Any takers? The Rendering Equation Assignments Projects • Checkpoint 3 • Project feedback – Graded – email sent • Approx 18 projects • Checkpoint 4 • Listing of projects now on Web – Due Wednesday • Presentation schedule – Inverse mapping in 2nd half – Presentations (15 min max) • Checkpoint 5 – Last 3 classes (week 10 + finals week) – To be given Wednesday – Sign up • RenderMan • Email me with 1 st , 2 nd , 3 rd choices – Due February 16th • First come first served. – License server problems Today’s Class Computer Graphics as Virtual Photography • The Rendering Equation real camera photo Photographic Photography: scene (captures processing print – What it is light) – Techniques for solving processing camera Computer 3D synthetic tone model Graphics: models reproduction image (focuses simulated lighting) 1

The Rendering Equation The Rendering Equation • General form [ ] ∫ • Kajiya: 1986 ′ = ′ ε ′ + ρ ′ ′ ′ ′ ′ ′ ′ ′ ( , ) ( , ) ( , ) ( , , ) ( , ) I x x g x x x x x x x I x x d x S • “Unified context for viewing rendering algorithms as more or less accurate approximations to the solution of a single equation” • Expresses the quantity of light transferred from one point x’ to another x , summed over all points. Ashdown The Rendering Equation The Rendering Equation • Transport Intensity • Geometry term [ ] [ ] ∫ ∫ ′ = ′ ε ′ + ρ ′ ′ ′ ′ ′ ′ ′ ′ ′ = ′ ε ′ + ρ ′ ′ ′ ′ ′ ′ ′ ′ I ( x , x ) g ( x , x ) ( x , x ) ( x , x , x ) I ( x , x ) d x I ( x , x ) g ( x , x ) ( x , x ) ( x , x , x ) I ( x , x ) d x S S I(x, x’) = Transport energy or intensity of light g(x, x’) = geometry term passing from point x’ to point x (unoccluded two = 0, if x is not visible from x’ point transport) = 1/d 2 otherwise The Rendering Equation The Rendering Equation • Emittance • Scattering [ ] ∫ [ ] ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ = ε + ρ I ( x , x ) g ( x , x ) ( x , x ) ( x , x , x ) I ( x , x ) d x ∫ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ = ε + ρ S I ( x , x ) g ( x , x ) ( x , x ) ( x , x , x ) I ( x , x ) d x S ρ (x, x ′ , x ′′ ) = light energy reflected off ε (x, x ′ ) = light energy emitted from point x ′ point x’ towards point x from light coming from x ′′ towards x . 2

The Rendering Equation The Rendering Equation • Incoming = direct + indirect (scattered) [ ] ∫ ′ = ′ ε ′ + ρ ′ ′ ′ ′ ′ ′ ′ ′ I ( x , x ) g ( x , x ) ( x , x ) ( x , x , x ) I ( x , x ) d x S = ε + ( I ) I g gR direct indirect ρ (x, x’, x’’) = light energy reflected from point x’ towards point x from light coming from x’’, i.e. BRDF The Rendering Equation The Rendering Equation • In short… – The transport of light from point x’ to point x is equal to the sum of • the light emitted from x’ in the direction of x and • the total light scattered from x’ towards x due to light from all other surfaces in the scene. The Rendering Equation The Rendering Equation • This can be expanded using the Neuman series for • Why is this important? implementation purposes: = ε + ε + ε + ε + … 2 3 I g g ( Rg ) g ( Rg ) g ( Rg ) = ε + ε + ε + ε + … 2 3 I g g ( Rg ) g ( Rg ) g ( Rg ) direct 1st scattering 2nd scattering 3rd scattering or Rendering methods can be ∞ ∑ = ε characterized by the number of i I g ( Rg ) scatterings considered = i 0 3

