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CS 551/651: Advanced Computer Graphics. Advanced Ray Tracing Radiosity. Administrivia. My penance: Ray tracing homeworks very slow to grade Many people didn’t include READMEs Many (most) people didn’t include workspace/project files or Makefiles Some people’s don’t work

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Cs 551 651 advanced computer graphics

CS 551/651: Advanced Computer Graphics

Advanced Ray Tracing


David Luebke 111/17/2014


  • My penance: Ray tracing homeworks very slow to grade

    • Many people didn’t include READMEs

    • Many (most) people didn’t include workspace/project files or Makefiles

    • Some people’s don’t work

    • Nobody’s works perfectly

  • Quiz 1: Tuesday, Feb 20

David Luebke 211/17/2014

Recap stochastic sampling
Recap: Stochastic Sampling

  • Sampling theory tells us that with a regular sampling grid, frequencies higher than the Nyquist limit will alias

  • Q: What about irregular sampling?

  • A: High frequencies appear as noise, not aliases

  • This turns out to bother our visual system less!

David Luebke 311/17/2014

Recap stochastic sampling1
Recap: Stochastic Sampling

  • Poisson distribution:

    • Completely random

    • Add points at random until area is full.

    • Uniform distribution: some neighboring samples close together, some distant

David Luebke 411/17/2014

Recap stochastic sampling2
Recap: Stochastic Sampling

  • Poisson disc distribution:

    • Poisson distribution, with minimum-distance constraint between samples

    • Add points at random, removing again if they are too close to any previous points

  • Jittered distribution

    • Start with regular grid of samples

    • Perturb each sample slightly in a random direction

    • More “clumpy” or granular in appearance

David Luebke 511/17/2014

Recap stochastic sampling3
Recap: Stochastic Sampling

  • Spectral characteristics of these distributions:

    • Poisson: completely uniform (white noise). High and low frequencies equally present

    • Poisson disc: Pulse at origin (DC component of image), surrounded by empty ring (no low frequencies), surrounded by white noise

    • Jitter: Approximates Poisson disc spectrum, but with a smaller empty disc.

David Luebke 611/17/2014

Recap nonuniform supersampling
Recap: Nonuniform Supersampling

  • To be correct, need to modify filtering step:

David Luebke 711/17/2014

Recap nonuniform supersampling1
Recap: Nonuniform Supersampling

  • Approximate answer: weighted average filter

  • Correct answer: multistage filtering

  • Real-world answer: ignore the problem

  I(i, j) h(x-i, y-j)


  h(x-i, y-j)

David Luebke 811/17/2014

Recap antialiasing and texture mapping
Recap: Antialiasing and Texture Mapping

  • Texture mapping is uniquely harder

    • Potential for infinite frequencies

  • Texture mapping is uniquely easier

    • Textures can be prefiltered

David Luebke 911/17/2014

Recap antialiasing and texture mapping1
Recap: Antialiasing and Texture Mapping

  • Issue in prefiltering texture is how much texture a pixel filter covers

  • Simplest prefiltering scheme: MIP-mapping

    • Idea: approximate filter size, ignore filter shape

    • Create a pyramid of texture maps

    • Each level doubles filter size

David Luebke 1011/17/2014

Recap mip mapping
Recap: Mip-Mapping

  • Tri-linear mip-mapping

    • Pixel size  depth d

    • Linearly interpolate colors within 2 levels closest to d

    • Linearly interpolate color between levels according to d

David Luebke 1111/17/2014

Distributed ray tracing
Distributed Ray Tracing

  • Distributed ray tracing is an elegant technique that tackles many problems at once

    • Stochastic ray tracing: distribute rays stochastically across pixel

    • Distributed ray tracing: distribute rays stochastically across everything

David Luebke 1211/17/2014

Distributed ray tracing1
Distributed Ray Tracing

  • Distribute rays stochastically across:

