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

Paper Summaries • Any takers? The Rendering Equation Assignments Projects • Project feedback • Checkpoint 3 • Approx 22 projects – Graded – email sent • Listing of projects now on Web • Checkpoint 4 • Presentation schedule – Due Wednesday – Presentations (15 min max) – Stick around after break for help – Last 3 classes (week 10 + finals week) • Checkpoint 5 – Sign up – To be given Wednesday • Email me with 1 st , 2 nd , 3 rd choices • RenderMan • First come first served. – Due May 3rd Projects Projects • Projects Mid-term report – due on web on Wed • Exam day presentations April 15 th (Wednesday) – Tuesday, May 18 th / 8-10am – Final chance to change specification of what will be • or presented and turned in – Wednesday, May 19 th / 10:15-12:15 – The spec upon which project will be judged – If no change from original proposal, simply say so. – Include explicit plans for presentation • ICL6 is available, we can project from there. • Need to know if other tools need to be installed. • Can also use other ICLs 1

Announcement Announcement • Student Meeting – Wednesday, April 14 th • 5pm • RIT at SIGGRAPH 2004? • Auditorium – Perhaps – Come wish Ken and Margaret Reek well – Interested? Contact me. – Plus free pizza 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 tone synthetic model Graphics: models image reproduction (focuses simulated lighting) 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 2

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 . 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 g gR ( I ) direct indirect ρ (x, x’, x’’) = light energy reflected from point x’ towards point x from light coming from x’’, i.e. BRDF 3

The Rendering Equation The Rendering Equation • This can be expanded using the Neuman series for • In short… implementation purposes: – The transport of light from point x’ to point x is = ε + ε + ε + ε + … equal to the sum of 2 3 I g g ( Rg ) g ( Rg ) g ( Rg ) • the light emitted from x’ in the direction of x and • the total light scattered from x’ towards x due to or light from all other surfaces in the scene. ∞ ∑ = ε i I g ( Rg ) = i 0 The Rendering Equation The Rendering Equation • Why is this important? • Local vs Global Illumination Models = ε + ε + ε + ε + … = ε + ε + ε + ε + … 2 3 I g g ( Rg ) g ( Rg ) g ( Rg ) 2 3 I g g ( Rg ) g ( Rg ) g ( Rg ) direct 1st scattering 2nd scattering 3rd scattering direct 1st scattering 2nd scattering 3rd scattering Local illumination - only considers direct component Rendering methods can be Global illumination - also considers other scattered characterized by the number of component scatterings considered The Rendering Equation The Rendering Equation • Local Illumination • Global Illumination = ε + ε + ε + ε + … = ε 2 3 ( ) ( ) ( ) I g I g g Rg g Rg g Rg direct direct 1st scattering 2nd scattering 3rd scattering Considers multiple scatterings Only object’s first contact with light is considered. • Ray-Tracing Lighting “simulated” by illumination model used. • Radiosity • Kajiya’s method NOTE: Kajiya does not include ambient light! 4

The Rendering Equation The Rendering Equation • Drawbacks • Summary – Uses geometric optics, based on light as rays – Equation for describing rendering algorithms. – Phase, diffraction, and transmission through – Describes light arriving at a point from another participatory media not considered (i.e., only point (and indirectly all other points) homogenous refraction considered) – Considers direct light and recursive scattering – Dependence on wavelength is implied – Not expressed using physical units • 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 Rendering Methods Rendering Methods • Ray Tracing • Ray Tracing – Light rays are traced from the eye, through a viewing plane, into scene – When rays strike an object, further rays are spawned representing reflection and refraction. – These newly spawned rays can strike objects and spawn more rays, etc. Watt/Watt Rendering Methods Rendering Method • Ray Tracing – Examples • 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 • video 5

Rendering Methods Rendering Methods • Radiosity • Radiosity – The problem with ray tracing – Based on the theory of heat transfer • Great for specular type reflections – Calculate lighting in a steady state • Awful for diffuse reflections. – Assumes that all surfaces are perfectly diffuse – Radiosity – Formulates a large systems of linear equations • From the guys who brought you Cook-Torrance illumination – Solution of these equations give you radiant model! exitance at each point • Not quite as elegant as ray tracing • More physically based Rendering Methods Rendering Methods • Radiosity • Radiosity – Radiant exitance - radiant flux out – Image is created by using calculated radiant existance values in standard rendering process. Φ d • In essence, radiosity defines a “made to fit” texture = M mapping dA dA Rendering Methods Rendering Methods • Radiosity • Radiosity – View dependence vs view independence – Not points -- But patches • Radiosity provides a view independent solution • Scene is subdivided into patches • Scene needs to be further rendered from a given • Radiant exitance will be calculated for each patch view point. 6

Rendering Methods Radiosity - Basics • Radiosity • Basic idea – Patches – Each patch will receive a certain amount of light from the environment – It will reflect fraction back into the environment – Keep track of amount of light reflected back – Continue till all light has been exhasted. [Ashdown94] Radiosity - Basics Rendering Methods • Key idea • Radiosity – Since all objects are perfectly diffuse, we can determine where light is coming from (and = ε + ε + ε + ε + … 2 3 I g g ( Rg ) g ( Rg ) g ( Rg ) going) by simply considering the geometry of the scene. direct 1st scattering 2nd scattering 3rd scattering – Calculation of radiant exitance per patch is can Mathematical derivation be detemined mathematically using a closed assumes limit as number of form solution. scatterings goes to Infinity Rendering Methods Remember BRDFs? • Radiosity – Examples Perfectly specular Perfectly diffuse Specular & diffuse Ray tracing Radiosity Reality • Bunny (Blue Sky Studios) 7