The Rendering Equation The Rendering Equation • Local vs Global Illumination Models • Local Illumination = ε + ε + ε + ε + … = ε 2 3 I g g ( Rg ) g ( Rg ) g ( Rg ) I g direct 1st scattering 2nd scattering 3rd scattering direct Local illumination - only considers direct component Only object’s first contact with light is considered. Lighting “simulated” by illumination model used. Global illumination - also considers other scattered component NOTE: Kajiya does not include ambient light! The Rendering Equation The Rendering Equation • Global Illumination • Drawbacks = ε + ε + ε + ε + … 2 3 – Uses geometric optics, based on light as rays I g g ( Rg ) g ( Rg ) g ( Rg ) – Phase, diffraction, and transmission through direct 1st scattering 2nd scattering 3rd scattering participatory media not considered (i.e., only homogenous refraction considered) Considers multiple scatterings • Ray-Tracing – Dependence on wavelength is implied • Radiosity – Not expressed using physical units • Kajiya’s method • Question: Isn’t I (x, x’) simply radiance at a point? • Yes! (with the exception of some geometric terms…) • More when we talk about radiosity The Rendering Equation Rendering Methods • Ray Tracing • Summary – Light rays are traced from the eye, through a – Equation for describing rendering algorithms. viewing plane, into scene – Describes light arriving at a point from another – When rays strike an object, further rays are point (and indirectly all other points) spawned representing reflection and refraction. – Considers direct light and recursive scattering – These newly spawned rays can strike objects and spawn more rays, etc. 4

Rendering Methods Rendering Methods • Ray Tracing • Ray tracing = ε + ε + ε + ε + … 2 3 I g g ( Rg ) g ( Rg ) g ( Rg ) direct 1st scattering 2nd scattering 3rd scattering Number of scatterings depend on recursion depth allowed by ray tracer Watt/Watt Rendering Method Rendering Methods • Ray Tracing – Examples • Radiosity – The problem with ray tracing • Great for specular type reflections • Awful for diffuse reflections. – Radiosity • From the guys who brought you Cook-Torrance illumination model! • Not quite as elegant as ray tracing • More physically based • video Rendering Methods Rendering Methods • Radiosity • Radiosity – Radiant exitance - radiant flux out – Based on the theory of heat transfer – Calculate lighting in a steady state Φ d = – Assumes that all surfaces are perfectly diffuse M – Formulates a large systems of linear equations dA – Solution of these equations give you radiant dA exitance at each point 5

Rendering Methods Rendering Methods • Radiosity • Radiosity – Image is created by using calculated radiant – View dependence vs view independence existance values in standard rendering process. • Radiosity provides a view independent solution • In essence, radiosity defines a “made to fit” texture • Scene needs to be further rendered from a given mapping view point. Rendering Methods Rendering Methods • Radiosity • Radiosity – Patches – Not points -- But patches • Scene is subdivided into patches • Radiant exitance will be calculated for each patch [Ashdown94] Radiosity - Basics Radiosity - Basics • Key idea • Basic idea – Since all objects are perfectly diffuse, we can – Each patch will receive a certain amount of determine where light is coming from (and light from the environment going) by simply considering the geometry of – It will reflect fraction back into the the scene. environment – Keep track of amount of light reflected back – Calculation of radiant exitance per patch is can – Continue till all light has been exhasted. be detemined mathematically using a closed form solution. 6

Rendering Methods Rendering Methods • Radiosity • Radiosity – Examples = ε + ε + ε + ε + … 2 3 I g g ( Rg ) g ( Rg ) g ( Rg ) direct 1st scattering 2nd scattering 3rd scattering Mathematical derivation assumes limit as number of scatterings goes to Infinity • Bunny (Blue Sky Studios) Remember BRDFs? Rendering Methods • Getting closer to reality – Ray tracing • Replace rays with cones/beams • Distributed ray tracing – Spawn more rays – Stochastic sampling • Supplement backward ray tracing with forward ray Perfectly specular Perfectly diffuse Specular & diffuse tracing (e.g. Photon Mapping) Ray tracing Radiosity Reality Rendering Methods Rendering Methods • Getting closer to reality • Kajiya’s solution (Path tracing) – Much like ray tracing – Radiosity – Determine the path of light that eventually reaches the • Two path method – combine ray tracing and eye. radiosity – Only 1 ray spawned per intersection • Reflection in specular • Reflection in diffuse • Refraction – Which ray to spawn determined via stochastic sampling. 7

Kajiya’s Method Summary • Rendering equation – Mathematical expression of rendering process • Solutions – Ray Tracing – Radiosity – Combination of both. • Questions? 8