    • Pixel for antialiasing

    • Light sourcefor soft shadows

    • Reflection function for soft (glossy) reflections

    • Time for motion blur

    • Lens for depth of field

  • Cook: 16 rays suffice for all of these

    • I done told you wrong: 4x4, not 8x8

David Luebke 1311/17/2014

Distributed ray tracing2
Distributed Ray Tracing

  • Distributed ray tracing is basically a Monte Carlo estimation technique

  • Practical details:

    • Use lookup tables (LUTs) for distributing rays across functions

    • See W&W Figure 10.14 p 263

David Luebke 1411/17/2014

Backwards ray tracing
Backwards Ray Tracing

  • Traditional ray tracing traces rays from the eye, through the pixel, off of objects, to the light source

  • Backwards ray tracing traces rays from the light source, into the scene, into the eye

  • Why might this be better?

David Luebke 1511/17/2014

Backwards ray tracing1
Backwards Ray Tracing

  • Backwards ray tracing can capture:

    • Indirect illumination

    • Color bleeding

    • Caustics

      • Remember what a caustic is?

      • Give some examples

      • Remember where caustics get the name?

David Luebke 1611/17/2014

Backwards ray tracing2
Backwards Ray Tracing

  • Usually implies two passes:

    • Rays are cast from light into scene

    • Rays are cast from the eye into scene, picking up illumination showered on the scene from the first pass

David Luebke 1711/17/2014

Backwards ray tracing3
Backwards Ray Tracing

  • Q: How might these two passes “meet in the middle?”

David Luebke 1811/17/2014

Backwards ray tracing4
Backwards Ray Tracing

  • Arvo: illumination mapstile surfaces with regular grids, like texture maps

    • Shoot rays outward from lights

    • Every ray hit deposits some of its energy into surface’s illumination map

      • Ignore first generation hits that directly illuminate surface (Why?)

    • Eye rays look up indirect illumination using bilinear interpolation

David Luebke 1911/17/2014

Backwards ray tracing5
Backwards Ray Tracing

  • Watt & Watt, Plate 34

    • Illustrates Arvo’s method of backwards ray tracing using illumination maps

    • Illustrates a scene with caustics

  • Related idea: photon maps

David Luebke 2011/17/2014

Advanced ray tracing wrapup
Advanced Ray Tracing Wrapup

  • Backwards ray tracing accounts for indirect illumination by considering more general paths from light to eye

  • Distributed ray tracing uses a Monte Carlo sampling approach to solve many ray-tracing aliasing problems

David Luebke 2111/17/2014

Next up radiosity
Next Up: Radiosity

  • Ray tracing:

    • Models specular reflection easily

    • Diffuse lighting is more difficult

  • Radiosity methods explicitly model light as an energy-transfer problem

    • Models diffuse interreflection easily

    • Shiny, specular surfaces more difficult

David Luebke 2211/17/2014

Radiosity introduction
Radiosity Introduction

  • First lighting model: Phong

    • Still used in interactive graphics

    • Major shortcoming: local illumination!

  • After Phong, two major approaches:

    • Ray tracing

    • Radiosity

David Luebke 2311/17/2014

Radiosity introduction1
Radiosity Introduction

  • Ray tracing: ad hoc approach to simulating optics

    • Deals well with specular reflection

    • Trouble with diffuse illumination

  • Radiosity: theoretically rigorous simulation of light transfer

    • Very realistic images

    • But makes simplifying assumption: only diffuse interaction!

David Luebke 2411/17/2014

Radiosity introduction2
Radiosity Introduction

  • Ray-tracing:

    • Computes a view-dependent solution

    • End result: a picture

  • Radiosity:

    • Models only diffuse interaction, so can compute a view-independent solution

    • End result: a 3-D model

David Luebke 2511/17/2014


  • Basic idea: represent surfaces in environment as many discrete patches

  • A patch, or element, is a polygon over which light intensity is constant

David Luebke 2611/17/2014


  • Model light transfer between patches as a system of linear equations

  • Solving this system gives the intensity at each patch

  • Solve for R, G, B intensities and get color at each patch

  • Render patches as colored polygons in OpenGL. Voila!

