Slide 1Lastra, 2/14/2016 Monte-Carlo Methods

Slide 2Lastra, 2/14/2016 Topics Kajiya’s paper –Showed that existing rendering methods are approximations of rendering equation. –Introduced path tracing. More recent work – Lafortune, Veach.

Slide 3Lastra, 2/14/2016 Kajiya’s Rendering Equation Expressed as point-to-point transfer Integral is over S, union of all surfaces

Slide 4Lastra, 2/14/2016 Rendering Equation Terms is unoccluded two-point transport intensity Energy per unit time per unit area of source per unit area of target

Slide 5Lastra, 2/14/2016 Rendering Equation Terms is geometry term 0 if x not visible from x’, 1/r 2 if visible.

Slide 6Lastra, 2/14/2016 Rendering Equation Terms is unoccluded emittance term

Slide 7Lastra, 2/14/2016 Rendering Equation Terms is unoccluded three-point transport reflectance (scattering term) Intensity scattered to x by x’ originating from x”

Slide 8Lastra, 2/14/2016 Why express as point-to-point? Wants to set up for a path tracing solution –Point-to-point transfer of energy

Slide 9Lastra, 2/14/2016 Relationship to Other Rendering Methods Compared rendering eqn. to conventional polygon rendering, ray tracing and distributed ray tracing. Let scattering,

Slide 10Lastra, 2/14/2016 Relationship to “Utah Approx.” Geometry term, g, only computed to eye. g  is ambient term. Scattering operator, M, only operates on point sources (  0 ), ignores visibility to light (no shadows). M is now only sum over lights, not integration Later extensions for shadows and area lights

Slide 11Lastra, 2/14/2016 Relationship to Ray Tracing M 0 now one reflection, one refraction, and cosine for diffuse term Computes visibility, g, of point lights (generates shadow) M still small sum

Slide 12Lastra, 2/14/2016 Relationship to Distributed Ray Tracing Similar equation to Whitted’s Now need to evaluate M as integral M now distribution around reflection, refraction, and shadow ray Ambient term still “elusive”

Slide 13Lastra, 2/14/2016 Relationship to Radiosity Solves energy balance, but only for diffuse. Derives equations for radiosity from the one presented. Points out that solving visibility is expensive and that you may not need radiosity for all surfaces (on other hand, you might).

Slide 14Lastra, 2/14/2016 Kajiya’s Path Tracing

Slide 15Lastra, 2/14/2016 Method At each hit, –One ray cast based on specular, diffuse, and transmission coefficients –One random ray per light Constant number of rays per pixel (40)

Slide 16Lastra, 2/14/2016 Markov Chains for Solution Absorbing state p = 0.11 p = 0.08 p = 0.02 p = 1

Slide 17Lastra, 2/14/2016 Algorithm 1.Choose pt. x’ visible from eye 2.Add in radiated intensity 3.For length of Markov path 1.Select pt. x” and compute g(x’, x”) 2.Calculate reflectance  (x, x’, x”), multiply by  (x’, x”) 3.Add contribution to pixel

Slide 18Lastra, 2/14/2016 Sampling Most important factor in Monte Carlo –Need to avoid bias –But also need to make most out of few rays –Otherwise, noisy images Kajiya discusses several ways. Better to look in more modern reference –Glassner –Recent dissertations

Slide 19Lastra, 2/14/2016 Path Tracing Postulates that even for ray tracing, following one path (probabilistically) is better. Why? Most contribution from first ray. Need to be careful about proportion of reflection, refraction, and shadow rays.

Slide 20Lastra, 2/14/2016 Results 401 minutes 533 minutes 256 x 256 image Ray Traced (no ambient) Path Traced Light scattered by sphere

Slide 21Lastra, 2/14/2016 Results Objects are gray, except for spheres and base. Color bleeding Caustics

Slide 22Lastra, 2/14/2016 Current Methods Bi-directional path tracing (Lafortune and Veach) Metropolis (Veach)

Slide 23Lastra, 2/14/2016 Pure Path Tracing

Slide 24Lastra, 2/14/2016 Pure Path Tracing Best for big luminaires. If lights small, few hits and large variance.

Slide 25Lastra, 2/14/2016 With Shadow Ray to Lights

Slide 26Lastra, 2/14/2016 With Shadow Ray to Lights Small lights OK. Best for specular surfaces.

Slide 27Lastra, 2/14/2016 Light Tracing

Slide 28Lastra, 2/14/2016 Light Tracing Small lights OK. Best for caustics.

Slide 29Lastra, 2/14/2016 Bi-Directional Path Tracing

Slide 30Lastra, 2/14/2016 Bi-Directional Path Tracing

Slide 31Lastra, 2/14/2016 Generating Samples Generated as groups Prefix from light joined with suffix from (to) eye (if edge is not obstructed) Russian roulette to cut off path Contribution must be multiplied by probability of generating path

Slide 32Lastra, 2/14/2016 Results (from Veach)

Slide 33Lastra, 2/14/2016 Metropolis Method for importance sampling A path is a sequence of points from a light to the eye. Let be the image contribution function, a measure of contribution over path is flux contributed by paths D Strategy: generate sequence of paths with probability proportional to ƒ

Slide 34Lastra, 2/14/2016 Mutations New path X i+1 mutated from X i Probability of rejecting each mutation keeps paths distributed according to contribution Discard start-up to reduce bias Can be run from a set of seed paths Run some bi-directional paths to find good seeds

Slide 35Lastra, 2/14/2016 Implementation Path selection important –minimize rejected paths, but not too correlated –finds sub-paths and replace –or perturb vertices of path (for caustics, etc) Mutation of “lens” path, (L|D)DS*E, to cover image pixels Adds direct lighting –rejects path if found by Metropolis

Slide 36Lastra, 2/14/2016 Results Light for this example comes only through crack in doorway

Slide 37Lastra, 2/14/2016 Results There are specific mutations to capture caustics.

Slide 38Lastra, 2/14/2016 Advantages Works well for difficult lighting –because it “stays” in important area Like bi-directional path tracing there’s just a little work to get a new path

Slide 39Lastra, 2/14/2016 References Kajiya, Jim, “The Rendering Equation”, SIGGRAPH ‘86. Lafortune papers and thesis Veach papers and thesis Jensen papers Shirley draft of MC book on his web page Kalos & Whitlock, Monte Carlo Methods, 1986.