David Luebke 2711/17/2014


  • Theoretical foundation: heat transfer

  • Need system of equations that describes surface interreflections

  • Simplifying assumptions:

    • Environment is closed

    • All surfaces are Lambertian reflectors

David Luebke 2811/17/2014


  • The radiosity of a surface is the rate at which energy leaves the surface

  • Radiosity = rate at which the surface emits energy + rate at which the surface reflects energy

    • Notice: previous methods distinguish light sources from surfaces

    • In radiosity all surfaces can emit light

    • Thus: all emitters inherently have area

David Luebke 2911/17/2014


  • Break environment up into a finite number n of discrete patches

    • Patches are opaque Lambertian surfaces of finite size

    • Patches emit and reflect light uniformly over their entire surface

  • Q: What’s wrong with this model?

  • Q: What can we do about it?

David Luebke 3011/17/2014


  • Then for each surface i:

    Bi = Ei + i Bj Fji(Aj / Ai)


    Bi, Bj= radiosity of patch i, j

    Ai, Aj= area of patch i, j

    Ei= energy/area/time emitted by i

    i = reflectivity of patch i

    Fji = Form factor from j to i

David Luebke 3111/17/2014

Form factors
Form Factors

  • Form factor: fraction of energy leaving the entirety of patch i that arrives at patch j, accounting for:

    • The shape of both patches

    • The relative orientation of both patches

    • Occlusion by other patches

David Luebke 3211/17/2014

Form factors1
Form Factors

  • Some examples…

Form factor:

nearly 100%

David Luebke 3311/17/2014

Form factors2
Form Factors

  • Some examples…

Form factor:

roughly 50%

David Luebke 3411/17/2014

Form factors3
Form Factors

  • Some examples…

Form factor:

roughly 10%

David Luebke 3511/17/2014

Form factors4
Form Factors

  • Some examples…

Form factor:

roughly 5%

David Luebke 3611/17/2014

Form factors5
Form Factors

  • Some examples…

Form factor:

roughly 30%

David Luebke 3711/17/2014

Form factors6
Form Factors

  • Some examples…

Form factor:

roughly 2%

David Luebke 3811/17/2014

Form factors7
Form Factors

  • In diffuse environments, form factors obey a simple reciprocity relationship:

    Ai Fij = Ai Fji

  • Which simplifies our equation:Bi = Ei + i Bj Fij

  • Rearranging to:Bi - i Bj Fij = Ei

David Luebke 3911/17/2014

Form factors8
Form Factors

  • So…light exchange between all patches becomes a matrix:

  • What do the various terms mean?

David Luebke 4011/17/2014

Form factors9
Form Factors

1 - 1F11 - 1F12 … - 1F1n B1E1

- 2F21 1 - 2F22 … - 2F2n B2E2

. . … . . .

. . … . . .

. . … . . .

- pnFn1- nFn2 … 1 - nFnn Bn En

  • Note: Ei values zero except at emitters

  • Note: Fii is zero for convex or planar patches

  • Note: sum of form factors in any row = 1 (Why?)

  • Note: n equations, n unknowns!

David Luebke 4111/17/2014


  • Now “just” need to solve the matrix!

    • W&W: matrix is “diagonally dominant”

    • Thus Guass-Siedel must converge

  • End result: radiosities for all patches

  • Solve RGB radiosities separately, color each patch, and render!

  • Caveat: actually, color vertices, not patches (see F&vD p 795)

David Luebke 4211/17/2014


  • Q: How many form factors must be computed?

  • A: O(n2)

  • Q: What primarily limits the accuracy of the solution?

  • A: The number of patches

David Luebke 4311/17/2014


  • Where we go from here:

    • Evaluating form factors

    • Progressive radiosity: viewing an approximate solution early

    • Hierarchical radiosity: increasing patch resolution on an as-needed basis

David Luebke 4411/17/2014

The end
The End

David Luebke 4511/17/2014